# Equivariant vector bundles on varieties with codimension-one orbits

Lucas Mason-Brown and James Tao

###### Abstract.

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two
orbits: an open orbit and a a closed orbit of codimension $1$. We give an
algebraic description of the category of $G$-equivariant vector bundles on $X$
under a mild technical hypothesis. We deduce simpler classifications in the
special cases of line bundles and vector bundles which are generically local
systems. We apply our results to the study of admissible representations of
semisimple Lie groups. Our main result gives a new set of constraints on the
associated cycles of unipotent representations.

###### Contents

  1. 1 Introduction
    1. 1.1 Summary
    2. 1.2 The idea of gluing
    3. 1.3 Applications
    4. 1.4 Acknowledgments
  2. 2 Chains
    1. 2.1 Basic definitions
    2. 2.2 Fastened chains
    3. 2.3 Fastening data
  3. 3 Equivariant sheaves on smooth simple chains
    1. 3.1 Overview
      1. 3.1.3 Further comments
    2. 3.2 The punctured ideal sheaf
    3. 3.3 The group scheme $G^{s}$ associated to a fastening datum
    4. 3.4 Descent along a degree-$n$ cover
    5. 3.5 A $\mathbb{G}_{m}$-action on $G^{s}$
    6. 3.6 Applying the criterion for descent
      1. 3.6.2 Proof of (i)
      2. 3.6.3 Proof of (ii)
    7. 3.7 The main result
  4. 4 Specialization maps
    1. 4.1 Line bundles
      1. 4.1.2 Proof that (i) implies (ii) in Proposition 4.1.1
      2. 4.1.4 Proof of Proposition 4.1.1(i)
    2. 4.2 Factoring through a quotient of $\tilde{G}^{s}$
    3. 4.3 Local systems
    4. 4.4 The universal tame quotient
    5. 4.5 The subcategory associated to a tame quotient
    6. 4.6 Application to equivariant twisted $\mathscr{D}$-modules
      1. 4.6.2 Definition of the tame quotient associated to $\chi$
  5. 5 Equivariant sheaves on weakly normal chains
    1. 5.1 Weak normality
    2. 5.2 Patching vector bundles on chains
      1. 5.2.1 The graph of a chain
    3. 5.3 The case of line bundles
      1. 5.3.1 An analogue of the nearby cycles functor
      2. 5.3.3 The category of line bundles
  6. 6 An application to the representation theory of real reductive groups
    1. 6.1 Admissibility
    2. 6.2 The chain associated to an irreducible $(\mathfrak{g},K)$-module
    3. 6.3 The Slodowy slice
    4. 6.4 Unipotent sheaves

##  1\. Introduction

###  1.1. Summary

Work over an algebraically closed field $k$ of characteristic zero. Consider a
variety $X$ acted on by an algebraic group $G$. Make the following
assumptions:

  * •

$X$ is smooth.

  * •

$X$ consists of one open $G$-orbit and one closed $G$-orbit of codimension
one.

Then our main result (Theorem 3.7.2) gives an algebraic description of the
category of $G$-equivariant vector bundles on $X$, provided that $X$ satisfies
a technical hypothesis called ‘fastenedness’ (Definition 2.2.1).

In the rest of the paper, we deduce simpler classifications in the special
cases of line bundles (Proposition 4.1.1) and vector bundles which are
generically $G$-equivariant local systems (Corollary 4.3.1). We also use the
main result to describe $G$-equivariant vector bundles on _weakly normal_
$G$-varieties obtained by gluing together those of the form described above
(Section 5). As an application, we study the coherent sheaves which arise as
associated graded modules of $(\mathfrak{g},K)$-modules, when the pair
$(\mathfrak{g},K)$ arises from a real reductive group (Section 6).

In the rest of this introduction, we discuss Theorem 3.7.2 in the larger
context of ‘gluing of categories,’ and we describe some of these applications
in more detail.

###  1.2. The idea of gluing

Let $X$ be as the first paragraph of 1.1, and write $X=U\cup Z$ where $U$ is
the open $G$-orbit and $Z$ is the closed $G$-orbit of codimension one,
equipped with the reduced subscheme structure. Upon choosing basepoints $z\in
Z$ and $u\in U$, it is easy to describe the categories of $G$-equivariant
vector bundles on $Z$ and $U$ separately:

$\displaystyle\mathrm{Vect}^{G}(Z)$
$\displaystyle\simeq\operatorname{Rep}(G^{z})$
$\displaystyle\mathrm{Vect}^{G}(U)$
$\displaystyle\simeq\operatorname{Rep}(G^{u}),$

where $G^{z}$ and $G^{u}$ are the stabilizer groups. It is natural to ask
whether $\mathrm{Vect}^{G}(X)$ can be constructed by ‘gluing’ the two
categories displayed above.

Motivation for this idea comes from Beilinson’s description [7] of perverse
(constructible) sheaves on a scheme $X$ which is decomposed as $U\cup Z$ where
$Z=V(f)$ for some regular function $f\in\mathscr{O}_{X}$. That description
says that a perverse sheaf $\mathscr{F}$ on $X$ can be encoded as a quadruple
$(\mathscr{F}_{U},\mathscr{F}_{Z},u,v)$ such that

  * •

$\mathscr{F}_{U}$ and $\mathscr{F}_{Z}$ are perverse sheaves on $U$ and $Z$,
respectively.

  * •

$u$ and $v$ are maps

$\Psi_{f}(\mathscr{F}_{U})\xrightarrow{u}\mathscr{F}_{Z}\xrightarrow{v}\Psi_{f}(\mathscr{F}_{U})$

where $\Psi_{f}(-)$ is the nearby cycles functor. (We have ignored the Tate
twist.) The composition $u\circ v$ must equal $1-(\text{monodromy
endomorphism})$.

This description has the good feature that it involves only the categories of
interest on $U$ and $Z$, along with a functor between them (and an
endomorphism of this functor).

For vector bundles, the situation is not as nice. The standard approaches to
gluing quasicoherent sheaves (e.g. recollement of categories) require knowing
about sheaves on $X_{\widehat{Z}}$ (the formal completion of $X$ along $Z$).
For example, Beauville–Laszlo gluing [6] implies that a $G$-equivariant vector
bundle on $X$ can be encoded as a triple
$(\mathscr{F}_{U},\mathscr{F}_{\widehat{Z}},\alpha)$ where

  * •

$\mathscr{F}_{U}$ and $\mathscr{F}_{\widehat{Z}}$ are $G$-equivariant vector
bundles on $U$ and $X_{\widehat{Z}}$, respectively.

  * •

$\alpha$ is an isomorphism between their restrictions to
$X_{\widehat{Z}}\smallsetminus Z$.

Since $X_{\widehat{Z}}$ is not easy to compute with in practice, we sought a
different description of $\mathrm{Vect}^{G}(X)$ which does not involve
$X_{\widehat{Z}}$.

In some special cases, knowledge about $X_{\widehat{Z}}$ can be gleaned from
looking at the action of $G$ on the normal bundle $\mathscr{N}_{Z/X}$. For
instance, if $G$ is reductive and $X$ is affine, then the Luna Slice Theorem
provides a one-dimensional $G^{z}$-invariant locally closed subvariety
$\ell\subset X$ which contains $z$ and is transverse to $Z$, together with a
$G^{z}$-equivariant étale map $\ell\to\mathscr{N}_{Z/X}|_{z}$. Combined with
the previous paragraph, this essentially reduces the problem to studying
$G$-equivariant vector bundles on $\mathscr{N}_{Z/X}$. However, in our desired
application $X$ is usually not affine, so this reduction does not apply.

In our desired application, something weaker is true (Proposition 6.3.7):
there is a line $\ell\subset X$ which intersects $Z$ transversely, is
invariant under a one-parameter subgroup $\gamma:\mathbb{G}_{m}\hookrightarrow
G^{z}$, and is acted upon nontrivially by $\gamma(\mathbb{G}_{m})$. The
existence of such a pair $(\ell,\gamma)$ is equivalent to the ‘fastenedness’
hypothesis of Theorem 3.7.2.

Starting from a choice of $(\ell,\gamma)$ as above, we arrive at a description
of $\mathrm{Vect}^{G}(X)$ which bears little resemblance to ‘gluing.’ Instead,
it should be thought of in the following terms. If we pull back
$\mathscr{F}\in\mathrm{Vect}^{G}(X)$ along $\ell\hookrightarrow X$, two
algebraic structures emerge:

  * •

$\mathscr{F}|_{\ell}$ is equivariant with respect to a $\mathbb{G}_{m}$-action
which comes from the subgroup $\gamma$. Recall that the category of
$\mathbb{G}_{m}$-equivariant vector bundles on $\mathbb{A}^{1}$ is equivalent
to the category of finite-dimensional vector spaces equipped with exhausting
filtrations, via the Artin–Rees construction (Proposition 3.1.1).

  * •

Each fiber of $\mathscr{F}|_{\ell}$ is acted on by the stabilizer group of the
corresponding point in $\ell$.

The classifying data defined in Theorem 3.7.2 are a combination of these two
bullet points.

However, in the case of line bundles, it is possible to reformulate the main
result in terms which more closely resemble ‘gluing via nearby cycles’ see
Lemma 5.3.2. The possibility of doing so reflects that equivariant
quasicoherent sheaves are more ‘rigid’ than arbitrary quasicoherent sheaves.

###  1.3. Applications

Suppose $G_{\mathbb{R}}$ is a real reductive group and $K_{\mathbb{R}}\subset
G_{\mathbb{R}}$ is a maximal compact subgroup. Let $K$ be the complexification
of $K_{\mathbb{R}}$ and let $\mathcal{N}$ be the cone of nilpotent elements in
the complexified Lie algebra of $G_{\mathbb{R}}$

The determination of the irreducible unitary representations of
$G_{\mathbb{R}}$ is one of the major unsolved problem in representation
theory. There is evidence to suggest that every such representation can be
constructed through a sequence of well-understood operations from a finite set
of building blocks, called the _unipotent representations_. These
representations are ‘attached’ (in a certain mysterious sense) to the
nilpotent orbits of $G_{\mathbb{R}}$ on the dual space of its Lie algebra.

Attached to every irreducible admissible representation $V$ of
$G_{\mathbb{R}}$ (for example, every irreducible unitary representation) is a
Harish-Chandra module $M$, which has the structure of an irreducible
$K$-equivariant $\mathfrak{g}$-module. Taking its associated graded gives a
well-defined class $[\operatorname{gr}(M)]$ in the Grothendieck group of
$K$-equivariant coherent sheaves on $\mathcal{N}$. This class provides
valuable information about $V$, including, for example, its
$K_{\mathbb{R}}$-types. Determining the classes $[\operatorname{gr}(M)]$
associated to unipotent representations is a fundamental problem with origins
in the seminal paper ([15]) of Vogan.

In Section 6, we define the $K$-chain $\overline{\mathrm{Ch}}(M)$ associated
to $M$. This is the variety with $K$-action obtained by first deleting the
orbits of codimension $\geq 2$ from the reduced support of
$\operatorname{gr}(M)$ and then passing to the projectivization (see 6.2 for
details). The class $[\operatorname{gr}(M)]$ gives rise to a class
$[\overline{\operatorname{gr}}(M)]$ on $\overline{\mathrm{Ch}}(M)$, and when
$V$ is unipotent, we conjecture that $[\overline{\mathrm{gr}}(M)]$ determines
$[\operatorname{gr}(M)]$. Our main result in Section 6 is that
$\overline{\mathrm{Ch}}(M)$ is fastened (Proposition 6.3.7). To prove this
remarkable fact, we use in an essential way the special structure of
$\mathcal{N}$ (e.g. the existence of ‘real’ Slodowy slices with nice
contracting $\mathbb{G}_{m}$-actions). Since $\overline{\mathrm{Ch}}(M)$ is
fastened, we can understand $\overline{\mathrm{gr}}(M)$ (and by extension
$\operatorname{gr}(M)$ and $V$) using Theorem 3.7.2 (and its corollaries).

###  1.4. Acknowledgments

The authors would like to thank Roman Bezrukavnikov and David Vogan for many
helpful discussions. The second author is supported by an NSF GRFP grant.

##  2\. Chains

###  2.1. Basic definitions

Let $k$ be an algebraically closed field of characteristic zero, and let $G$
be an affine algebraic group over $k$.

###### Definition 2.1.1.

A $G$-chain is a $k$-variety111For us, a ‘variety’ is a reduced, separated,
finite type $k$-scheme. It is not necessarily irreducible. $X$ equipped with
an action by $G$ such that

  1. (1)

$G$ acts on $X$ with finitely many orbits.

  2. (2)

$X$ is pure dimensional.

  3. (3)

Each $G$-orbit on $X$ has codimension $0$ or $1$.

These properties imply that every $G$-orbit on $X$ is either open or closed.
Each open orbit $U\subset X$ has codimension 0, and each closed orbit
$Z\subset X$ has codimension 1.

###### Example 2.1.2.

Let $G=\mathbb{G}_{m}$, and $X=\operatorname{Spec}\mathbb{C}[x,y]/(xy)$. Let
$G$ act on $X$ by the formulas

$t\cdot x=t^{2}x\qquad t\cdot y=t^{-2}y\qquad t\in\mathbb{G}_{m}$

Then $X$ is a $G$-chain. It has two open $G$-orbits (the punctured axes), and
one closed $G$-orbit ($\\{0\\}$). This example is closely related to the
(infinite-dimensional) representation theory of
$\operatorname{SL}_{2}(\mathbb{R})$, see Example 6.4.1.

We say that a $G$-chain is _irreducible_ if it has only one open $G$-orbit.
(Note that an irreducible $G$-chain need not be irreducible as a $k$-variety
if $G$ is not connected.) For an arbitrary $G$-chain $X$, we define its
_irreducible components_ to be the closures of the open orbits of $X$,
equipped with the reduced subscheme structures. These are precisely the
irreducible closed sub-chains of $X$ whose dimensions equal $\dim X$. We say
that a $G$-chain is _simple_ if it is irreducible and has at most one closed
$G$-orbit. If $X$ is irreducible, then the simple locally-closed sub-chains
form an open cover of $X$.

If $X$ is a $G$-chain, then its normalization $\tilde{X}$ is again a
$G$-chain. Any normal $G$-chain is smooth, since its singular locus is a
subset of codimension $\geq 2$ and hence empty. Any smooth $G$-chain is the
disjoint union of its irreducible components. In particular, the normalization
of an irreducible $G$-chain is a smooth, irreducible $G$-chain.

###  2.2. Fastened chains

###### Definition 2.2.1.

We define the property of ‘fastenedness’ for $G$-chains in three steps:

  * •

If $X$ is a smooth irreducible $G$-chain, we say that a closed $G$-orbit
$Z\subset X$ is _fastened_ to the open orbit $U\subset X$ if $G$ acts
transitively on the punctured normal bundle

$\mathscr{N}^{\circ}_{Z/X}:=\mathscr{N}_{Z/X}\smallsetminus(\text{zero
section}).$

  * •

If $X$ is a smooth irreducible $G$-chain, we say that $X$ is _fastened_ if
each closed $G$-orbit $Z\subset X$ is fastened to the open orbit.

  * •

If $X$ is any $G$-chain, we say that $X$ is _fastened_ if each irreducible
component of the normalization of $X$ is fastened.

The terminology is set up so that it is clear what it means for a closed orbit
$Z\subset X$ to be fastened to one open orbit which is adjacent to $Z$ but not
to another.

###### Remark 2.2.2.

Suppose that $X$ is a smooth irreducible $G$-chain and that $Z\subset X$ is a
closed $G$-orbit. For any closed point $z\in Z$, the stabilizer subgroup
$G^{z}\subset G$ acts linearly on the (one-dimensional) fiber
$\mathscr{N}_{Z_{i}/X}|_{z}$. This action defines a character
$G^{z}\to\mathbb{G}_{m}$, which is surjective if and only if $Z$ is fastened
to the open orbit of $X$.

###### Proposition 2.2.3.

Every affine $G$-chain is fastened.

_Proof._ Let $X$ be an affine $G$-chain. Since normalization maps of finite
type schemes are finite, hence affine, we may assume that $X$ is also smooth
and irreducible. Pick an arbitrary closed $G$-orbit $Z\subset X$. We shall
show that $Z$ is fastened to the open orbit of $X$.

Choose a point $z\in Z$. Assume for sake of contradiction that the character
$G^{z}\to\mathbb{G}_{m}$ defined in Remark 2.2.2 is not surjective. Then it
factors through some finite subgroup $\mu_{r}\subset\mathbb{G}_{m}$ where
$\mu_{r}$ is the group of $r$-th roots of unity. Choose a linear coordinate
$\mathscr{N}_{Z/X}|_{z}\simeq\mathbb{A}^{1}_{t}$, and define the regular
function

$f:\mathscr{N}_{Z/X}\to\mathbb{A}^{1}$

via the formula $f(t)=t^{r}$.

Since $\mathscr{N}_{Z/X}$ is the associated bundle produced by the homogeneous
space $G/G^{z}\simeq Z$ and the action
$G^{z}\curvearrowright\mathscr{N}_{Z/X}|_{z}$, the function $f$ defined above
extends to a unique $G$-invariant function $\tilde{f}$ on $\mathscr{N}_{Z/X}$.
By construction, it is nonconstant along the fibers of $\mathscr{N}_{Z/X}$.

Let

$X_{h}:=\operatorname{Bl}_{Z\times\\{0\\}}(X\times\mathbb{A}^{1}_{h})\smallsetminus(X\times\\{0\\})^{\sim}\to\mathbb{A}^{1}_{h}$

be the deformation to the normal cone of $Z\subset X$. (This construction
appears again in 3.2.) Since $X$ is affine, so is $X_{h}$. Since $G$ is
reductive, the functor of $G$-invariants is exact, so we may non-uniquely
extend $\tilde{f}$ to a $G$-invariant regular function $\tilde{f}_{h}$ on
$X_{h}$. Then $\tilde{f}_{h}$ is constant along the $G$-orbits
$U\times\\{a\\}\subset X_{h}$ and hence along their limit, which is
$\mathscr{N}_{Z/X}^{\circ}$.222This argument uses that
$X_{h}\to\mathbb{A}^{1}_{h}$ is flat. But
$\tilde{f}_{h}|_{\mathscr{N}_{Z/X}^{\circ}}=\tilde{f}$ by construction, so
this constancy contradicts the previous paragraph. ∎

Outside the world of affine varieties, there are many examples of non-fastened
chains. Here is one:

###### Example 2.2.4.

Let $G=SL_{2}(\mathbb{C})$ and let $V$ be its four-dimensional irreducible
representation. $V$ is identified with the space of homogeneous degree-3
polynomials in the variables $x$ and $y$.

Let $U$ be the 3-dimensional $G$-orbit passing through the element $x^{2}y\in
V$. There are two $G$-orbits in its boundary: $\\{0\\}$ and the 2-dimensional
$G$-orbit $Z$ passing through the element $z=x^{3}\in V$. Let $X$ be the
irreducible chain $U\cup Z$. Note that

$G^{z}=\left\\{\begin{pmatrix}\xi&a\\\
0&\xi^{-1}\end{pmatrix}:\xi^{3}=1\right\\}.$

The variety $X$ is singular, but its normalization $\tilde{X}$ is smooth. For
any closed orbit $\tilde{Z}\subset\tilde{X}$, the natural map $\tilde{Z}\to Z$
is a $G$-equivariant cover. In particular, for any $\tilde{z}\in\tilde{Z}$, we
have an inclusion of stabilizer groups

$G^{\tilde{z}}\subseteq G^{z}=\left\\{\begin{pmatrix}\xi&a\\\
0&\xi^{-1}\end{pmatrix}:\xi^{3}=1\right\\}.$

Thus $G^{\tilde{z}}$ does not contain a torus, so Remark 2.2.2 implies that
$X$ is not fastened.

###  2.3. Fastening data

We will show that fastened chains are glued together (or ‘fastened’) by
certain $\mathbb{G}_{m}$-invariant curves, as described below:

###### Definition 2.3.1.

Let $X$ be a smooth irreducible $G$-chain. Let $U$ be the open orbit of $X$,
and let $Z$ be some closed orbit. A _fastening datum_ for $(U,Z)$ is a pair
$(\gamma,\ell)$ given as follows:

  * •

$\gamma:\mathbb{G}_{m}\to G$ is a co-character.

  * •

$\ell:\mathbb{A}^{1}\hookrightarrow X$ is a locally-closed embedding.

We require $(\gamma,\ell)$ to satisfy the following properties:

  1. (1)

$\ell^{-1}(Z)=\\{0\\}$ and $\ell$ is transverse to $Z$.

  2. (2)

$\gamma(\mathbb{G}_{m})$ leaves $\ell(\mathbb{A}^{1})\subset X$ invariant and
acts transitively on $\ell(\mathbb{A}^{1}\smallsetminus\\{0\\})$.

###### Proposition 2.3.2.

In the setting of Definition 2.3.1, $Z$ is fastened to $U$ if and only if
there is a fastening datum $(\gamma,\ell)$ for $(U,Z)$.

_Proof._ First, assume that $Z$ is fastened to $U$. Fix a closed point $z\in
Z$. By Remark 2.2.2, the action $G^{z}\curvearrowright\mathscr{N}_{Z/X}|_{z}$
occurs via a surjective character $G^{z}\xrightarrow{\chi}\mathbb{G}_{m}$. Let
$L^{z}\subset G^{z}$ be the identity component of a Levi subgroup of $G$.
Since $\ker(\chi)$ contains the unipotent radical of $G^{z}$, the surjectivity
of $\chi$ implies that $L^{z}\xrightarrow{\chi}\mathbb{G}_{m}$ is also
surjective. Since one-dimensional tori are dense in reductive groups, there is
a one-dimensional torus $T^{z}\subset L^{z}$ such that
$T^{z}\xrightarrow{\chi}\mathbb{G}_{m}$ is nontrivial (hence surjective).
Since $T^{z}$ is reductive, the projection map
$\mathscr{T}_{z}X\to\mathscr{N}_{Z/X}|_{z}$ has a $T^{z}$-invariant section.
The image of this section is a line $k\cdot v\subset\mathscr{T}_{z}X$ on which
$T^{z}$ acts by the same nontrivial character $\chi$.

By Sumihiro’s theorem on torus actions, which applies since $X$ is smooth and
hence normal, there exists a $T^{z}$-invariant open affine subscheme
$W_{z}\subset X$ which contains $z$. Applying the Luna Slice Theorem to the
$T^{z}$-orbit given by $\\{z\\}\subset W_{z}$, we conclude (possibly upon
replacing $W_{z}$ by a smaller $T^{z}$-invariant affine open subscheme) that
there is a $T^{z}$-equivariant étale map $\varphi:W_{z}\to\mathscr{T}_{z}X$
which induces an isomorphism on tangent spaces at $z\in W_{z}$. Consider the
base change

${C}$${W_{z}}$${k\cdot v}$${\mathscr{T}_{z}X}$$\scriptstyle{\varphi}$cl.emb.

where the bottom horizontal map was constructed in the previous paragraph.
Certainly $C$ contains $z\in W_{z}$ because $k\cdot v$ contains
$0\in\mathscr{T}_{z}X$. Let $C_{z}\subset C$ be the connected component of $C$
which contains $z$. The map $C_{z}\to k\cdot v$ is étale and
$\mathbb{G}_{m}$-equivariant, so the previous two sentences imply that it is
an isomorphism.333Proof: Since $C_{z}$ is a smooth affine connected curve
equipped with a $\mathbb{G}_{m}$-action with a closed orbit $\\{z\\}$, it is
isomorphic to $\mathbb{A}^{1}$ with some multiple of the standard
$\mathbb{G}_{m}$-action. Since the map $C_{z}\to k\cdot v$ induces an
isomorphism on tangent spaces at $z\in C_{z}$, this multiple must be the same
as that given by $\chi|_{T^{z}}$. Now the map $C_{z}\to k\cdot v$ is
determined by its behavior on the open $\mathbb{G}_{m}$-orbits; there it is an
isomorphism, and one can easily check that this implies the claim.

We define $\ell$ to be the composition

$\mathbb{A}^{1}\simeq k\cdot v\simeq
C_{z}\xhookrightarrow{\text{cl.emb.}}W_{z}\xhookrightarrow{\text{open
emb.}}X,$

and we define $\gamma$ to be the composition

$\mathbb{G}_{m}\simeq T^{z}\hookrightarrow G^{z}\hookrightarrow G$

where $\mathbb{G}_{m}\simeq T^{z}$ is an arbitrary isomorphism. Then
$(\gamma,\ell)$ is a fastening datum for $(U,Z)$.

Conversely, assume that $(\gamma,\ell)$ is a fastening datum for $(U,Z)$.
Definition 2.3.1(2) implies that the map $\mathbb{A}^{1}\xrightarrow{\ell}X$
is $\mathbb{G}_{m}$-equivariant with respect to some nontrivial linear action
$\mathbb{G}_{m}\curvearrowright\mathbb{A}^{1}$ (which is not necessarily the
standard one). Passing to tangent spaces, we get a
$\mathbb{G}_{m}$-equivariant map

$\mathbb{A}^{1}\simeq\mathscr{T}_{0}\mathbb{A}^{1}\xrightarrow{d\ell}\mathscr{T}_{z}X$

where the action of $\mathbb{G}_{m}\curvearrowright\mathscr{T}_{z}X$ is via
the cocharacter $\gamma:\mathbb{G}_{m}\to G^{z}$ (this uses Definition
2.3.1(1)). By the transversality statement in Definition 2.3.1(2), the
composition

$\mathbb{A}^{1}\simeq\mathscr{T}_{0}\mathbb{A}^{1}\xrightarrow{d\ell}\mathscr{T}_{z}X\to\mathscr{N}_{Z/X}|_{z}$

is nonzero, hence an isomorphism. It is also $\mathbb{G}_{m}$-equivariant, so
we conclude that the action
$\gamma(\mathbb{G}_{m})\curvearrowright\mathscr{N}_{Z/X}|_{z}$ is nontrivial.
Hence, the action $G^{z}\curvearrowright\mathscr{N}_{Z/X}|_{z}$ is nontrivial.
By Remark 2.2.2, this implies that $Z$ is fastened to the open orbit in $X$. ∎

##  3\. Equivariant sheaves on smooth simple chains

###  3.1. Overview

In this section, $X=U\cup Z$ is a smooth $G$-chain with one open and one
closed orbit. Assume that $X$ is fastened, and let $(\gamma,\ell)$ be a
fastening datum (Definition 2.3.1).

Given this datum, we will describe the category
$\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$ of $G$-equivariant torsion-free
quasicoherent sheaves on $X$ in algebraic terms (Theorem 3.7.2). The main idea
is that the line $\ell:\mathbb{A}^{1}\to X$ functions as a ‘basepoint’ which
sees both the open and the closed orbit of $X$. More precisely, we will
explain how $X$ is related to the ‘homogeneous space’ given by
$G\times\mathbb{A}^{1}$ modulo the stabilizer scheme of
$\ell(\mathbb{A}^{1})$, see Lemma 3.3.1.

The special case when $G\curvearrowright X$ is the standard (weight one)
action $\mathbb{G}_{m}\curvearrowright\mathbb{A}^{1}$ will feature prominently
in our analysis. Here, the desired classification is elementary and well-
known:

###### Proposition 3.1.1 (Artin–Rees).

Let $\mathrm{Vect}^{\mathrm{filt}}$ be the category of vector spaces equipped
with exhausting (but not necessarily separating) $\mathbb{Z}$-filtrations.
There is a natural equivalence of monoidal categories:

$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1})\simeq\mathrm{Vect}^{\mathrm{filt}}$

_Proof._ The category $\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathbb{A}^{1})$
is equivalent (via global sections) to the category of graded modules over the
algebra $k[x]$, where the generator $x$ is placed in degree 1. Each such
module is encoded by a diagram

$\cdots\to V_{i}\to V_{i+1}\to\cdots$

of vector spaces, where the maps are arbitrary. The module is torsion-free if
and only if the maps are all injective. In this case, the diagram of vector
spaces is equivalent to the datum of a single vector space
$V=\operatorname*{colim}_{i}V_{i}$ equipped with the exhausting filtration $F$
defined by $F^{\geq i}V=V_{i}$. The reverse direction is clear. ∎

We will often refer to this equivalence as the Artin–Rees construction. Since
filtrations already appear here, it is unsurprising that they also appear in
the statement of the main classification result (Theorem 3.7.2).

###### Remark 3.1.2.

We now explain how we propose to get ‘algebraic’ descriptions of categories of
equivariant sheaves. The appearance of the word ‘monoidal’ in the proposition
implies that we get analogous equivalences involving algebro-geometric objects
defined over $\mathbb{A}^{1}/\mathbb{G}_{m}$. For example, considering algebra
objects on both sides yields the equivalence

$\operatorname{Sch}^{\operatorname{aff},\mathrm{tf}}_{\mathbb{A}^{1}/\mathbb{G}_{m}}\simeq(\text{filtered
rings})^{\mathrm{op}}$

where the left hand side is the category of affine schemes over
$\mathbb{A}^{1}/\mathbb{G}_{m}$ for which the structure map
$S\to\mathbb{A}^{1}/\mathbb{G}_{m}$ is flat.

This equivalence intertwines the Cartesian monoidal structure on the left hand
side with the $\otimes$ monoidal structure on the right hand side. Passing to
group objects with respect to these monoidal structures, we obtain a
(contravariant) equivalence between affine group schemes flat over
$\mathbb{A}^{1}/\mathbb{G}_{m}$ and filtered Hopf algebras. This list of
equivalences continues indefinitely, but essentially we will only need one
more observation: given such a group scheme
$\mathscr{G}\to\mathbb{A}^{1}/\mathbb{G}_{m}$ corresponding to a filtered Hopf
algebra $A$, the category of $\mathbb{G}_{m}$-equivariant torsion-free
quasicoherent sheaves on the stack $\mathbb{A}^{1}/\mathscr{G}$ is equivalent
to the category of filtered comodules for $A$. This is applied in Definition
3.7.1.

####  3.1.3. Further comments

In the above proposition, the ‘torsion-free’ requirement is equivalent to
flatness. Similarly, a $G$-equivariant sheaf on $X$ is torsion-free if and
only if it is flat, because $Z$ has codimension one. In the rest of this
section, we use ‘torsion-free’ and ‘flat’ interchangeably.

In this section, a (back)slash means a quotient in the stack-theoretic sense.

To clarify what is canonical and what is not, let us explain where the
fastening datum $(\gamma,\ell)$ is used. In 3.2, the fastening datum is not
used, but Lemma 3.2.2 relies on the assumption that $X$ is fastened. In 3.3,
the line $\ell$ (satisfying Definition 2.3.1(1)) is used to define the
stabilizer scheme $G^{s}$. From 3.5 onward, we use the full datum
$(\gamma,\ell)$.

###  3.2. The punctured ideal sheaf

Let $\mathscr{I}_{Z/X}$ be the ideal sheaf of $Z\subset X$, which is a line
bundle on $X$. Consider its total space
$\mathcal{S}\kern-0.5ptpec(\mathcal{S}\kern-0.5ptym_{\mathscr{O}_{X}}^{\cdot}\mathscr{I}_{Z/X}^{\vee})$
which we (abusively) also denote $\mathscr{I}_{Z/X}$. Let

$\mathscr{I}^{\circ}_{Z/X}:=\mathscr{I}_{Z/X}\smallsetminus(\text{zero
section})$

be the punctured ideal sheaf. Since $Z$ is $G$-invariant in $X$, there are
commuting actions

$G\curvearrowright\mathscr{I}^{\circ}_{Z/X}\curvearrowleft\mathbb{G}_{m}$

where $\mathbb{G}_{m}$ acts by dilation on the fibers. Since
$\mathscr{I}^{\circ}_{Z/X}/\mathbb{G}_{m}\simeq X$, we deduce that

(1)
$\operatorname{QCoh}^{G}(X)\simeq\operatorname{QCoh}^{G\times\mathbb{G}_{m}}(\mathscr{I}^{\circ}_{Z/X})\simeq\operatorname{QCoh}^{\mathbb{G}_{m}}(G\backslash\mathscr{I}^{\circ}_{Z/X}).$

Our goal in this and the next subsection is to explain how
$G\backslash\mathscr{I}^{\circ}_{Z/X}$ is related to the stabilizer scheme of
the line $\ell:\mathbb{A}^{1}\to X$.

###### Remark 3.2.1.

We point out two facts about the geometry of $\mathscr{I}^{\circ}_{Z/X}$.

  1. (i)

$\mathscr{I}^{\circ}_{Z/X}$ is closely related to the deformation of $X$ to
the normal cone of $Z\subset X$. To see this, note that the injection of
sheaves $\mathscr{I}_{Z/X}\hookrightarrow\mathscr{O}_{X}$ defines a
(noninjective) map of varieties

$\mathscr{I}_{Z/X}\xrightarrow{p}X\times\mathbb{A}^{1}.$

It is easy to check that this is isomorphic to the map

$\operatorname{Bl}_{Z\times\\{0\\}}(X\times\mathbb{A}^{1})\smallsetminus(Z\times\mathbb{A}^{1})^{\sim}\to
X\times\mathbb{A}^{1}$

where the left hand side is obtained by taking the blow-up of
$X\times\mathbb{A}^{1}$ along $Z\times\\{0\\}$ and deleting the proper
transform of $Z\times\mathbb{A}^{1}$. On the other hand, the deformation to
the normal cone is given by

$\operatorname{Bl}_{Z\times\\{0\\}}(X\times\mathbb{A}^{1})\smallsetminus(X\times\\{0\\})^{\sim}\to
X\times\mathbb{A}^{1}.$

This is not the same as the previous map, but the deformation to the normal
cone contains the open subscheme

$\operatorname{Bl}_{Z\times\\{0\\}}(X\times\mathbb{A}^{1})\smallsetminus\Big{(}(X\times\\{0\\})^{\sim}\sqcup(Z\times\mathbb{A}^{1})^{\sim}\Big{)}$

which is isomorphic to $\mathscr{I}^{\circ}_{Z/X}$ as schemes over
$X\times\mathbb{A}^{1}$.

  2. (ii)

Note that $\mathscr{I}_{Z/X}^{\circ}|_{Z}$ is the punctured conormal bundle of
$Z$, while the fiber over $0\in\mathbb{A}^{1}$ of the scheme displayed in the
last line above is the punctured normal bundle of $Z$. This apparent
contradiction is resolved by the observation that the punctured total spaces
of $\mathscr{L}$ and $\mathscr{L}^{\vee}$ are canonically isomorphic (but not
in a $\mathbb{G}_{m}$-equivariant way), for any line bundle $\mathscr{L}$.
Thus, there is no harm in writing
$\mathscr{I}_{Z/X}^{\circ}|_{Z}\simeq\mathscr{N}_{Z/X}^{\circ}$.

Define the ‘projection’ map $\pi=\mathrm{pr}_{2}\circ
p:\mathscr{I}^{\circ}_{Z/X}\to\mathbb{A}^{1}.$

###### Lemma 3.2.2.

We have the following:

  1. (i)

$\pi$ is smooth and $G\times\mathbb{G}_{m}$-equivariant, where the action
$(G\times\mathbb{G}_{m})\curvearrowright\mathbb{A}^{1}$ is given by
$(\operatorname{triv},\operatorname{standard})$.

  2. (ii)

As $G$-varieties, the closed fibers of $\pi$ are given by

$\pi^{-1}(c)\simeq\begin{cases}U&\text{ if }c\neq 0\\\
\mathscr{N}^{\circ}_{Z/X}&\text{ if }c=0.\end{cases}$

In particular, $G$ acts transitively on the fibers of $\pi$.

_Proof._ The map $\pi$ is $G\times\mathbb{G}_{m}$-equivariant because $p$ and
$\operatorname{pr}_{2}$ are both $G\times\mathbb{G}_{m}$-equivariant. The
description of fibers of $\pi$ follows from Remark 3.2.1 and the transitivity
of the $G$-action on fibers follows from the assumption that $X$ is fastened.
This proves (ii).

Since the domain and target of $\pi$ are smooth, and each fiber has dimension
equal to $\dim X$, the Miracle Flatness Theorem implies that $\pi$ is flat.
Since each fiber is also smooth, this implies that $\pi$ is smooth (since
$\pi$ is finitely presented), which proves (i). ∎

###  3.3. The group scheme $G^{s}$ associated to a fastening datum

The choice of a line $\ell:\mathbb{A}^{1}\to X$ which satisfies Definition
2.3.1(1) is equivalent to the choice of a section
$s:\mathbb{A}^{1}\to\mathscr{I}^{\circ}_{Z/X}$ of the projection map $\pi$.
The forward direction of this equivalence sends $\ell$ to the proper transform
of $(\ell,\operatorname{id}_{\mathbb{A}^{1}}):\mathbb{A}^{1}\hookrightarrow
X\times\mathbb{A}^{1}$ under the blow-up considered in Remark 3.2.1(i), and
the transversality hypothesis is needed to ensure that the proper transform
lands in the open subscheme $\mathscr{I}^{\circ}_{Z/X}$.

In the rest of this section, we will work in the category of schemes (or
stacks) over $\mathbb{A}^{1}$. (Starting from the next subsection, these
objects will also be $\mathbb{G}_{m}$-equivariant, i.e. they descend to
$\mathbb{A}^{1}/\mathbb{G}_{m}$.) From this point of view, the section $s$
functions as a ‘basepoint’ for $\mathscr{I}^{\circ}_{Z/X}$, which we view as a
homogeneous space for the group scheme $G\times\mathbb{A}^{1}$. The rest of
this subsection makes this statement precise.

Define the map $a$ via the commutative diagram

${G\times\mathbb{A}^{1}}$${G\times\mathscr{I}^{\circ}_{Z/X}}$${\mathscr{I}^{\circ}_{Z/X}}$$\scriptstyle{\operatorname{id}_{G}\times
s}$$\scriptstyle{a}$$\scriptstyle{\operatorname{act}}$

where $\operatorname{act}$ is the $G$-action on $\mathscr{I}^{\circ}_{Z/X}$.
Define $G^{s}$ as the fibered product

${G^{s}}$${G\times\mathbb{A}^{1}}$${\mathbb{A}^{1}}$${\mathscr{I}^{\circ}_{Z/X}}$$\scriptstyle{a}$$\scriptstyle{s}$

###### Lemma 3.3.1.

Work in the category of schemes over $\mathbb{A}^{1}$.

  1. (i)

$G^{s}$ is smooth over $\mathbb{A}^{1}$.

  2. (ii)

The map $G^{s}\hookrightarrow G\times\mathbb{A}^{1}$ expresses the former as a
group subscheme of the latter.

  3. (iii)

The quotient $(G\times\mathbb{A}^{1})/G^{s}$ taken in the smooth topology (or
any finer one) identifies with $\mathscr{I}^{\circ}_{Z/X}$ via the map $a$.

_Proof._ (ii). The indicated map $G^{s}\hookrightarrow G\times\mathbb{A}^{1}$
is a closed embedding because the map $s$ is a closed embedding. The ‘group
subscheme’ property follows from the definition of group action applied to
$(G\times\mathbb{A}^{1})\curvearrowright\mathscr{I}^{\circ}_{Z/X}$.

(i). The Miracle Flatness Theorem implies that $a$ is flat, since the source
and target of $a$ are smooth varieties, and Lemma 3.2.2(ii) ensures that the
fibers have the correct dimensions. Furthermore, $a$ is finitely presented,
and its fibers are smooth because they are isomorphic to stabilizer group
schemes, and group schemes are smooth in characteristic zero. Therefore $a$ is
smooth. Since $G^{s}\to\mathbb{A}^{1}$ is obtained from $a$ by base change, it
is smooth as well.

(iii). The following statement is well-known in algebraic geometry:

  * •

Let $S$ be a scheme, and work in the category of sheaves of groupoids on the
category of affine schemes over $S$ (denoted
$\operatorname{Sch}^{\operatorname{aff}}_{/S}$) with respect to some
Grothendieck topology. Let $\mathscr{G}$ be a group object, and let
$\mathscr{G}\curvearrowright\mathscr{I}$ be a left action on some other object
$\mathscr{I}$. Let $s:S\to\mathscr{I}$ be a section, let
$a:\mathscr{G}\to\mathscr{I}$ be the action map, and let $\mathscr{G}^{s}$ be
the stabilizer subgroup of $s$ (defined as above). If $a$ is a cover, then
$\mathscr{G}/\mathscr{G}^{s}\simeq\mathscr{I}$.

This implies (iii) by taking $S=\mathbb{A}^{1}$, working in the smooth
topology, and taking $\mathscr{G}=G\times\mathbb{A}^{1}$ and
$\mathscr{I}=\mathscr{I}^{\circ}_{Z/X}$. Our proof of (i) shows that $a$ is a
smooth cover, so the statement applies.

For sake of completeness, we prove the statement in the bullet point. Since
$a$ is a cover, the geometric realization of the Cech nerve of $a$ is
isomorphic to $\mathscr{I}$. The Cech nerve of $a$ is the simplicial object

${\cdots}$${\mathscr{G}\underset{\mathscr{I}}{\times}\mathscr{G}}$${\mathscr{G}}$

It suffices to show that this is isomorphic to the simplicial object which
encodes the right action of $\mathscr{G}^{s}$ on $\mathscr{G}$, shown below:

${\cdots}$${\mathscr{G}\times\mathscr{G}^{s}}$${\mathscr{G}}$

At the level of objects, this isomorphism is defined by the maps

$\mathscr{G}\times\mathscr{G}^{s}\times\cdots\times\mathscr{G}^{s}\to\mathscr{G}\underset{\mathscr{I}}{\times}\mathscr{G}\underset{\mathscr{I}}{\times}\cdots\underset{\mathscr{I}}{\times}\mathscr{G}$

given by the formula

$(g_{1},g_{2},\ldots,g_{n})\mapsto(g_{1},g_{1}g_{2},\ldots,g_{1}g_{2}\cdots
g_{n}),$

where $g_{1}$ is a $T$-point of $\mathscr{G}$ and $g_{2},\ldots,g_{n}$ are
$T$-points of $\mathscr{G}^{s}$, for any test object
$T\in\operatorname{Sch}^{\operatorname{aff}}_{/S}$. It is straightforward to
make these maps compatible with the maps in the simplicial objects. ∎

###### Remark 3.3.2.

For $c\in\mathbb{A}^{1}\smallsetminus\\{0\\}$, we have $G/G^{s(c)}\simeq U$
where $G^{s(c)}$ means the stabilizer of $s(c)\in U$. For
$0\in\mathbb{A}^{1}$, we similarly have
$G/G^{s(0)}\simeq\mathscr{N}^{\circ}_{Z/X}$. The preceding lemma smoothly
interpolates between these two cases, since $G^{s}|_{c}\simeq G^{s(c)}$.

In 3.2(1), replacing $\mathscr{I}^{\circ}_{Z/X}$ by
$(G\times\mathbb{A}^{1})/G^{s}$ yields a preliminary result:

$\operatorname{QCoh}^{G}(X)\simeq\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathbb{A}^{1}/G^{s}).$

The category $\operatorname{QCoh}(\mathbb{A}^{1}/G^{s})$ is just the category
of sheaves on $\mathbb{A}^{1}$ equipped with an action by the group scheme
$G^{s}$. The problem is that the action
$\mathbb{A}^{1}/G^{s}\curvearrowleft\mathbb{G}_{m}$ is inexplicit. It does
_not_ arise from an action $G^{s}\curvearrowleft\mathbb{G}_{m}$ via
automorphisms of $G^{s}$ as a group scheme over $\mathbb{A}^{1}$. We will
rectify this problem in 3.5.

###  3.4. Descent along a degree-$n$ cover

Let $\mathbb{A}^{1}_{h}$ be the affine line with coordinate $h$, let $n$ be a
positive integer, and consider the degree-$n$ map

$\tau_{n}:\mathbb{A}^{1}_{h^{1/n}}\to\mathbb{A}^{1}_{h}$

defined by $c\mapsto c^{n}$.444We will use the notations
$\mathbb{A}^{1}_{h^{1/n}}$ and $\mathbb{A}^{1}_{h}$ to distinguish between the
domain and target of the map $\tau_{n}$. Of course, both are abstractly
isomorphic to $\mathbb{A}^{1}$. Consider the standard $\mathbb{G}_{m}$-actions
on the domain and target. The map $\tau_{n}$ intertwines these actions, up to
the degree-$n$ group homomorphism
$\mathbb{G}_{m}\xrightarrow{p_{n}}\mathbb{G}_{m}$. The goal of this subsection
is to relate the categories
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}_{h})$ and
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}_{h^{1/n}})$.

###### Lemma 3.4.1.

Consider the commutative diagram of stacks

${\mathrm{pt}/\mathbb{G}_{m}}$${\mathrm{pt}/\mathbb{G}_{m}}$${\mathbb{A}^{1}_{h^{1/n}}/\mathbb{G}_{m}}$${\mathbb{A}^{1}_{h}/\mathbb{G}_{m}}$$\scriptstyle{p_{n}}$$\scriptstyle{\tau_{n}}$

where the vertical maps go to $0\in\mathbb{A}^{1}$. The induced diagram of
categories

${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}_{h})}$${\operatorname{QCoh}_{\mathrm{tf}}^{\mathbb{G}_{m}}(\mathbb{A}^{1}_{h^{1/n}})}$${\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathrm{pt})}$${\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathrm{pt})}$$\scriptstyle{\tau_{n}^{*}}$$\scriptstyle{p_{n}^{*}}$

is Cartesian. Equivalently, $\tau_{n}^{*}$ is fully faithful, and its
essential image is the subcategory

$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}_{h^{1/n}})\subset\operatorname{QCoh}_{\mathrm{tf}}^{\mathbb{G}_{m}}(\mathbb{A}^{1}_{h^{1/n}})$

consisting of $\mathbb{G}_{m}$-equivariant sheaves $\mathscr{F}$ for which the
weights of the action $\mathbb{G}_{m}\curvearrowright\mathscr{F}|_{0}$ are all
multiples of $n$.

_Proof._ We apply Proposition 3.1.1 to the above diagram of categories. Under
this reformulation, the map $\tau_{n}^{*}$ sends a filtered vector space
$(V,F)$ to $(V,F^{\prime})$ where $F^{\prime}$ is the filtration defined by
$(F^{\prime})^{\geq m}V=F^{\geq nm}V$. The map $p_{n}^{*}$ sends a graded
vector space $\oplus_{i}V_{i}$ to the same vector space but with the degrees
multiplied by $n$. The vertical maps send a filtered vector space to its
associated graded.

Now the claim follows because a filtered vector space is of the form
$(V,F^{\prime})$ if and only if its associated graded has nonzero components
only in degrees which are multiples of $n$. ∎

###### Remark 3.4.2.

The restriction to torsion-free sheaves is necessary. An example of a torsion
sheaf on $\mathbb{A}^{1}_{h^{1/n}}$ which satisfies the condition that the
weights of the $\mathbb{G}_{m}$-action on the zero fiber are multiples of $n$
but which does not lie in the essential image of $\tau_{n}^{*}$ is a
skyscraper sheaf at $0\in\mathbb{A}^{1}_{h^{1/n}}$ with
$\mathbb{G}_{m}$-weight 0.

Since the functors $\tau_{n}^{*}$ and $p_{n}^{*}$, one can immediately
bootstrap Lemma 3.4.1 from a statement about sheaves on
$\mathbb{A}^{1}_{h}/\mathbb{G}_{m}$ to a statement about stacks (and sheaves
on stacks) over $\mathbb{A}^{1}_{h}/\mathbb{G}_{m}$, by the reasoning in
Remark 3.1.2. This yields the following:

###### Proposition 3.4.3.

Let $\mathscr{Y}$ be a stack with $\mathbb{G}_{m}$-action, and let
$\mathscr{Y}\to\mathbb{A}^{1}_{h}$ be a $\mathbb{G}_{m}$-equivariant map. Let
$\tau_{n}^{*}\mathscr{Y}$ be the base change along
$\tau_{n}:\mathbb{A}^{1}_{h^{1/n}}\to\mathbb{A}^{1}_{h}$. Then the following
diagram is Cartesian and the horizontal functors are fully faithful:

${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathscr{Y})}$${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\tau_{n}^{*}\mathscr{Y})}$${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathscr{Y}|_{0})}$${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}((\tau_{n}^{*}\mathscr{Y})|_{0})}$$\scriptstyle{\tau_{n}^{*}}$$\scriptstyle{p_{n}^{*}}$

Here the subscript ‘$\mathrm{tf}$’ refers to sheaves which are flat over the
base $\mathbb{A}^{1}$, not necessarily flat over $\mathscr{Y}$. The diagram of
categories is induced from an analogous diagram of stacks, and the symbols
$\mathscr{Y}|_{0}$ and $(\tau_{n}^{*}\mathscr{Y})|_{0}$ refer to the fibers
over $0\in\mathbb{A}^{1}_{h}$ and $0\in\mathbb{A}^{1}_{h^{1/n}}$,
respectively. Note that
$\mathscr{Y}|_{0}\simeq(\tau_{n}^{*}\mathscr{Y})|_{0}$, but the natural
$\mathbb{G}_{m}$-actions on them differ by $p_{n}$, which is why we denote the
bottom horizontal map by $p_{n}^{*}$.

Applying Proposition 3.4.3 to $\mathscr{Y}=\mathbb{A}^{1}/G^{s}$ yields the
following:

###### Corollary 3.4.4.

We have an equivalence

$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}_{h}/G^{s})\xrightarrow{\tau_{n}^{*}}\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}),$

where the right hand side is the full subcategory of
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s})$
consisting of sheaves whose pullback to the zero fiber $\mathrm{pt}/G^{s(0)}$
lies in the essential image of

$\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathrm{pt}/G^{s(0)})\xrightarrow{p_{n}^{*}}\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathrm{pt}/G^{s(0)}).$

Again, note that the two $\mathbb{G}_{m}$-actions on $\mathrm{pt}/G^{s(0)}$
differ by $p_{n}$.

###  3.5. A $\mathbb{G}_{m}$-action on $G^{s}$

From this point onward, we use the cocharacter $\gamma:\mathbb{G}_{m}\to G$
included in the fastening datum $(\gamma,\ell)$. This cocharacter will allow
us to resolve the issue mentioned at the end of 3.3.

The conditions of Definition 2.3.1 imply that there is a unique _nontrivial_
action $\mathbb{G}_{m}\curvearrowright\mathbb{A}^{1}$ for which the map
$\mathbb{A}^{1}\xrightarrow{\ell}X$ is $\mathbb{G}_{m}$-equivariant, where the
action $\mathbb{G}_{m}\curvearrowright X$ occurs via $\gamma$. Indeed, $\ell$
induces an isomorphism between $\mathbb{A}^{1}$ and the locally closed
subscheme $\ell(\mathbb{A}^{1})\subset X$, and the latter is invariant under
the action of $\gamma(\mathbb{G}_{m})$.

Furthermore, Definition 2.3.1(1) implies that this action
$\mathbb{G}_{m}\curvearrowright\mathbb{A}^{1}$ fixes $0\in\mathbb{A}^{1}$, so
this action is linear. Thus it is given by $t\cdot h=t^{n}h$ for some nonzero
integer $n$. Replacing $\gamma$ with $\gamma^{-1}$ if necessary, we can assume
that $n>0$. We specialize the general constructions of 3.4 to the case of this
particular $n$.

Define a $\mathbb{G}_{m}$-action on $G\times\mathbb{A}^{1}_{h^{1/n}}$ by the
formula

$(g,x)\cdot t=(\gamma(t)g\gamma(t)^{-1},tx)\qquad t\in\mathbb{G}_{m},g\in
G,x\in\mathbb{A}^{1}_{h^{1/n}}.$

The compatibility between $\gamma$ and $\ell$ implies that the group subscheme
$\tau_{n}^{*}G^{s}\subset G\times\mathbb{A}^{1}_{h^{1/n}}$ is invariant under
this action. Hence, we obtain an action

(2) $\tau_{n}^{*}G^{s}\curvearrowleft\mathbb{G}_{m}.$

###### Lemma 3.5.1.

The action
$\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}\curvearrowleft\mathbb{G}_{m}$
induced from (2) agrees with the pullback of the action
$\mathbb{A}^{1}_{h}/G^{s}\curvearrowleft\mathbb{G}_{m}$ along the map
$\tau_{n}$.

_Proof._ Let us start with the action
$\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}\curvearrowleft\mathbb{G}_{m}$
obtained by pullback along $\tau_{n}$. We will define a
$\mathbb{G}_{m}$-equivariant map

$\mathbb{A}^{1}_{h^{1/n}}\xrightarrow{\phi}\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}$

of stacks over $\mathbb{A}^{1}_{h^{1/n}}$ such that the resulting
$\mathbb{G}_{m}$-action on the relative stabilizer scheme obtained by the
fibered product

${\tau_{n}^{*}G^{s}}$${\mathbb{A}^{1}_{h^{1/n}}}$${\mathbb{A}^{1}_{h^{1/n}}}$${\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}}$$\scriptstyle{\phi}$$\scriptstyle{\phi}$

coincides with the action of (2). This proves the lemma.

Recall from 3.3 that the action
$\mathbb{A}^{1}_{h}/G^{s}\curvearrowleft\mathbb{G}_{m}$ was defined using the
isomorphism

$G\backslash\mathscr{I}^{\circ}_{Z/X}\simeq\mathbb{A}^{1}_{h}/G^{s}$

and the action $\mathscr{I}^{\circ}_{Z/X}\curvearrowleft\mathbb{G}_{m}$ which
scales the fibers of the line bundle. We display these actions in the
following diagram:

${G}$${G\times\mathbb{A}^{1}_{h}}$${G}$${\mathscr{I}^{\circ}_{Z/X}}$${\mathbb{G}_{m}}$$\scriptstyle{\curvearrowright}$$\scriptstyle{a}$$\scriptstyle{\curvearrowright}$$\scriptstyle{\curvearrowleft}$

Consider the base change along $\tau_{n}$:

${G}$${G\times\mathbb{A}^{1}_{h^{1/n}}}$${G}$${\mathscr{I}^{\circ}_{Z/X}\underset{\mathbb{A}^{1}_{h}}{\times}\mathbb{A}^{1}_{h^{1/n}}}$${\mathbb{G}_{m}}$$\scriptstyle{\curvearrowright}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\curvearrowright}$$\scriptstyle{\curvearrowleft}$

Here, $\mathbb{G}_{m}$ acts on $\mathscr{I}^{\circ}_{Z/X}$ via the $n$-th
power of the old action, and it acts on $\mathbb{A}^{1}_{h^{1/n}}$ via the
standard action.

We define an action
$(G\times\mathbb{A}^{1}_{h^{1/n}})\curvearrowleft\mathbb{G}_{m}$ via the
following formula:

(3) $(g,c)\cdot t=(g\cdot\gamma^{-1}(t),tc).$

We claim that this action ensures that $a^{\prime}$ is
$\mathbb{G}_{m}$-equivariant. Since the $\mathbb{G}_{m}$-actions take place on
irreducible varieties, it suffices to check this agreement on a dense open
subset of the domain. Therefore, we can restrict to the locus
$G\times(\mathbb{A}^{1}_{h^{1/n}}\smallsetminus\\{0\\})$ and use the
isomorphism
$\mathscr{I}^{\circ}_{Z/X}|_{\mathbb{A}^{1}_{h}\smallsetminus\\{0\\}}\simeq
U\times(\mathbb{A}^{1}_{h}\smallsetminus\\{0\\})$ which was noted in Remark
3.2.1. With this identification in hand, the desired commutativity follows
from a direct calculation:

${(g,c)}$${(g\cdot\gamma^{-1}(t),tc)}$${\big{(}(g\cdot\ell(c^{n}),c^{n}),c\big{)}}$${\big{(}(g\cdot\ell(c^{n}),t^{n}c^{n}),tc\big{)}}$${\big{(}(g\cdot\gamma^{-1}(t)\cdot\ell(t^{n}c^{n}),t^{n}c^{n}),tc\big{)}}$$\scriptstyle{\operatorname{act}_{t}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\operatorname{act}_{t}}$

for $g\in G,\ c\in\mathbb{A}^{1}_{h^{1/n}}\smallsetminus\\{0\\},\
t\in\mathbb{G}_{m}$. Thus, we get a diagram

${G}$${G\times\mathbb{A}^{1}_{h^{1/n}}}$${\mathbb{G}_{m}}$${G}$${\mathscr{I}^{\circ}_{Z/X}\underset{\mathbb{A}^{1}_{h}}{\times}\mathbb{A}^{1}_{h^{1/n}}}$${\mathbb{G}_{m}}$$\scriptstyle{\curvearrowright}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\curvearrowleft}$$\scriptstyle{\curvearrowright}$$\scriptstyle{\curvearrowleft}$

Upon passing to $G$-quotients, the map $a^{\prime}$ becomes the desired map
$\phi$.

###### Remark 0.

Let us summarize the previous construction in more conceptual language. The
map $\phi$ corresponds to the choice of a $\mathbb{G}_{m}$-equivariant
$\tau_{n}^{*}G^{s}$-torsor on $\mathbb{A}^{1}_{h^{1/n}}$. From this point of
view, the original map $\mathbb{A}^{1}_{h}\to\mathbb{A}^{1}_{h}/G^{s}$
corresponds to the (trivial) $G^{s}$-torsor on $\mathbb{A}^{1}_{h}$ obtained
by taking the pullback of $a$ along $s$. Unfortunately, this $G^{s}$-torsor
cannot be made $\mathbb{G}_{m}$-equivariant over $\mathbb{A}^{1}_{h}$.
However, once we pull everything back along $\tau_{n}$, the resulting
(trivial) $\tau_{n}^{*}G^{s}$-torsor does admit a structure of
$\mathbb{G}_{m}$-equivariance, which yields $\phi$.

It remains to check that the action
$\tau_{n}^{*}G^{s}\curvearrowleft\mathbb{G}_{m}$ coincides with the action of
(2). We have a fibered diagram555This diagram is the $\tau_{n}$-pullback of
part of the (augmented) simplicial object in Lemma 3.3.1(iii) which, to use
the terminology in that lemma, encodes the right action of $\mathscr{G}^{s}$
on $\mathscr{G}$.

${G\times\tau_{n}^{*}G^{s}}$${G\times\mathbb{A}^{1}_{h^{1/n}}}$${G\times\mathbb{A}^{1}_{h^{1/n}}}$${\mathscr{I}^{\circ}_{Z/X}\underset{\mathbb{A}^{1}_{h}}{\times}\mathbb{A}^{1}_{h^{1/n}}}$$\scriptstyle{\operatorname{id}_{G}\times\pi^{\prime}}$$\scriptstyle{(\operatorname{mult},\pi^{\prime})}$$\scriptstyle{a^{\prime}}$$\scriptstyle{a^{\prime}}$

Here $\pi^{\prime}$ is the projection to the base $\mathbb{A}^{1}_{h^{1/n}}$
and $\operatorname{mult}$ is the group multiplication. The
$\mathbb{G}_{m}$-actions on the three lower factors collectively induce a
$\mathbb{G}_{m}$-action on the upper-left factor, which one checks is given by
the formula

$\big{(}g,(g^{\prime},c)\big{)}\cdot
t=\big{(}g\,\gamma^{-1}(t),(\gamma(t)\,g^{\prime}\,\gamma^{-1}(t),tc)\big{)}$

for $g\in G,\ c\in\mathbb{A}^{1}_{h^{1/n}},\
g^{\prime}\in\tau_{n}^{*}G^{s}|_{c}\simeq G^{s(c^{n})},\ t\in\mathbb{G}_{m}$.
On the upper-left factor, passing to the $G$-quotient amounts to forgetting
the first coordinate, and the resulting $\mathbb{G}_{m}$-action is exactly
that of (2). ∎

###  3.6. Applying the criterion for descent

Let
$\phi:\mathbb{A}^{1}_{h^{1/n}}\to\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}$
be the $\mathbb{G}_{m}$-equivariant map constructed in the proof of Lemma
3.5.1. Given a quasicoherent sheaf
$\mathscr{F}\in\operatorname{QCoh}^{\mathbb{G}_{m}}(\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s})$,
the pullback $\phi^{*}\mathscr{F}$ is a sheaf on $\mathbb{A}^{1}_{h^{1/n}}$
which is equivariant with respect to $\mathbb{G}_{m}$ and the group scheme
$\tau_{n}^{*}G^{s}$. These two actions are intertwined (in an obvious manner)
with the action $\tau_{n}^{*}G^{s}\curvearrowleft\mathbb{G}_{m}$ defined in
3.5(2).666One could formulate this as an action of the semidirect product
$\tau_{n}^{*}G^{s}\rtimes\mathbb{G}_{m}$ on the sheaf $\phi^{*}\mathscr{F}$ on
$\mathbb{A}^{1}_{h^{1/n}}$.

We can now characterize the subcategory
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s})$
in concrete terms:

###### Corollary 3.6.1.

  1.   2. (i)

Let $G^{s(0)}\rtimes\mathbb{G}_{m}$ be defined using the $\gamma$-conjugation
action of 3.5(2). The image of the group homomorphism
$G^{s(0)}\rtimes\mathbb{G}_{m}\xrightarrow{(\iota,\gamma)}G$ is $G^{\ell(0)}$.

  3. (ii)

The subcategory
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s})$
defined in Corollary 3.4.4 consists of quasicoherent sheaves $\mathscr{F}$
satisfying the following condition:

     * •

The action $G^{s(0)}\rtimes\mathbb{G}_{m}\curvearrowright\mathscr{F}|_{0}$
factors through the map $(\iota,\gamma)$ in (i).

####  3.6.2. Proof of (i)

Consider the $G$-equivariant projection map $\mathscr{N}^{\circ}_{Z/X}\to Z$.
Since $\gamma(\mathbb{G}_{m})$ acts transitively on the fiber over $\ell(0)\in
Z$, and since this fiber contains $s(0)\in\mathscr{N}^{\circ}_{Z/X}$, we have
$G^{\ell(0)}=\gamma(\mathbb{G}_{m})\cdot G^{s(0)}$ as subgroups of $G$. This
proves (i).

For expository purposes, let us record a few additional relations between
these groups. We have a diagram of exact sequences

(4)
${\mu_{n}}$${\mu_{n}}$${G^{s(0)}}$${G^{s(0)}\rtimes\mathbb{G}_{m}}$${\mathbb{G}_{m}}$${G^{s(0)}}$${G^{\ell(0)}}$${\mathbb{G}_{m}}$$\scriptstyle{(\gamma\circ\operatorname{inv},\iota)}$$\scriptstyle{(\iota,\gamma)}$$\scriptstyle{n}$$\scriptstyle{\operatorname{act}}$

Here, the map $\operatorname{act}$ encodes the action of $G^{\ell(0)}$ on the
normal line to $z\in Z\subset X$ (i.e. the fiber mentioned in the previous
paragraph), and the map ‘inv’ is group inversion.

####  3.6.3. Proof of (ii)

To unpack the definition of
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s})$,
we need to describe the map of stacks

$(\mathrm{pt}/G^{s(0)})/\mathbb{G}_{m}^{(n)}\xrightarrow{p_{n}}(\mathrm{pt}/G^{s(0)})/\mathbb{G}_{m}$

which appears in Corollary 3.4.4.777Here, and in what follows, we use the
notation $\mathbb{G}_{m}^{(n)}$ to indicate a copy of $\mathbb{G}_{m}$ which
acts via $n$ times the standard action. This is to avoid collision of notation
for nonisomorphic stacks. As noted there, the map is the identity on the
underlying stack $\mathrm{pt}/G^{s(0)}$, but the two $\mathbb{G}_{m}$-actions
differ by $p_{n}$, the $n$-th power group homomorphism
$\mathbb{G}_{m}\to\mathbb{G}_{m}$. In view of Lemma 3.3.1 and Remark 3.3.2,
this map is given by

$G\ \backslash\ (\mathscr{N}^{\circ}_{Z/X})\ /\ \mathbb{G}_{m}^{(n)}\to G\
\backslash\ (\mathscr{N}^{\circ}_{Z/X})\ /\ \mathbb{G}_{m}.$

The $\mathbb{G}_{m}$-action on the right hand side is the standard one which
scales the fibers of the normal bundle, while the $\mathbb{G}_{m}$-action on
the left hand side is $n$ times the former. Choosing the basepoint
$s(0)\in\mathscr{N}^{\circ}_{Z/X}$, this becomes the map

$\mathrm{pt}/\operatorname{Stab}_{G\times\mathbb{G}_{m}^{(n)}}(s(0))\to\mathrm{pt}/\operatorname{Stab}_{G\times\mathbb{G}_{m}}(s(0))$

which is induced by the map of groups $\psi_{1}$ in this Cartesian diagram:

${\operatorname{Stab}_{G\times\mathbb{G}_{m}^{(n)}}(s(0))}$${\operatorname{Stab}_{G\times\mathbb{G}_{m}}(s(0))}$${G^{\ell(0)}}$${G\times\mathbb{G}_{m}^{(n)}}$${G\times\mathbb{G}_{m}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{id}_{G}\times
p_{n}}$

This diagram contains the additional observation that
$\operatorname{Stab}_{G\times\mathbb{G}_{m}}(s(0))\simeq G^{\ell(0)}$, which
follows because $Z\simeq\mathscr{N}^{\circ}_{Z/X}/\mathbb{G}_{m}$ using the
standard $\mathbb{G}_{m}$-action, and under this quotient map
$s(0)\in\mathscr{N}^{\circ}_{Z/X}$ projects to $\ell(0)\in Z$.

We extend this by another Cartesian square

${G^{s(0)}\rtimes\mathbb{G}_{m}^{(n)}}$${\operatorname{Stab}_{G\times\mathbb{G}_{m}^{(n)}}(s(0))}$${\operatorname{Stab}_{G\times\mathbb{G}_{m}}(s(0))}$${G^{\ell(0)}}$${G\rtimes\mathbb{G}_{m}^{(n)}}$${G\times\mathbb{G}_{m}^{(n)}}$${G\times\mathbb{G}_{m}}$$\scriptstyle{\psi_{2}^{\prime}}$$\scriptstyle{\sim}$$\scriptstyle{\iota\times\operatorname{id}_{\mathbb{G}_{m}}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\sim}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{id}_{G}\times
p_{n}}$

Here, the map $\psi_{2}$ sends $(g,t)\mapsto(g\,\gamma^{-1}(t),t)$, which is a
group isomorphism by definition of the action 3.5(2). The pullback of
$\operatorname{Stab}_{G\times\mathbb{G}_{m}^{(n)}}(s(0))$ under $\psi_{2}$ is
indeed $G^{s(0)}\rtimes\mathbb{G}_{m}^{(n)}$, because the formula for
$\psi_{2}$ shows that the whole subgroup $\mathbb{G}_{m}^{(n)}\subset
G\rtimes\mathbb{G}_{m}^{(n)}$ lies in the stabilizer. And it is
straightforward to check that the upper horizontal composition
$G^{s(0)}\rtimes\mathbb{G}_{m}^{(n)}\twoheadrightarrow G^{\ell(0)}$ coincides
with the map $(\iota,\gamma)$ from (i), by comparing the formula for
$\psi_{2}$ with the formula 3.5(3).

Thus, the map of stacks written at the beginning of this subsubsection
identifies (via the isomorphisms of the previous paragraph) with the map

$\mathrm{pt}/(G^{s(0)}\rtimes\mathbb{G}_{m}^{(n)})\to\mathrm{pt}/G^{\ell(0)}$

which is induced by the group homomorphism $(\iota,\gamma)$ from (i). Tracing
through the construction of Lemma 3.5.1, we find that the action
$G^{s(0)}\rtimes\mathbb{G}_{m}\curvearrowright\mathscr{F}|_{0}$ mentioned in
(ii) coincides with the action obtained by pulling back along

$\mathrm{pt}/(G^{s(0)}\rtimes\mathbb{G}_{m}^{(n)})\simeq(\mathrm{pt}/G^{s(0)})/\mathbb{G}_{m}^{(n)}\hookrightarrow\mathbb{A}^{1}_{h^{1/n}}/\tau_{n}^{*}G^{s}.$

This proves (ii). $\square$

###  3.7. The main result

In this subsection, we assemble the previous constructions to obtain an
algebraic description of the category
$\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$.

###### Definition 3.7.1.

Let $A$ be the filtered Hopf algebra corresponding to the group scheme
$\tau_{n}^{*}G^{s}\to\mathbb{A}^{1}_{h^{1/n}}$ under Proposition 3.1.1 and
Remark 3.1.2. This map is flat by Lemma 3.3.1(i), and 3.5(2) gives a
$\mathbb{G}_{m}$-action on $\tau_{n}^{*}G^{s}$, so the Artin–Rees construction
applies. As is always the case with Artin–Rees, we have

$\displaystyle\operatorname{Spec}A$
$\displaystyle\simeq\big{(}(\tau_{n}^{*}G^{s})|_{\mathbb{A}^{1}_{h^{1/n}}\smallsetminus\\{0\\}}\big{)}/\mathbb{G}_{m}$
$\displaystyle\operatorname{Spec}\operatorname{gr}A$
$\displaystyle\simeq\tau_{n}^{*}G^{s}|_{0}\simeq G^{s(0)}.$

In the second identification, the $\mathbb{G}_{m}$-action on the left hand
side given by the grading corresponds to the usual $\mathbb{G}_{m}$-action on
the right hand side (via $\gamma$).

###### Theorem 3.7.2.

Let $\mathscr{C}_{(n)}(X)$ be the category whose objects are triples
$(V,F,\alpha)$ as follows:

  * •

$(V,F)$ is an exhaustively filtered vector space, and $\alpha:V\to V\otimes A$
is a coaction map which makes $(V,F)$ a filtered comodule for $A$.

  * •

We furthermore require that the associated graded action

$\mathbb{G}_{m}\ltimes G^{s(0)}\curvearrowright\operatorname{gr}^{F}V$

factors through the quotient $\mathbb{G}_{m}\ltimes G^{s(0)}\twoheadrightarrow
G^{\ell(0)}$ from Corollary 3.6.1(i).

The morphisms in this category are those of filtered $A$-comodules. Then there
are equivalences of categories

$\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)\simeq\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/\tau_{n}^{*}G^{s})\simeq\mathcal{C}_{(n)}(X).$

_Proof._ The first equivalence follows from Corollary 3.4.4. The second
equivalence follows from the definition of $A$ and the last sentence of Remark
3.1.2. The reformulation of the $(-)_{(n)}$ condition given in the theorem
statement follows from Corollary 3.6.1(ii). ∎

###### Remark 3.7.3.

If one desires to remove the torsion-free requirement, then in Proposition
3.1.1 the notion of ‘filtered vector space’ must be replaced with that of
‘graded vector space equipped with a degree 1 endomorphism’ and the criterion
for descent given in Lemma 3.4.1 must be reformulated accordingly.

###### Remark 3.7.4.

Fixing the nonzero point $1\in\mathbb{A}^{1}_{h^{1/n}}$ allows one to view
these constructions more concretely (albeit less canonically):

  1. (i)

We get identifications

$\displaystyle\operatorname{Spec}A$ $\displaystyle\simeq G^{s(1)}$
$\displaystyle\operatorname{Spec}\operatorname{gr}A$ $\displaystyle\simeq
G^{s(0)},$

where the identifications in the second line are $\mathbb{G}_{m}$-equivariant.
The filtration $F^{\geq m}$ induced on the ring of functions
$\mathscr{O}_{G^{s(1)}}$ is given as follows: for a function $f$, we have
$f\in F^{\geq m}$ if and only if the limit

$\lim_{t\to 0}\big{(}t^{-m}\operatorname{act}_{t^{-1}}^{*}f\big{)}$

exists as a function on $G^{s(0)}$. Here $\operatorname{act}_{t^{-1}}$ is the
isomorphism $G^{s(t^{n})}\to G^{s(1)}$ given by the $\mathbb{G}_{m}$-action on
$G^{s}$.

  2. (ii)

Given an object $\mathscr{F}\in\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$
corresponding to $(V,F,\alpha)$ via Theorem 3.7.2, the fiber
$\mathscr{F}|_{\ell(1)}$ identifies with $V$, and the action of $G^{\ell(1)}$
on this fiber is given by the coaction $\alpha$ (forgetting the filtrations).
The fiber $\mathscr{F}|_{\ell(0)}$ is given by $\operatorname{gr}V$.

  3. (iii)

The action of $G^{\ell(0)}$ on the fiber $\mathscr{F}|_{\ell(0)}$ is
determined as follows. The action of $G^{s(0)}$ on $\mathscr{F}|_{\ell(0)}$ is
given by $\operatorname{gr}\alpha$. Since this coaction map is graded, the
action of $G^{s(0)}$ extends to an action by $G^{s(0)}\rtimes\mathbb{G}_{m}$.
The second bullet point in Theorem 3.7.2 says that the action descends to one
by the quotient group $G^{\ell(0)}$.

##  4\. Specialization maps

Retain the notations and assumptions of Section 3. This means that $X=U\cup Z$
is a smooth fastened chain equipped with fastening datum $(\gamma,\ell)$,
which determines the group schemes $G^{s}$ and $\tau_{n}^{*}G^{s}$, the latter
being $\mathbb{G}_{m}$-equivariant over $\mathbb{A}^{1}$. Theorem 3.7.2 gives
us an equivalence between $\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$ and a
category of filtered co-modules for the filtered Hopf algebra
$A=\mathscr{O}_{G^{s(1)}}$. In this section, we show that some objects in
$\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$ can be presented more economically,
without referring explicitly to $A$.

In the first half of this section, we give nicer classifications of line
bundles (Proposition 4.1.1) and sheaves whose restrictions to $U$ are
$G$-equivariant local systems (Corollary 4.3.1). These classifications are
‘nicer’ in the sense that the filtered Hopf algebra $A$ does not explicitly
appear; instead, in each case $\tau_{n}^{*}G^{s}$ acts through a certain
quotient

$q:\tau_{n}^{*}G^{s}\to H$

(see 4.2), where $H\to\mathbb{A}^{1}$ is a $\mathbb{G}_{m}$-equivariant group
scheme with the property that the action $\mathbb{G}_{m}\curvearrowright
H|_{0}$ is trivial. This implies that the specialization to the zero fiber for
$H$ is encoded in a _bona fide_ map of algebraic groups $H|_{0}\to H|_{1}$.

In the second half of this section, we investigate the largest subcategory of
$\mathrm{Vect}^{G}(X)$ for which this works. We show that there is a largest
possible quotient of $\tau_{n}^{*}G^{s}$ which admits a ‘specialization map’
description (Corollary 4.4.6) and we state the corresponding classification of
vector bundles in Corollary 4.5.1. Lastly, in 4.6, we apply this idea to the
problem of classifying sheaves whose restrictions to $U$ are _twisted_
$G$-equivariant local systems, which is the case of interest for our
application (Section 6).

_Notations._ For the remainder of this section, we adopt the more concise
notation $\tilde{G}^{s}=\tau_{n}^{*}G^{s}$. Recall from Remark 3.3.2 that
$G^{s(0)}\simeq G^{s}|_{0}$ and $G^{s(1)}\simeq G^{s}|_{1}$. As a matter of
convention, any scheme written as $S\times\mathbb{A}^{1}$ (for some $S$) will
be equipped with the $\mathbb{G}_{m}$-action given by the standard action on
the second factor. For an arbitrary map $H\to\mathbb{A}^{1}$, we let $H_{0}$
and $H_{1}$ denote the fibers over $0,1\in\mathbb{A}^{1}$, respectively. For a
map $q$ of schemes over $\mathbb{A}^{1}$, we let $q_{0}$ and $q_{1}$ denote
the corresponding fibers of this map.

###  4.1. Line bundles

In this subsection, we give a very simple description of the category of
$G$-equivariant line bundles (Proposition 4.1.1(ii)), which will be applied in
5.3. Along the way, we prove a technical result which may be of independent
interest: the specialization of a character of $G^{s(1)}$ to the zero fiber
$G^{s(0)}$ is always well-defined (Proposition 4.1.1(i)). This result is
applied again in 4.6.

Let $\mathrm{Pic}^{G}(X)$ denote the category of $G$-equivariant line bundles
on $X$. For any character $\chi:G^{s(1)}\to\mathbb{G}_{m}$, let
$\mathrm{Pic}^{G}_{\chi}(X)$ be the full subcategory consisting of line
bundles $\mathscr{L}$ for which the action
$G^{s(1)}\curvearrowright\mathscr{L}|_{s(1)}$ factors through $\chi$. We have
a decomposition

$\mathrm{Pic}^{G}(X)\simeq\bigsqcup_{\chi}\mathrm{Pic}^{G}_{\chi}(X).$

###### Proposition 4.1.1.

Fix a character $\chi:G^{s(1)}\to\mathbb{G}_{m}$.

  1. (i)

We have $\chi\in F^{\geq 0}A$. Concretely, this means that $\lim_{t\to
0}\operatorname{act}_{t^{-1}}^{*}\chi$ exists as a character of $G^{s(0)}$.

Let $q:\tilde{G}^{s}\to\mathbb{G}_{m}\times\mathbb{A}^{1}$ be the
$\mathbb{G}_{m}$-equivariant map of group schemes given by (i). The map $q$ is
uniquely determined by the requirement that $q_{1}:G^{s(1)}\to\mathbb{G}_{m}$
equals $\chi$.

  1. (ii)

We have a canonical equivalence of groupoids

$\mathrm{Pic}^{G}_{\chi}(X)\simeq(n\mathbb{Z}+m)\times(\mathrm{pt}/\mathbb{C}^{\times}).$

The residue class $m\bmod{n}$ is determined by the following property: if
$\mu_{n}\simeq G^{s(0)}\cap\gamma(\mathbb{G}_{m})$ (see 3.6.2), then the
restriction of the character

${\mu_{n}}$${G^{s(0)}}$${\mathbb{G}_{m}}$$\scriptstyle{\operatorname{inclusion}}$$\scriptstyle{\lim_{t\to
0}\mathrm{Ad}_{\gamma(t^{-1})}^{*}\chi}$

is given by $\zeta\mapsto\zeta^{m}$.

####  4.1.2. Proof that (i) implies (ii) in Proposition 4.1.1

The first equivalence in Theorem 3.7.2 implies that an object
$\mathscr{L}\in\mathrm{Pic}^{G}(X)$ is equivalent to a pair
$(\varphi,\mathscr{L}^{\prime})$ defined as follows:

  * •

$q:\tilde{G}^{s}\to\mathbb{G}_{m}\times\mathbb{A}^{1}$ is a
$\mathbb{G}_{m}$-equivariant homomorphism of group schemes.

  * •

$\mathscr{L}^{\prime}$ is a $\mathbb{G}_{m}$-equivariant line bundle on
$\mathbb{A}^{1}$.

  * •

We require that the restriction of
$\mathbb{G}_{m}\curvearrowright\mathscr{L}^{\prime}|_{0}$ to $\mu_{n}$ is
given by the group homomorphism
$\mu_{n}\xrightarrow{\gamma}G^{s(0)}\xrightarrow{q_{0}}\mathbb{G}_{m}$.

This is because, for any $\mathbb{G}_{m}$-equivariant line bundle
$\mathscr{L}^{\prime}$ on $\mathbb{A}^{1}$, the group scheme
$\mathcal{A}\kern-0.5ptut_{\mathbb{A}^{1}}(\mathscr{L}^{\prime})$ is
$\mathbb{G}_{m}$-equivariantly isomorphic to
$\mathbb{G}_{m}\times\mathbb{A}^{1}$.

Furthermore, $\mathscr{L}\in\mathrm{Pic}^{G}_{\chi}(X)$ if and only if
$q_{1}=\chi$. Thus $q$ is determined, and the remaining datum
$\mathscr{L}^{\prime}$ amounts to the choice of a one-dimensional vector space
(which plays the role of $\mathscr{L}^{\prime}|_{1}$) and an integer (which
determines the filtration). The third bullet point above says that this
integer lies in $n\mathbb{Z}+m$, because $q_{1}=\chi$ implies that
$q_{0}=\lim_{t\to 0}\operatorname{act}_{t^{-1}}^{*}\chi$. ∎

The rest of this subsection is devoted to the proof of Proposition 4.1.1(i).

###### Lemma 4.1.3.

Let $\pi:H\to\mathbb{A}^{1}$ be a flat $\mathbb{G}_{m}$-equivariant group
scheme of relative dimension 1. If there is an isomorphism
$\varphi:H_{1}\simeq\mathbb{G}_{m}$, then the following are true:

  1. (i)

The $\mathbb{G}_{m}$-action on $H_{0}$ is (weakly) expanding as $t\to
0$.888This means that $\mathscr{O}_{H_{0}}$ lives in nonnegative degrees with
respect to the $\mathbb{G}_{m}$-action. We omit the word ‘weakly’ from now on,
because we will not use the ‘strong’ notion anywhere.

  2. (ii)

There exists a (unique) $\mathbb{G}_{m}$-equivariant map
$q:H\to\mathbb{G}_{m}\times\mathbb{A}^{1}$ of group schemes with the property
that $q_{1}:H_{1}\to\mathbb{G}_{m}$ equals $\varphi$.

_Proof._ (i). Suppose for sake of contradiction that the action
$\mathbb{G}_{m}\curvearrowright H_{0}$ is not expanding. This implies that all
weights of the $\mathbb{G}_{m}$-action on the (two-dimensional) tangent space
$\mathscr{T}_{1_{H_{0}}}H$ are positive (here $1_{H_{0}}\in H_{0}$ is the
identity element). (It also implies that the identity component of $H_{0}$ is
isomorphic to $\mathbb{G}_{a}$, but we will not use this explicitly.)

Applying the Luna Slice Theorem to the $\mathbb{G}_{m}$-orbit
$\\{1_{H_{0}}\\}\subset H$ gives us a $\mathbb{G}_{m}$-invariant affine open
neighborhood $W\subset H$ containing $1_{H_{0}}$, along with a
$\mathbb{G}_{m}$-equivariant étale map

$\psi:W\to\mathscr{T}_{1_{H_{0}}}H$

which induces an isomorphism on tangent spaces at $1_{H_{0}}\in W$. (The last
phrase uses that $\pi$ is smooth, which is true because $\pi$ is flat,
finitely presented, and has smooth fibers.) The $\mathbb{G}_{m}$-equivariance
of $\psi$, along with the previous paragraph, implies that $\psi$ is
surjective. This in turn implies that $H_{1}\cap W\simeq\mathbb{A}^{1}$,
contradicting the assumption that $H_{1}\simeq\mathbb{G}_{m}$.

(ii). Point (i) says that
$\mathscr{O}_{H_{0}}\simeq\operatorname{gr}\mathscr{O}_{H_{1}}$ lives in
nonnegative degrees. In view of Remark 3.7.4(i), this implies that any regular
function on $H_{1}$ extends to a $\mathbb{G}_{m}$-invariant function on $H$.
Applying this to the character $\varphi$ yields the desired result. ∎

One can describe all possibilities for $\pi:H\to\mathbb{A}^{1}$ satisfying the
hypotheses of this lemma. See the second paragraph of Remark 4.6.4 for more
details.

####  4.1.4. Proof of Proposition 4.1.1(i)

Consider the codimension-one subgroup $\ker(\chi)\subset G^{s(1)}$, and extend
it to a $\mathbb{G}_{m}$-equivariant sub group scheme $K\subset\tilde{G}^{s}$
which is flat over $\mathbb{A}^{1}$. (Concretely, $K$ is obtained by taking
the flat limit of the $\mathbb{G}_{m}$-saturation of $\ker(\chi)$. It
satisfies the sub group scheme property because this is a closed condition,
hence this condition is preserved under taking the flat limit.) Let
$\tilde{G}^{s}\twoheadrightarrow H$ be the quotient by $K$, so that $H$ is a
flat $\mathbb{G}_{m}$-equivariant group scheme of relative dimension $\leq 1$.
By construction, there is a group map
$H_{1}\xhookrightarrow{\operatorname{cl.emb.}}\mathbb{G}_{m}$ which expresses
$H_{1}$ as the image of $\chi$.

Assume that $H$ has relative dimension $1$, so that
$H_{1}\hookrightarrow\mathbb{G}_{m}$ is an isomorphism. Then Lemma 4.1.3(ii)
gives a map $q:H\to\mathbb{G}_{m}\times\mathbb{A}^{1}$ which witnesses the
fact that the limit $\lim_{t\to 0}\operatorname{act}_{t^{-1}}^{*}\chi$ exists.

If $H$ has relative dimension $0$, then $\chi$ is locally constant. Now the
desired limit exists because two different connected components of $G^{s(1)}$
cannot specialize to the same component of $G^{s(0)}$. Indeed, this would
contradict the fact that $G^{s(0)}$ is a group scheme and hence reduced. (This
case can also be handled using 4.3.) ∎

###  4.2. Factoring through a quotient of $\tilde{G}^{s}$

Forget the notation of 4.1. Now let $H$ be any $\mathbb{G}_{m}$-equivariant
group scheme over $\mathbb{A}^{1}$, and let $q:\tilde{G}^{s}\to H$ be a map of
$\mathbb{G}_{m}$-equivariant group schemes over $\mathbb{A}^{1}$.

###### Definition 4.2.1.

Let $\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/H)$
be the full subcategory of
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}/H)$
consisting of sheaves $\mathscr{F}$ for which the restriction of the action
$\mathbb{G}_{m}\ltimes H_{0}\curvearrowright\mathscr{F}|_{0}$ along the group
map

$q_{0}:\mathbb{G}_{m}\ltimes G^{s(0)}\to\mathbb{G}_{m}\ltimes H_{0}$

factors through the quotient $\mathbb{G}_{m}\ltimes G^{s(0)}\to G^{\ell(0)}$.

We would like to think of
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/H)$ as
encoding a subcategory of $\operatorname{QCoh}_{\mathrm{tf}}^{G}(X)$ which
‘corresponds’ to the quotient $q$. The next result makes this precise. In 4.5,
we explain how to classify the objects of this subcategory.

###### Proposition 4.2.2.

We have the following:

  1. (i)

If $q_{1}:G^{s(1)}\to H_{1}$ is surjective and $H$ is flat (i.e. torsion-free)
over $\mathbb{A}^{1}$, then the pullback functor

$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/H)\xrightarrow{q^{*}}\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/\tilde{G}^{s})$

is fully faithful.

  2. (ii)

If furthermore $q_{0}:G^{s(0)}\to H_{0}$ is surjective, then the image of this
functor is the subcategory of sheaves $\mathscr{F}$ for which the action
$G^{s(1)}\curvearrowright\mathscr{F}|_{1}$ factors through $q_{1}$.

_Proof._ (i). Since $H$ is flat, an object in the left hand side is given by
$(V,F,\beta)$ where $(V,F)$ is a filtered vector space and $\beta:V\to
V\otimes\mathscr{O}_{H_{1}}$ is a filtered coaction. Since $q_{1}$ is
surjective, $\mathscr{O}_{H_{1}}\to\mathscr{O}_{G^{s(1)}}$ is injective, so
whether or not a filtered coaction $V\to V\otimes\mathscr{O}_{G^{s(1)}}$
factors through a coaction $V\to V\otimes\mathscr{O}_{H_{1}}$ is a property,
not additional structure.

(ii). The crux of the proof is the following lemma, which the second author
learned from Dori Bejleri:

###### Lemma 0.

Let $f:X\to Y$ be a map of schemes over a base $S$. Let $x\in X$ be a point,
and let $y\in Y$ and $s\in S$ be its images. Then the following two statements
are equivalent:

  * •

$X\to S$ is flat at $x$, and $f|_{s}:X|_{s}\to Y|_{s}$ is flat at $x$.

  * •

$Y\to S$ is flat at $y$, and $f$ is flat at $x$.

Since $q_{0}$ is a surjective map of algebraic groups, it is flat. Since
$\tilde{G}^{s}$ is flat over $\mathbb{A}^{1}$, we may apply the lemma and
conclude that $q$ is flat. Since flat maps are preserved by base change, we
conclude that the scheme $\ker(q)$ is flat over $\mathbb{A}^{1}$.

The image of the functor is obviously contained in the indicated subcategory.
For the reverse containment, suppose that the sheaf
$\mathscr{F}\in\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/\tilde{G}^{s})$
lies in the indicated subcategory. The action of $\tilde{G}^{s}$ corresponds
to a map
$\tilde{G}^{s}\xrightarrow{\varphi}\mathcal{E}\kern-0.5ptnd_{\mathbb{A}^{1}}(\mathscr{F})$.
The closed subscheme $\ker(\varphi)|_{\mathbb{A}^{1}\smallsetminus\\{0\\}}$
contains $\ker(q)|_{\mathbb{A}^{1}\smallsetminus\\{0\\}}$ by the hypothesis on
$\mathscr{F}$. The flatness proved in the previous paragraph implies that
$\ker(\varphi)$ contains $\ker(q)$. Therefore the action of $\tilde{G}^{s}$
factors through $q$, as desired. ∎

###### Definition 4.2.3.

As a notational convenience, when the hypotheses of Proposition 4.2.2(i) are
satisfied, we define $\operatorname{QCoh}^{G}_{\mathrm{tf},q}(X)$ via the
following Cartesian diagram:

${\operatorname{QCoh}^{G}_{\mathrm{tf},q}(X)}$${\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)}$${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/H)}$${\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/\tilde{G}^{s})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$Thm.
3.7.2$\scriptstyle{q^{*}}$

Note that this subcategory of $\operatorname{QCoh}^{G}_{\mathrm{tf},(n)}(X)$
_a priori_ depends on the choice of fastening $(\gamma,\ell)$ as well as the
map $q$. But when Proposition 4.2.2(ii) applies, it tells us that this
subcategory consists exactly of those sheaves
$\mathscr{F}\in\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$ for which the action
$G^{s(1)}\curvearrowright\mathscr{F}|_{\ell(1)}$ factors through $q_{1}$.

This definition will appear only in Corollary 4.5.1.

###  4.3. Local systems

We apply 4.2 to the problem of describing the subcategory

$\operatorname{QCoh}_{\mathrm{tf},\operatorname{LocSys}}^{G}(X)\hookrightarrow\operatorname{QCoh}_{\mathrm{tf}}^{G}(X)$

consisting of sheaves $\mathscr{F}$ for which the restriction
$\mathscr{F}_{U}$ is a $G$-equivariant local system.

We construct a quotient map $q:\tilde{G}^{s}\to H_{\operatorname{LocSys}}$ of
group schemes over $\mathbb{A}^{1}/\mathbb{G}_{m}$ as follows. For each
connected (equivalently, irreducible) component $C\subset\tilde{G}^{s}$, let
$H_{C}\to\mathbb{A}^{1}$ be the image of the projection map
$C\to\mathbb{A}^{1}$. Since $\tilde{G}^{s}$ is flat over $\mathbb{A}^{1}$,
each $H_{C}$ is either $\mathbb{A}^{1}$ or
$\mathbb{A}^{1}\smallsetminus\\{0\\}$. We define

$H_{\operatorname{LocSys}}:=\bigsqcup_{\begin{subarray}{c}C\subset\tilde{G}^{s}\\\
\text{components}\end{subarray}}H_{C}.$

Then $H_{\operatorname{LocSys}}$ is obviously a group scheme over
$\mathbb{A}^{1}/\mathbb{G}_{m}$, and the desired map $q$ is defined on
components by the tautological map $C\to H_{C}$. Note that
$H_{\operatorname{LocSys}}|_{1}\simeq\operatorname{Com}(G^{s(1)})$ where
$\operatorname{Com}(-)$ means the group of components.

Since each component of $G^{s(0)}$ lies in a unique component of
$\tilde{G}^{s}$, and the latter components are in bijection with those of
$G^{s(1)}$, we obtain a canonical map of groups

$\sigma:\operatorname{Com}(G^{s(0)})\to\operatorname{Com}(G^{s(1)}).$

###### Corollary 4.3.1.

Let $\mathscr{C}_{\operatorname{LocSys},(n)}(X)$ be the category whose objects
are triples $(V,F,\rho)$ as follows:

  * •

$V$ is a vector space, $\rho:\operatorname{Com}(G^{s(1)})\curvearrowright V$
is a representation, and $F$ is an exhausting filtration on $V$ which is
invariant under $\rho$.

  * •

We furthermore require that the restriction of the associated graded action

$\mathbb{G}_{m}\ltimes
G^{s(0)}\to\mathbb{G}_{m}\times\operatorname{Com}(G^{s(0)})\xrightarrow{\sigma}\mathbb{G}_{m}\times\operatorname{Com}(G^{s(1)})\curvearrowright\operatorname{gr}^{F}V$

factors through the quotient $\mathbb{G}_{m}\ltimes G^{s(0)}\twoheadrightarrow
G^{\ell(0)}$.

The morphisms in this category are maps of filtered representations. Then
there is an equivalence of categories

$\operatorname{QCoh}^{G}_{\mathrm{tf},\operatorname{LocSys}}(X)\simeq\mathscr{C}_{\operatorname{LocSys},(n)}(X).$

_Proof._ Since Proposition 4.2.2(ii) applies to $q:\tilde{G}^{s}\to
H_{\operatorname{LocSys}}$, we conclude that the category
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/H_{\operatorname{LocSys}})$
is equivalent to the subcategory of
$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/\tilde{G}^{s})$
consisting of sheaves $\mathscr{F}$ for which the action
$G^{s(1)}\curvearrowright\mathscr{F}|_{1}$ factors through
$G^{s(1)}\twoheadrightarrow\operatorname{Com}(G^{s(1)})$. But this subcategory
is also equivalent to
$\operatorname{QCoh}^{G}_{\mathrm{tf},\operatorname{LocSys}}(X)$ by Theorem
3.7.2. Thus, to finish the proof, it suffices to show that

$\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},(n)}(\mathbb{A}^{1}/H_{\operatorname{LocSys}})\simeq\mathscr{C}_{\operatorname{LocSys},(n)}(X).$

Using the Artin–Rees construction, we interpret the left hand side as
consisting of triples $(V,F,\beta)$ where $(V,F)$ is as before, and
$\beta:V\to V\otimes\mathscr{O}_{H_{\operatorname{LocSys}}|_{1}}$ is a
filtered coaction. We observed above that
$H_{\operatorname{LocSys}}|_{1}\simeq\operatorname{Com}(G^{s(1)})$. If
$I\subset\mathscr{O}_{\operatorname{Com}(G^{s(1)})}$ is the ideal of the
subscheme $\operatorname{Im}(\sigma)$, then the filtration on
$\mathscr{O}_{H_{\operatorname{LocSys}}|_{1}}$ (given by the definition of
$H_{\operatorname{LocSys}}$) corresponds to the filtration

(5) $F^{\geq
m}\mathscr{O}_{\operatorname{Com}(G^{s(1)})}=\begin{cases}\mathscr{O}_{\operatorname{Com}(G^{s(1)})}&\text{
if }m\leq 0\\\ I&\text{ if }m>0.\end{cases}$

This filtration implies that the filtered coaction $\beta$ is equivalent to
the data in the first bullet point in the definition of
$\mathscr{C}_{\operatorname{LocSys},(n)}(X)$. It is easy to check that the
second bullet point corresponds to the $(-)_{(n)}$ condition on the left hand
side. ∎

###  4.4. The universal tame quotient

The analysis in 4.1 and 4.3 is especially nice because in both cases we
considered a quotient $q:\tilde{G}^{s}\to H$ (in the framework of 4.2) with
the property that $H\to\mathbb{A}^{1}$ is ‘almost’ a constant group scheme.
Indeed, in 4.1, $H$ was the constant group scheme
$H\simeq\mathbb{G}_{m}\times\mathbb{A}^{1}$. And in 4.3,
$H_{\operatorname{LocSys}}$ is obtained from the constant group scheme
$\operatorname{Com}(G^{s(1)})\times\mathbb{A}^{1}$ by replacing the zero fiber
by its closed subscheme $\operatorname{Im}(\sigma)$.

One thing these two group schemes have in common is that the
$\mathbb{G}_{m}$-action on their zero fibers is trivial. In what follows, we
explain why this property implies that a $\mathbb{G}_{m}$-equivariant scheme
over $\mathbb{A}^{1}$ is ‘almost constant’ in a sense that will be made
precise (see Lemma 4.4.2).

###### Definition 4.4.1.

Define the full subcategory

$\operatorname{QCoh}_{\mathrm{tf},{\operatorname{tame}}}^{\mathbb{G}_{m}}(\mathbb{A}^{1})\subset\operatorname{QCoh}_{\mathrm{tf}}^{\mathbb{G}_{m}}(\mathbb{A}^{1})$

to consist of sheaves
$\mathscr{F}\in\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1})$
for which the action $\mathbb{G}_{m}\curvearrowright\mathscr{F}|_{0}$ is
trivial. We call such a sheaf _tame_. Under Artin–Rees, tame sheaves
correspond to filtered vector spaces $(V,F)$ with the property that $F^{\geq
0}V=V$ and $F^{\geq 1}V=F^{\infty}V$.999By definition,
$F^{\infty}V=\bigcap_{m\in\mathbb{Z}}F^{\geq m}V$ and
$F^{-\infty}V=\bigcup_{m\in\mathbb{Z}}F^{\geq m}V$. The filtration $F$ is
exhausting if and only if $F^{-\infty}V=V$. Such vector spaces we call _tamely
filtered_.

Since this is a monoidal subcategory, we obtain an analogous notion of tame
affine schemes over $\mathbb{A}^{1}/\mathbb{G}_{m}$ as in Remark 3.1.2. In the
rest of the section, we study tame affine schemes in more detail.

Any (flat) affine scheme over $\mathbb{A}^{1}/\mathbb{G}_{m}$ is given by

$\underline{\operatorname{Spec}}(R,F):=\operatorname{Spec}\oplus_{m}F^{\geq
m}R\xrightarrow{\pi}\operatorname{Spec}k[h]=\mathbb{A}^{1}$

for some filtered ring $(R,F)$. Here $\oplus_{m}F^{\geq m}R$ and $k[h]$ are
graded rings with $\oplus_{m}F^{\geq m}R$ in degree $m$ and $h$ in degree
$-1$. (This convention ensures that the action
$\mathbb{G}_{m}\curvearrowright\mathbb{A}^{1}_{h}$ is standard.) Then $\pi$ is
defined by $h\mapsto 1_{R}\in F^{\geq-1}R$.

Now $S=\underline{\operatorname{Spec}}(R,F)$ is _tame_ if $(R,F)$ is tamely
filtered, or equivalently if the action $\mathbb{G}_{m}\curvearrowright
S|_{0}$ is trivial. Note that any tame affine scheme is by definition torsion-
free (equivalently flat) over $\mathbb{A}^{1}$.

Next, we observe that tame affine schemes are ‘almost constant’ as described
in the first paragraph of this subsection. For a vector space $V$, let $F_{0}$
be the filtration for which $F_{0}^{\geq 0}=V$ and $F_{0}^{\geq 1}V=0$.

###### Lemma 4.4.2.

We have the following.

  1. (i)

The map of filtered rings $(F^{\geq 0}R,F_{0})\to(R,F)$ yields a
$\mathbb{G}_{m}$-equivariant map

$\underline{\operatorname{Spec}}(R,F)\xrightarrow{p}(\operatorname{Spec}R)\times\mathbb{A}^{1}.$

  2. (ii)

If $(R,F)$ is tamely filtered, then the base change of $p$ along
$\mathbb{A}^{1}\smallsetminus\\{0\\}\hookrightarrow\mathbb{A}^{1}$ is an
isomorphism and the base change along $\\{0\\}\hookrightarrow\mathbb{A}^{1}$
is the closed embedding $\operatorname{Spec}(R/F^{\geq
1})R\hookrightarrow\operatorname{Spec}R$. In particular, there is a canonical
map from the zero fiber of $\underline{\operatorname{Spec}}(R,F)$ to the
general fiber.

_Proof._ Point (i) is a tautology. For point (ii), if $(R,F)$ is tamely
filtered, then $R=F^{\geq 0}R$, from which it follows that $p$ is an
isomorphism over $\mathbb{A}^{1}\smallsetminus\\{0\\}$. The base change of $p$
along $\\{0\\}\hookrightarrow\mathbb{A}^{1}$ is the induced map on associated
graded rings, which is $R\twoheadrightarrow R/F^{\geq 1}R$ if $(R,F)$ is
tamely filtered. ∎

Tame affine schemes over $\mathbb{A}^{1}$ are usually not finite type. For
example, let $(R,F)$ be the tamely filtered ring such that $R=k[x]$ and
$F^{\geq 1}R=\langle x\rangle$. Then the Artin–Rees ring is $\oplus_{m}F^{\geq
m}R\simeq k[h,h^{-1}x,h^{-2}x,\cdots]$, which is not finitely generated.
Geometrically, this ring can be constructed from the affine plane
$\operatorname{Spec}k[h,x]$ by taking an infinite sequence of affine charts of
blow-ups at the origin, the first of which is $k[h,x]\to k[h,h^{-1}x]$. The
following lemma tells us what finite type tame affine schemes look like:

###### Lemma 4.4.3.

Let $(R,F)$ be a tamely filtered $k$-algebra.

  1. (i)

$\oplus_{m}F^{\geq m}R$ is finitely generated (as a $k$-algebra) if and only
if $R$ is finitely generated and the ideal $F^{\geq 1}R$ is idempotent.

  2. (ii)

If the situation of (i) obtains, then the map $p$ from Lemma 4.4.2 looks as
follows. For each connected component $C\subset\operatorname{Spec}R$, the map
$p$ is either the identity map of $C\times\mathbb{A}^{1}$ or the open
embedding $C\times(\mathbb{A}^{1}\smallsetminus\\{0\\})\hookrightarrow
C\times\mathbb{A}^{1}$. The former occurs if and only if the ideal $F^{\geq
1}R$ vanishes on $C$.

_Proof._ (i). If $\oplus_{m}F^{\geq m}R$ is finitely generated, choose a
finite set of homogeneous generators, and partition them into two groups
$\\{r_{i}\\}\cup\\{s_{j}\\}$ where the $r_{i}$ live in nonpositive degrees and
the $s_{j}$ live in positive degrees. Let $n$ be the largest degree which
occurs among the generators. For degree reasons, every polynomial in the
generators which lies in the summand $F^{\geq n+1}R$ must be a sum of
monomials each of which contains at least two $s_{j}$’s. The $s_{j}$’s lie in
$F^{\geq 1}R$ when viewed as elements of $R$ (i.e. forgetting the grading).
Therefore

$F^{\geq 1}R=F^{\geq n+1}R\subset(F^{\geq 1}R)^{2}$

as ideals of $R$. (The equality uses the tamely filtered hypothesis.) Hence
$F^{\geq 1}R$ is idempotent. Also, the fiber of
$\operatorname{Spec}\oplus_{m}F^{\geq
m}R\xrightarrow{\pi}\operatorname{Spec}k[h]$ at $h=1$ is
$\operatorname{Spec}R$, so $R$ is also finitely generated.

Conversely, assume that $F^{\geq 1}R$ is idempotent and $R$ is finitely
generated. Since $R$ is Noetherian, $F^{\geq 1}R$ is a finitely generated
ideal. A set of generators for $\oplus_{m}F^{\geq m}R$ is given by taking a
set of generators for $R$ placed in degree 0, a set of ideal generators for
$F^{\geq 1}R$ placed in degree $1$, and the element $1_{R}$ placed in degree
$-1$.

Point (ii) is deduced from Lemma 4.4.2 and the fact that an idempotent ideal
of a Noetherian ring vanishes on a union of connected components. ∎

Next, we observe that every filtered ring has a universal tame quotient.

###### Lemma 4.4.4.

There are adjoint functors

$\iota:\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf},{\operatorname{tame}}}(\mathbb{A}^{1})\rightleftarrows\operatorname{QCoh}^{\mathbb{G}_{m}}_{\mathrm{tf}}(\mathbb{A}^{1}):\tau_{{\operatorname{tame}}}$

where $\iota$ is the inclusion and $\tau_{{\operatorname{tame}}}$ sends
$(V,F)$ to $(F^{\geq 0}V,F_{1})$ where $F_{1}$ is the tame filtration defined
by $F_{1}^{\geq m}=F^{\infty}V$ for all $m>0$. Both of these functors are
monoidal.

_Proof._ Clear. ∎

Let
$\operatorname{Sch}^{\operatorname{aff},\mathrm{tf}}_{\mathbb{A}^{1}/\mathbb{G}_{m}}$
be the category of $\mathbb{G}_{m}$-equivariant affine schemes flat over
$\mathbb{A}^{1}$, and let
$\operatorname{Sch}_{\mathbb{A}^{1}/\mathbb{G}_{m}}^{\operatorname{aff},{\operatorname{tame}}}$
be the subcategory of tame affine schemes.

###### Corollary 4.4.5.

There are adjoint functors

$\tau^{{\operatorname{tame}}}:\operatorname{Sch}_{\mathbb{A}^{1}/\mathbb{G}_{m}}^{\operatorname{aff}}\rightleftarrows\operatorname{Sch}_{\mathbb{A}^{1}/\mathbb{G}_{m}}^{\operatorname{aff},{\operatorname{tame}}}:\iota$

where $\iota$ is the inclusion, and both functors are monoidal with respect to
the Cartesian monoidal structures. The functor $\tau^{{\operatorname{tame}}}$
preserves the class of affine schemes whose every connected component is
integral and finite type.

_Proof._ The last sentence is not obvious. It suffices to show that, if
$\oplus_{m}F^{\geq m}R$ is finitely generated and integral, then the following
statements hold:

  1. (i)

$F^{\geq 0}R$ is a finitely generated $k$-algebra.

  2. (ii)

The ideal $F^{\infty}R$ is either $\langle 0\rangle$ or $F^{\geq 0}R$.101010If
$F^{\infty}R=F^{\geq 0}R$, then $F$ must be the trivial filtration on $R$,
that is $F^{\geq m}R=R$ for all $m$.

This is because of Lemma 4.4.3(i) and the fact that we can consider each
connected component separately.

Let $s_{i}$ for $i=1,\ldots,r$ be a finite set of homogeneous generators for
the graded ring $\oplus_{m}F^{\geq m}R$, where $F^{\geq m}R$ lives in degree
$m$. Then we have

$F^{\geq m}R=\underset{\begin{subarray}{c}e_{1},\ldots,e_{r}\geq 0\\\
\Sigma_{i}\,e_{i}\deg(s_{i})=m\end{subarray}}{\operatorname{span}}\left(\underset{i=1}{\overset{r}{\Pi}}s_{i}^{e_{i}}\right)$

where the notation means the $k$-linear span. In particular, the monomials of
degree zero (whose $k$-span is $F^{\geq 0}R$) are in bijection with the kernel
of the map of monoids $\mathbb{N}^{\oplus
r}\xrightarrow{\operatorname{deg}}\mathbb{Z}$ which sends
$(e_{1},\ldots,e_{r})\mapsto\sum_{i}e_{i}\deg(s_{i})$. Gordan’s Lemma says
that the submonoid of $\mathbb{N}^{\oplus r}$ determined by the equation
$\operatorname{deg}(e_{1},\ldots,e_{r})=0$ is finitely generated. This
provides a finite set of $k$-algebra generators of $F^{\geq 0}R$, which proves
(i).

Similarly, applying the Gordon’s lemma to the submonoid of $\mathbb{N}^{\oplus
r}$ cut out by the inequality $\deg\geq 0$ yields a finite set of generators
of the graded subring $\oplus_{m\geq 0}F^{\geq m}R$. Let use denote these
generators by $t_{1},\ldots,t_{\ell}$, so that $\operatorname{deg}(t_{i})\geq
0$. If all these degrees are zero, then $F^{\geq m}R=\langle 0\rangle$ for
$m>0$, which proves (ii). Therefore, we may assume that not all degrees are
zero. Partition the generators into two parts

$\\{t_{1},\ldots,t_{\ell}\\}=\\{t_{1},\ldots,t_{\ell_{0}}\\}\sqcup\\{t_{\ell_{0}+1},\ldots,t_{\ell}\\}$

so that $\deg(t_{i})>0$ if and only if $i\leq\ell_{0}$. Then

$F^{\geq m}R=\sum_{\begin{subarray}{c}e_{1},\ldots,e_{\ell_{0}}\geq 0\\\
\Sigma_{i}\,e_{i}\deg(t_{i})=m\end{subarray}}\left\langle\underset{i=1}{\overset{\ell_{0}}{\Pi}}t_{i}^{e_{i}}\right\rangle$

where $\langle-\rangle$ denotes an ideal in $F^{\geq 0}R$. (Note that the sum
ranges over possible exponents for $t_{1}$ through $t_{\ell_{0}}$. The
remaining generators $t_{\ell_{0}+1},\ldots,t_{\ell}$ appear as coefficients
when one constructs the $F^{\geq 0}R$-ideal generated by the previous
elements.) On the other hand, define $I=\langle
t_{1},\ldots,t_{\ell_{0}}\rangle$, so that

$I^{m}=\sum_{\begin{subarray}{c}e_{1},\ldots,e_{\ell_{0}}\geq 0\\\
\Sigma_{i}\,e_{i}=m\end{subarray}}\left\langle\underset{i=1}{\overset{\ell_{0}}{\Pi}}t_{i}^{e_{i}}\right\rangle.$

Let $d:=\max_{i}\deg(t_{i})>0$. Then we have

$I^{m}\subset F^{\geq m}R\subset I^{\lfloor\frac{m}{d}\rfloor},$

because we have the implications

$\Sigma_{i}\,e_{i}\geq
m\quad\Longrightarrow\quad\Sigma_{i}\,e_{i}\deg(t_{i})\geq
m\quad\Longrightarrow\quad\Sigma_{i}\,e_{i}d\geq m.$

Taking intersections over all $m\geq 0$ implies that
$F^{\infty}R=\cap_{m}I^{m}$. Since $R$ is integral by hypothesis, the Krull
Intersection Theorem implies that $F^{\infty}R$ is either $\langle 0\rangle$
or $F^{\geq 0}R$, which proves (ii). ∎

We shall omit instances of $\iota$ in the subsequent notation.

###### Corollary 4.4.6.

There is a finite type flat affine group scheme
$\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$ over
$\mathbb{A}^{1}/\mathbb{G}_{m}$ and a map
$q:\tilde{G}^{s}\to\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$ of group
schemes over $\mathbb{A}^{1}/\mathbb{G}_{m}$ which is initial for all such
maps from $\tilde{G}^{s}$ to a tame affine group scheme over
$\mathbb{A}^{1}/\mathbb{G}_{m}$. The map $q$ is dominant because $q_{1}$ is
surjective.

_Proof._ This follows from Corollary 4.4.5. For the last sentence, note that
$q_{1}$ is dominant because $F^{\geq 0}R\to R$ is an injection for any
filtered ring $(R,F)$, and this implies that $q_{1}$ is surjective because it
is a map of group schemes. ∎

In view of Lemma 4.4.3(ii), $\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$ can
be described very concretely: it is defined by a group
$G^{{\operatorname{tame}}}_{1}$ and a subgroup $G^{{\operatorname{tame}}}_{0}$
corresponding to a subset of the connected components. The former is the
general fiber and the latter is the special fiber of
$\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$. There is a canonical surjective
map $G^{s(1)}\twoheadrightarrow G^{{\operatorname{tame}}}_{1}$ and a canonical
$\mathbb{G}_{m}$-equivariant map $G^{s(0)}\to G^{{\operatorname{tame}}}_{0}$,
where the $\mathbb{G}_{m}$-action on the target is trivial.

${G^{s(1)}}$${G^{s(0)}}$${G^{{\operatorname{tame}}}_{1}}$${G^{{\operatorname{tame}}}_{0}}$$\scriptstyle{\mathrm{surjective}}$$\scriptstyle{\mathbb{G}_{m}\text{-equivariant}}$incl.
of componentsLemma 4.4.2(ii)

And $\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$ is the ‘universal’ example
of a family of quotients of the fibers of $\tilde{G}^{s}$ which admits such a
description.

###### Remark 4.4.7.

By the universal property of $\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$,
Proposition 4.1.1(i) is equivalent to the assertion that every character of
$G^{s(1)}$ factors through the quotient map $G^{s(1)}\twoheadrightarrow
G_{1}^{\operatorname{tame}}$. In addition, the character $\lim\chi$ of
$G^{s(0)}$ obtained by taking the limit under specialization can be computed
as the composition

$G^{s(0)}\to G_{0}^{\operatorname{tame}}\hookrightarrow
G_{1}^{\operatorname{tame}}\xrightarrow{\chi}\mathbb{G}_{m}.$

In effect, the computation of the limiting character is encapsulated in the
construction of the tame quotient, which contains information about the limits
of all functions on $G^{s(1)}$ under specialization toward the zero fiber.

###  4.5. The subcategory associated to a tame quotient

Let us apply 4.2 to a map $q:\tilde{G}^{s}\to H$ where $H$ is tame. For
example one could take $q$ to be the universal map produced by Corollary
4.4.6. We deduce the following generalization of Corollary 4.3.1:

###### Corollary 4.5.1.

Let $\mathscr{C}_{q,(n)}(X)$ be the category whose objects are triples
$(V,F,\rho)$ as follows:

  * •

$V$ is a vector space, $\rho:H_{1}\curvearrowright V$ is a representation, and
$F$ is an exhausting filtration on $V$ which is invariant under $\rho$.

  * •

We furthermore require that the restriction of the associated graded action

$\mathbb{G}_{m}\ltimes G^{s(0)}\to\mathbb{G}_{m}\times
H_{0}\xrightarrow{\sigma}\mathbb{G}_{m}\times
H_{1}\curvearrowright\operatorname{gr}^{F}V$

factors through the quotient $\mathbb{G}_{m}\ltimes G^{s(0)}\twoheadrightarrow
G^{\ell(0)}$. Here the specialization map $\sigma$ arises from Lemma
4.4.2(ii).

The morphisms in this category are maps of filtered representations. Then
there is an equivalence of categories

$\operatorname{QCoh}^{G}_{\mathrm{tf},q}(X)\simeq\mathscr{C}_{q,(n)}(X)$

where the left hand side is as in Definition 4.2.3.

_Proof._ The proof is identical to that of Corollary 4.3.1. ∎

Recall that Proposition 4.2.2(ii) gives a criterion under which the
subcategory

$\operatorname{QCoh}^{G}_{\mathrm{tf},q}(X)\subset\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$

can be characterized in terms of the behavior of a sheaf
$\mathscr{F}\in\operatorname{QCoh}^{G}_{\mathrm{tf}}(X)$ on $U$. Thus, when
this criterion is satisfied, we have a subcategory for which it is easy to
detect membership and to classify objects. Unfortunately, the closer $q$ is to
being ‘universal,’ the less likely it is to satisfy this criterion.

###### Example 4.5.2.

In closing, we illustrate the preceding notions by constructing an example of
a family of groups over $\mathbb{A}^{1}$ which plays the role of
$\tilde{G}^{s}$ and a tame quotient of this family which fails the criterion
of Proposition 4.2.2(ii). Let $G=\operatorname{GL}_{2}$ and define the
$\mathbb{G}_{m}$-action on $G\times\mathbb{A}^{1}$ via conjugation by the
subgroup $\gamma:\mathbb{G}_{m}\hookrightarrow G$ defined by
$\gamma(t)=\begin{pmatrix}1&0\\\ 0&t\end{pmatrix}$. Define the closed
subscheme $\tilde{G}^{s}\subset G\times\mathbb{A}^{1}$ by requiring that

$\tilde{G}^{s}|_{1}=\left\\{\left.\begin{pmatrix}a&b\\\ b&a\end{pmatrix}\
\right|\ a^{2}-b^{2}=1\right\\},$

that $\tilde{G}^{s}$ is invariant under the $\mathbb{G}_{m}$-action, and that
it is flat over $\mathbb{A}^{1}$. This implies that

$\tilde{G}^{s}|_{t}=\left\\{\left.\begin{pmatrix}a&bt^{-1}\\\
bt&a\end{pmatrix}\ \right|\ a^{2}-b^{2}=1\right\\}$

for all $t\neq 0$, and

$\tilde{G}^{s}|_{0}=\left\\{\left.\begin{pmatrix}a&c\\\ 0&a\end{pmatrix}\
\right|\ a^{2}=1\right\\}.$

Thus, $\tilde{G}^{s}$ is a degeneration from the group $\mathbb{G}_{m}$ to the
group $\mathbb{Z}/2\ltimes\mathbb{G}_{a}$.

Next, we find the universal tame quotient
$\tilde{G}^{s}\to\tau^{{\operatorname{tame}}}(\tilde{G}^{s})$. We have
$A\simeq\mathscr{O}_{\tilde{G}^{s}|_{1}}\simeq k[a,b]/\langle
a^{2}-b^{2}-1\rangle$, and the filtration on $A$ is the $\langle
b\rangle$-adic filtration, meaning that

$F^{\geq m}A=\begin{cases}A&\text{ if }m\leq 0\\\ \langle b\rangle^{m}&\text{
if }m>0.\end{cases}$

Since $\cap_{m}\langle b\rangle^{m}=\langle 0\rangle$, we have

$F^{\geq m}\tau^{{\operatorname{tame}}}A=\begin{cases}A&\text{ if }m\leq 0\\\
\langle 0\rangle&\text{ if }m>0.\end{cases}$

Thus
$\tau^{{\operatorname{tame}}}(\widetilde{G}^{s})\simeq\mathbb{G}_{m}\times\mathbb{A}^{1}$.
The map $G^{s}|_{1}\twoheadrightarrow G_{1}^{{\operatorname{tame}}}$ is an
isomorphism, while the map $G^{s}|_{0}\to G_{0}^{{\operatorname{tame}}}$ is
the composition

$\mathbb{Z}/2\ltimes\mathbb{G}_{a}\twoheadrightarrow\mathbb{Z}/2\hookrightarrow\mathbb{G}_{m}.$

Thus the universal tame quotient is _not_ flat. In fact, the only flat tame
quotient of $\tilde{G}^{s}$ is the trivial one.

The characterization of the filtration on $A$ as the $\langle b\rangle$-adic
filtration allows one to say a bit more. The group scheme $\widetilde{G}^{s}$
is obtained from the constant family $\mathbb{G}_{m}\times\mathbb{A}^{1}$ via
deformation to the normal cone applied to the subgroup

$\\{\pm 1\\}=V(b)\hookrightarrow\operatorname{Spec}A\simeq\mathbb{G}_{m}.$

In general, given a group $G$ and a subgroup $H$, the deformation to the
normal cone of $H$ in $G$ yields a $\mathbb{G}_{m}$-equivariant degeneration
from $G$ to the normal bundle $\mathscr{N}_{H/G}$. (The group structure of the
normal bundle is given by its realization as $\mathscr{N}_{H/G}|_{1}\rtimes
H$.) The degeneration from the Poincaré group to the Galilean group is
realized in this way.

###  4.6. Application to equivariant twisted $\mathscr{D}$-modules

Let us relate the tame quotient idea to the application discussed in Section
6. In 6.1, we explain why the following two subcategories of
$\mathrm{Vect}^{G}(X)$ are equal:

  * •

The subcategory consisting of sheaves $\mathscr{F}$ whose restriction
$\mathscr{F}|_{U}$ has the property of being a strongly $G$-equivariant
twisted $\mathscr{D}$-module with respect to a fixed $G$-equivariant twisting
on $U$.

  * •

The subcategory consisting of sheaves $\mathscr{F}$ for which the restricted
action

$(G^{s(1)})_{\circ}\hookrightarrow
G^{s(1)}\curvearrowright\mathscr{F}|_{\ell(1)}$

factors through a fixed character $\chi:(G^{s(1)})_{\circ}\to\mathbb{G}_{m}$
corresponding to the aforementioned twisting.111111We emphasize that $\chi$
need not be defined on $G^{s(1)}$, but only on its identity component.

We call this category $\mathrm{Vect}^{G}_{\chi}(X)$. This notation generalizes
the notation $\mathrm{Pic}^{G}_{\chi}(X)$ which was introduced in 4.1. The
most important case (in view of Theorem 6.1.2) is when the twisting is a power
of the determinant line bundle on $U$.

In this subsection, we construct a tame quotient of $\tilde{G}^{s}$
corresponding to $\chi$ as above, and we reformulate the criterion of
Proposition 4.2.2(ii) more concretely. The main difference between this
subsection and 4.1 is that here our character $\chi$ need only be defined on
the identity component of $G^{s(1)}$.

The case when $\chi$ is trivial corresponds to untwisted
$\mathscr{D}$-modules. Since this case has already been discussed in 4.3, we
assume that $\chi$ is nontrivial.

###### Lemma 4.6.1.

If $\mathrm{Vect}^{G}_{\chi}(X)$ is nonzero, then $\ker(\chi)$ is normal in
$G^{s(1)}$, not just in $(G^{s(1)})_{\circ}$. Let $q_{1}:G^{s(1)}\to Q_{1}$ be
the quotient by the subgroup $\ker(\chi)$. Then there is a
$\mathbb{G}_{m}$-equivariant map $\tilde{G}^{s}\to Q_{1}\times\mathbb{A}^{1}$
whose fiber at 1 equals $q_{1}$.

_Proof._ The first sentence follows from the fact that, if $G^{s(1)}$ has a
nonzero representation for which the action of $(G^{s(1)})_{\circ}$ factors
through $\chi$, then the intersection of the kernel of this representation
with $(G^{s(1)})_{\circ}$ must be $\ker(\chi)$. Thus, $\ker(\chi)$ is an
intersection of two normal subgroups, so it is normal.

Now we prove the last statement. If $G^{s(1)}$ is connected, it follows from
Proposition 4.1.1(i). For the general case, we shall modify the proof, which
uses Lemma 4.1.3. Consider the codimension-one subgroup $\ker(\chi)\subset
G^{s(1)}$, extend it to a $\mathbb{G}_{m}$-equivariant sub group scheme
$K\subset\tilde{G}^{s}$ which is flat over $\mathbb{A}^{1}$, and let
$\tilde{G}^{s}\twoheadrightarrow\tilde{Q}$ be the quotient by $K$. Lemma
4.1.3(i) applies to the identity component of $\tilde{Q}$, and it tells us
that the action of $\mathbb{G}_{m}$ on the identity component of
$\tilde{Q}_{0}$ is expanding.

If $(\tilde{Q}_{0})_{\circ}\simeq\mathbb{G}_{m}$, then the action
$\mathbb{G}_{m}\curvearrowright\tilde{Q}_{0}$ must be trivial (by rigidity of
tori), so the proof of (i) implies (ii) in Lemma 4.1.3 gives a
$\mathbb{G}_{m}$-equivariant map $\tilde{Q}\to Q_{1}\times\mathbb{A}^{1}$, and
we are done.

If $(\tilde{Q}_{0})_{\circ}\simeq\mathbb{G}_{a}$, then since every action
$\mathbb{G}_{m}\curvearrowright\mathbb{A}^{1}$ is equal to a translate of a
power of the standard action, and since the action
$\mathbb{G}_{m}\curvearrowright\tilde{Q}_{0}$ respects the group structure, it
follows that the action $\mathbb{G}_{m}\curvearrowright\tilde{Q}_{0}$ is
expanding. Now we finish as in the previous paragraph. ∎

####  4.6.2. Definition of the tame quotient associated to $\chi$

In the notation of Lemma 4.6.1, let $Q$ be the group scheme obtained from
$Q_{1}\times\mathbb{A}^{1}$ by deleting from $Q_{1}\times\\{0\\}$ the
components which are not in the image of $G^{s(0)}$. Then consider the map
$q:\tilde{G}^{s}\to Q$ obtained by factoring the map $\tilde{G}^{s}\to
Q_{1}\times\mathbb{A}^{1}$ through $Q$. This is a tame quotient in the sense
of 4.4, and in view of Corollary 4.4.6 it must factor through the universal
tame quotient of $\tilde{G}^{s}$.

It is also related to the tame quotient of components considered in 4.3. We
have a map of exact sequences

${\mathbb{G}_{m}}$${\mathbb{G}_{m}}$${Q_{1}}$${Q_{0}}$${\operatorname{Com}(G^{s(1)})}$${\operatorname{Im}(\sigma)}$

where $\sigma$ was defined in 4.3. The bottom row is the ‘specialization map’
for $H_{\operatorname{LocSys}}$ as defined in 4.3.

When the twisting is given by a power of the determinant line, there is a nice
criterion for testing whether this tame quotient satisfies Proposition
4.2.2(ii).

###### Lemma 4.6.3.

If the character $\chi$ is a (nonzero) rational power of the character
$(G^{s(1)})_{\circ}\curvearrowright\wedge^{\operatorname{top}}\mathscr{T}_{\ell(1)}U$,
then the tame quotient $\tilde{G}^{s}\to Q$ constructed above is flat if and
only if the action
$(G^{s(0)})_{\circ}\curvearrowright\wedge^{\operatorname{top}}\mathscr{T}_{\ell(0)}Z$
is nontrivial.121212Since $G^{s(0)}$ acts trivially on the fibers of
$\mathscr{N}_{Z/X}$, it does not matter whether $\mathscr{T}_{\ell(0)}Z$ is
interpreted as the tangent space taken in $Z$ or in $X$.

_Proof._ Let $\chi:G^{s(1)}\to\mathbb{G}_{m}$ be the character given by acting
on the line $\wedge^{\operatorname{top}}\mathscr{T}_{\ell(1)}U$. First, we
claim the following:

  * •

The map $G^{s(0)}\to Q_{0}$ is surjective if and only if the character
$\lim\chi$ of $G^{s(0)}$ is nontrivial on $(G^{s(0)})_{\circ}$.

Since the map $G^{s(0)}\to Q_{0}$ is surjective on components (by definition
of $Q$), it suffices to test whether the map
$(G^{s(0)})_{\circ}\to(Q_{0})_{\circ}\simeq\mathbb{G}_{m}$ is a surjection.
This occurs if and only if it is nontrivial. Furthermore, the construction of
Lemma 4.6.1 shows that the latter map is given by a nonzero rational multiple
of $\lim\chi$. This character is surjective if and only if $\lim\chi$ is, so
the bullet point statement holds.

To finish, we show that $\lim\chi$ equals the character with which $G^{s(0)}$
acts on the line $\wedge^{\operatorname{top}}\mathscr{T}_{\ell(0)}Z$. The line
bundle $\wedge^{\operatorname{top}}\ell^{*}\mathscr{T}_{X}$ on
$\mathbb{A}^{1}$ is acted on by $G^{s}$ and $\mathbb{G}_{m}$. By 4.1.2, this
defines a (tame) quotient
$q:\tilde{G}^{s}\to\mathbb{G}_{m}\times\mathbb{A}^{1}$. By construction,
$q_{1}=\chi$, so $q_{0}=\lim\chi$, and the claim follows. ∎

###### Remark 4.6.4.

  1.   2. (i)

When the criterion of Lemma 4.6.3 is satisfied, then the category
$\mathrm{Vect}^{G}_{\chi}(X)$ can be described using Corollary 4.5.1.

Even when the criterion is not satisfied, it is always true (by continuity)
that the action $\tilde{G}^{s}\curvearrowright\mathscr{F}|_{\ell}$
corresponding to $\mathscr{F}\in\mathrm{Vect}^{G}_{\chi}(X)$ factors through
the (non-tame) quotient $\tilde{G}^{s}\to\tilde{Q}$ constructed in the proof
of Lemma 4.6.1. Using the method of Lemma 4.1.3, one can prove that
$\tilde{Q}$ is obtained from $Q_{1}\times\mathbb{A}^{1}$ by iterating the
following two operations:

     * •

One may delete some connected components from the special fiber.

     * •

One may apply the blow-up construction at the end of Example 4.5.2 to a (zero-
dimensional) subgroup of the special fiber.

We do not need this result, so we did not include the proof.

  3. (ii)

By the way, the criterion is not satisfied in Examples 6.4.1 and 6.4.2. But
the present subsection does not apply to those examples, because in those
examples the character $\chi:(G^{s(1)})_{\circ}\to\mathbb{G}_{m}$ is trivial.
In those examples, we study line bundles so we are free to use the material of
4.1 instead. For vector bundles, we could use 4.3.

##  5\. Equivariant sheaves on weakly normal chains

In this section, we explain how to classify vector bundles on _weakly normal_
$G$-chains. These are the $G$-chains obtained by gluing smooth simple ones (as
considered in Section 3) as transversely as possible along their codimension-
one orbits.

In 5.1 and 5.2, we describe how to construct vector bundles on weakly normal
$G$-chains by patching together vector bundles on the normalizations of their
irreducible components. This material does not use Theorem 3.7.2, but that
theorem (or any other classification result) can be used to describe vector
bundles on the normalizations of the irreducible components. In 5.3, we
specialize to the case of line bundles and apply Theorem 3.7.2 to arrive at a
description of $\mathrm{Vect}^{G}(X)$ in terms of a ‘nearby cycles’ or
‘gluing’ functor, see Lemma 5.3.2.

###  5.1. Weak normality

The condition of _weak normality_ was originally defined (for complex analytic
spaces) by Andreotti and Norguet in [2] and (for schemes) by Andreotti and
Bombieri in [1]. A variety $X$ is weakly normal if every finite, birational,
bijective map $Y\to X$ is an isomorphism (deleting the word ’bijective’ from
this definition recovers the usual notion of normality). All normal schemes
are weakly normal, but many others are as well. For example, a nodal cubic is
weakly normal but not normal. As proved in [11], every Noetherian scheme $X$
admits a _weak normalization_ , i.e. a finite, birational map
$X^{\mathrm{wn}}\to X$ satisfying the obvious universal property.

Let $G\curvearrowright X$ be a chain, let $Z\hookrightarrow X$ denote the
closed embedding of the disjoint union of the reduced codimension one orbits,
let $\nu:\widetilde{X}\to X$ be the normalization, and consider the following
Cartesian diagram:

(6)
${\widetilde{Z}}$${\widetilde{X}}$${Z}$${X}$$\scriptstyle{\nu}$$\scriptstyle{\iota}$

Since $\tilde{X}$ is a normal $G$-chain, it is smooth (see the last paragraph
of 2.1).

###### Proposition 5.1.1.

If $X$ is weakly normal, then the following are true:

  1. (i)

$\widetilde{Z}$ is reduced.

  2. (ii)

The diagram (6) is a Milnor square, meaning that these two equivalent
conditions are satisfied:

     * •

(6) is a pushout square in the category of schemes.

     * •

The map

$\mathscr{O}_{X}\to\mathscr{O}_{\widetilde{X}}\underset{\mathscr{O}_{\widetilde{Z}}}{\times}\mathscr{O}_{Z}$

is an isomorphism, where the terms on the right hand side are interpreted as
sheaves of algebras on $X$, and the fibered product is computed in the
category of such.

  3. (iii)

The following diagram of categories is Cartesian

${\mathrm{Vect}^{G}(X)}$${\mathrm{Vect}^{G}(Z)}$${\mathrm{Vect}^{G}(\widetilde{X})}$${\mathrm{Vect}^{G}(\widetilde{Z})}$$\scriptstyle{\nu^{*}}$$\scriptstyle{\iota^{*}}$

_Proof._ (i). Let $C_{X}\hookrightarrow X$ and
$C_{\widetilde{X}}\hookrightarrow\widetilde{X}$ be the conductor subschemes
associated to $\nu$. Proposition 53 from [11] implies that $C_{\widetilde{X}}$
and $C_{X}$ are reduced. Hence $C_{X}$ is a union of components of $Z$. The
relation $C_{\widetilde{X}}=\nu^{*}C_{X}$ (as closed subschemes of
$\widetilde{Z}$) implies that the components of $\widetilde{Z}$ which lie over
$C_{X}$ are reduced. The map $\widetilde{Z}\to Z$ must be an isomorphism over
the components of $Z$ which are not contained in the conductor $C_{X}$, so the
components of $\widetilde{Z}$ which do not lie over $C_{X}$ are also reduced.

Point (ii) follows from the definition of the conductor ideal and the previous
paragraph.

Point (iii) follows from Milnor’s Patching Theorem, as stated in [16].
Although this theorem is usually stated for affine schemes (i.e. rings), it
also applies to non-affine schemes because vector bundles are a sheaf of
categories over a scheme. ∎

###  5.2. Patching vector bundles on chains

####  5.2.1. The graph of a chain

Preserve the notation of 5.1, so $G\curvearrowright X$ is a chain. Let
$\\{U_{j}\\}_{j\in S_{U}},\\{Z_{i}\\}_{i\in S_{Z}},\\{\tilde{Z}_{k}\\}_{k\in
S_{\tilde{Z}}}$ be the irreducible components131313This term was defined in
2.1. of $U$, $Z$, and $\tilde{Z}$, respectively. Here
$S_{U},S_{Z},S_{\tilde{Z}}$ are sets that index the irreducible components. In
this subsection we describe a diagram in the category of $G$-chains consisting
of the objects $U_{j},Z_{i},\tilde{Z}_{k}$, and $U_{j}\cup\tilde{Z}_{k}$
whenever these two $G$-orbits are adjacent in $\tilde{X}$. The maps of this
diagram will be given by the obvious ones shown below:

${U_{j}\cup\tilde{Z}_{k}}$${U_{j}}$${\tilde{Z}_{k}}$${Z_{i}}$inclusioninclusion$\scriptstyle{\text{restriction
of }\nu}$

First, we define maps of sets

${S_{U}}$${S_{\tilde{Z}}}$${S_{Z}}$$\scriptstyle{a}$$\scriptstyle{\nu}$

The map $a$ sends $k\in S_{\tilde{Z}}$ to the element $j\in S_{U}$ such that
$\tilde{Z}_{k}$ is adjacent to $U_{j}$ in $\tilde{X}$. This $j$ is unique
because $\tilde{X}$ is smooth (see 2.1). The map $\nu$ sends $k\in
S_{\tilde{Z}}$ to the element $i\in S_{Z}$ such that the normalization map
$\nu:\tilde{X}\to X$ sends $\tilde{Z}_{k}$ to $Z_{i}$.

Next, we define a directed graph $\Gamma$ as follows:

  * •

The vertex set is $S_{U}\sqcup S_{\tilde{Z}}^{(1)}\sqcup
S_{\tilde{Z}}^{(2)}\sqcup S_{Z}$. As indicated here, we will use superscripts
$(-)^{(1)}$ and $(-)^{(2)}$ to differentiate between the two copies of
$S_{\tilde{Z}}$ in the vertex set.

  * •

For every $k\in S_{\tilde{Z}}$, there are arrows

${k^{(1)}}$${a(k)}$${k^{(2)}}$${\nu(k)}$

Finally, we define a functor $F:\Gamma\to G\mathrm{\text{-}Chains}$ which
sends $j\in S_{U}$ to $U_{j}$, sends $k^{(1)}\in S_{\tilde{Z}}^{(1)}$ to
$U_{a(k)}\cup\tilde{Z}_{k}$, sends $k^{(2)}\in S_{\tilde{Z}}^{(2)}$ to
$\tilde{Z}_{k}$, and sends $i\in S_{Z}$ to $Z_{i}$. The behavior on arrows is
given by the first diagram in this subsubsection.

###### Proposition 5.2.2.

If $X$ is weakly normal, then the limit of the diagram of categories given by
$\mathrm{Vect}^{G}(-)\circ
F^{\mathrm{op}}:\Gamma^{\mathrm{op}}\to\operatorname{Cat}$ is equivalent to
$\mathrm{Vect}^{G}(X)$.

_Proof._ For the duration of this paragraph, fix $j\in S_{U}$ and let $k$
range over the subset $a^{-1}(j)\subset S_{\tilde{Z}}$. The subgraph of
$\Gamma$ induced by the vertices $j$ and $k^{(1)}$ is connected to the rest of
$\Gamma$ only via the edges $k^{(1)}\leftarrow k^{(2)}$ for each $k$. The
limit of categories associated to this induced subgraph is
$\mathrm{Vect}^{G}(\overline{U}_{j})$ because the $U_{j}\cup\tilde{Z}_{k}$
constitute an open cover of $\overline{U}_{j}$. Thus, we may modify $F$ and
$\Gamma$ (without changing the limit) by contracting this subgraph to a single
vertex $v_{j}$ which maps to the category
$\mathrm{Vect}^{G}(\overline{U}_{j})$.

Now we vary $j\in S_{U}$ and perform this replacement for all $j\in S_{U}$. We
obtain a new directed graph $\Gamma^{\prime}$ as follows:

  * •

The vertex set is $\\{v_{j}\,|\,j\in S_{U}\\}\sqcup S_{\tilde{Z}}^{(2)}\sqcup
S_{Z}$.

  * •

For each $k\in S_{\tilde{Z}}$, there are edges $v_{a(k)}\leftarrow
k^{(2)}\to\nu(k)$.

We also obtain a new diagram
$(\Gamma^{\prime})^{\mathrm{op}}\to\operatorname{Cat}$ which sends
$v_{j}\mapsto\mathrm{Vect}^{G}(\overline{U}_{j})$ for each $j\in S_{U}$. By
construction, this diagram has the same limit as the old one.

The product of the categories associated to the $v_{j}$ is equivalent to
$\mathrm{Vect}^{G}(\tilde{X})$. Similarly, the product of the categories
associated to the $k\in S_{\tilde{Z}}$ is equivalent to
$\mathrm{Vect}^{G}(\tilde{Z})$, and the product of the categories associated
to the $i\in S_{Z}$ is equivalent to $\mathrm{Vect}^{G}(Z)$. Now it is easy to
see (using the definition of $\Gamma^{\prime}$) that the desired limit of
categories coincides with that of Proposition 5.1.1(iii), as desired. ∎

###### Remark 5.2.3.

Concretely, this means that a $G$-equivariant vector bundle on $X$ is
equivalent to the following data:

  * •

We choose $\mathscr{V}_{i}\in\mathrm{Vect}^{G}(Z_{i})$ for each $i\in S_{Z}$.

  * •

We choose $\mathscr{W}_{j}\in\mathrm{Vect}^{G}(U_{j})$ for each $j\in S_{U}$.

  * •

For each $k\in S_{\tilde{Z}}$, we choose
$\mathscr{E}\in\mathrm{Vect}^{G}(U_{a(k)}\cup\tilde{Z}_{k})$ along with
isomorphisms $\mathscr{E}|_{U_{a(k)}}\simeq\mathscr{W}$ and
$\mathscr{E}|_{\tilde{Z}_{k}}\simeq\nu^{*}(\mathscr{V}_{\nu(k)})$, where
$\nu:\tilde{Z}_{k}\to Z_{\nu(k)}$ is the restriction of the normalization map
of $X$.

Since there are no two-dimensional cells in $\Gamma^{\prime}$, cocycle
conditions do not appear in the above list. This is a convenient consequence
of the ‘codimension zero or one’ requirement in the definition of a chain.

This gives a purely algebraic description of $\mathrm{Vect}^{G}(X)$. Indeed,
if we choose basepoints $z_{i}\in Z_{i}$ and $u_{j}\in U_{j}$, then we have
equivalences
$\mathrm{Vect}^{G}(Z_{i})\simeq\operatorname{Rep}(\operatorname{Stab}_{G}(z_{i}))$
and
$\mathrm{Vect}^{G}(U_{j})\simeq\operatorname{Rep}(\operatorname{Stab}_{G}(u_{j}))$.
Theorem 3.7.2 can be used to describe the category
$\mathrm{Vect}^{G}(U_{j}\cup\tilde{Z}_{i})$.

###### Corollary 5.2.4.

If $X$ is weakly normal and $\Gamma$ is contractible,141414This means that the
topological space associated to $\Gamma$ is contractible, i.e. the directed
graph $\Gamma$ does not contain a ‘zig-zag cycle.’ then the assertion of
Proposition 5.2.2 also holds at the level of isomorphism classes.

###  5.3. The case of line bundles

####  5.3.1. An analogue of the nearby cycles functor

For this subsubsection only, we return to the setting of Section 3, so that
$X=U\cup Z$ is a smooth simple fastened $G$-chain with one closed orbit. We
shall describe the category $\mathrm{Pic}^{G}(X)$ in a way which more closely
resembles the ‘gluing data’ approach mentioned in 1.2.

Choose a fastening datum $(\gamma,\ell)$ for $X$. This allows one to define a
‘nearby cycles’ functor

$\Psi:\mathrm{Pic}^{G}(U)\to\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{Z/X}),$

given by the composition

${\mathrm{Pic}^{G}(U)}$${\mathrm{Pic}(\mathrm{pt}/G^{s(0)})}$${\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{Z/X})}$restrict$\scriptstyle{\sim}$

To define the unlabeled arrow, interpret an object of $\mathrm{Pic}^{G}(U)$ as
a pair $(\chi,V)$ where $\chi:G^{s(1)}\to\mathbb{G}_{m}$ is a character and
$V$ is a 1-dimensional vector space. This maps to
$(\lim\chi,V)\in\mathrm{Pic}(\mathrm{pt}/G^{s(0)})$ where $\lim\chi$ is the
character of $G^{s(0)}$ obtained by taking a limit of $\chi$ under
$\gamma$-conjugation. (Note that $\lim\chi$ exists by Proposition 4.1.1(i).)

###### Lemma 5.3.2.

We have a Cartesian diagram

${\mathrm{Pic}^{G}(X)}$${\mathrm{Pic}^{G}(Z)}$${\mathrm{Pic}^{G}(U)}$${\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{Z/X})}$$\scriptstyle{\mathrm{pullback}}$$\scriptstyle{\Psi}$

_Proof._ An object in the Cartesian product of the labeled maps is given by a
triple $(\chi,V,\chi^{\prime})$ as follows:

  * •

$\chi:G^{s(1)}\to\mathbb{G}_{m}$ is a character, $V$ is a 1-dimensional vector
space, and $\chi^{\prime}:G^{\ell(0)}\to\mathbb{G}_{m}$ is a character.

  * •

These data must satisfy the requirement that the restriction of
$\chi^{\prime}$ along $G^{s(0)}\xrightarrow{\iota}G^{\ell(0)}$ equals
$\lim\chi$.

On the other hand, an object in $\mathrm{Pic}^{G}(X)$ corresponds (via 4.1.2)
to a triple $(V,F,\chi)$ as follows:

  * •

$V$ is a 1-dimensional vector space, $F$ is an exhausting filtration on $V$
(i.e. an integer $d=\deg V$), and $\chi:G^{s(1)}\to\mathbb{G}_{m}$ is a
character.

  * •

These data must satisfy the requirement that the character of
$\mathbb{G}_{m}\ltimes G^{s(0)}$ defined by $(d,\lim\chi)$ descends along the
map $\mathbb{G}_{m}\ltimes G^{s(0)}\twoheadrightarrow G^{\ell(0)}$.

We define an equivalence between these two categories by sending

$(V,F,\chi)\mapsto(\chi,V,\chi^{\prime})$

where $\chi^{\prime}$ is the character of $G^{\ell(0)}$ given by the last
bullet point above, which in particular depends on $F$. The inverse
equivalence is defined by sending $(\chi,V,\chi^{\prime})$ to the triple
$(V,F,\chi)$ where $F$ places $V$ in degree $d$, for the integer $d$ defined
by the weight of the action
$\mathbb{G}_{m}\xrightarrow{\gamma}G^{\ell(0)}\curvearrowright V$. It is
straightforward to check that these are mutually inverse equivalences, and
that there are natural isomorphisms which make this equivalence compatible
with the projections to $\mathrm{Pic}^{G}(U)$ and $\mathrm{Pic}^{G}(Z)$. For
the latter, it is important to note that the vector spaces $V$ and
$\operatorname{gr}^{F}V$ are _canonically_ isomorphic, because $V$ is
1-dimensional. ∎

####  5.3.3. The category of line bundles

By combining Proposition 5.2.2 and Lemma 5.3.2, we obtain a characterization
of $\mathrm{Pic}^{G}(X)$ as the limit of a diagram of categories with maps as
follows, where $\Psi_{j,k}$ is the nearby cycles functor $\Psi$ for the smooth
simple chain $U_{j}\cup\tilde{Z}_{k}$:

${\mathrm{Pic}^{G}(U_{j})}$${\mathrm{Pic}^{G}(Z_{\nu(k)})}$${\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{\tilde{Z}_{k}/(U_{j}\cup\tilde{Z}_{k})})}$$\scriptstyle{\Psi_{j,k}}$pullback

In what follows, we explain this observation in more detail.

Let $\Gamma_{\operatorname{pic}}$ be the directed graph defined as follows:

  * •

The vertex set is $S_{U}\sqcup S_{\tilde{Z}}\sqcup S_{Z}$.

  * •

For every $k\in S_{\tilde{Z}}$, there are arrows $a(k)\to k\leftarrow\nu(k)$.

Let $F_{\operatorname{pic}}:\Gamma_{\operatorname{pic}}\to\operatorname{Cat}$
be the diagram defined as follows:

  * •

It sends $j\in S_{U}$ to $\mathrm{Pic}^{G}(U_{j})$.

  * •

It sends $k\in S_{\tilde{Z}}$ to
$\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{\tilde{Z}_{k}/(U_{a(k)}\cup\tilde{Z}_{k})})$.

  * •

It sends $i\in S_{Z}$ to $\mathrm{Pic}^{G}(Z_{i})$.

  * •

Its behavior on arrows is given by the above diagram.

###### Corollary 5.3.4.

If $X$ is weakly normal, then we have $\lim
F_{\operatorname{pic}}\simeq\mathrm{Pic}^{G}(X)$.

_Proof._ This follows from similar ‘diagram modifications’ as were used in the
proof of Proposition 5.2.2. Starting with the diagram of Proposition 5.2.2 but
with $\mathrm{Pic}^{G}(-)$ in place of $\mathrm{Vect}^{G}(-)$, one first
replaces subdiagrams of the form

${\mathrm{Pic}^{G}(U_{a(k)}\cup\tilde{Z}_{k})}$${\mathrm{Pic}^{G}(U_{a(k)})}$${\mathrm{Pic}^{G}(\tilde{Z}_{k})}$

with diagrams of the form

${\mathrm{Pic}^{G}(U_{a(k)})}$${\mathrm{Pic}^{G}(\tilde{Z}_{k})}$${\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{\tilde{Z}_{k}/(U_{a(k)}\cup\tilde{Z}_{k})})}$$\scriptstyle{\Psi_{j,k}}$

Lemma 5.3.2 tells us that this does not change the resulting limit of
categories. Lastly, for each $i\in S_{Z}$, one may replace the subdiagram
induced by $\mathrm{Pic}^{G}(Z_{i})$ and $\mathrm{Pic}^{G}(\tilde{Z}_{k})$ for
all $k\in\nu^{-1}(i)$ by the single object $\mathrm{Pic}^{G}(Z_{i})$. This is
because the limit does not change if we pass to an initial subdiagram. ∎

###### Remark 5.3.5.

Concretely, this means that a $G$-equivariant line bundle on $X$ is equivalent
to the following data:

  * •

We choose $\mathscr{V}_{i}\in\mathrm{Pic}^{G}(Z_{i})$ for each $i\in S_{Z}$.

  * •

We choose $\mathscr{W}_{j}\in\mathrm{Pic}^{G}(U_{j})$ for each $j\in S_{U}$.

  * •

For each $k\in S_{\tilde{Z}}$, we choose an isomorphism
$\eta:\Psi(\mathscr{W}_{i})\simeq\pi^{*}\mathscr{V}_{\nu(k)}$ of objects in
$\mathrm{Pic}^{G}(\mathscr{N}^{\circ}_{\tilde{Z}_{k}/(U_{a(k)}\cup\tilde{Z}_{k})})$,
where $\pi$ is the composition

$\mathscr{N}^{\circ}_{\tilde{Z}_{k}/(U_{a(k)}\cup\tilde{Z}_{k})}\to\tilde{Z}_{k}\to
Z_{\nu(k)}.$

As was the case in Proposition 5.2.2, there are no cocycle conditions because
$\Gamma_{\operatorname{pic}}$ contains no 2-cells.

##  6\. An application to the representation theory of real reductive groups

Let $G_{\mathbb{R}}$ be the real points of a connected reductive algebraic
group and let $G$ be the complexification of $G_{\mathbb{R}}$. Choose a Cartan
involution $\theta$ of $G$ and let $K=G^{\theta}$, the fixed-points of
$\theta$. By definition, $K\cap G_{\mathbb{R}}\subset G_{\mathbb{R}}$ is a
maximal compact subgroup. If we write $\mathfrak{g}$ and $\mathfrak{k}$ for
the Lie algebras of $G$ and $K$, respectively, the differenital of $\theta$
provides a decomposition

$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$

into $+1$ and $-1$ eigenspaces.

Let $\mathcal{N}\subset\mathfrak{g}$ be the set of nilpotent elements of
$\mathfrak{g}$. Then $\mathcal{N}$ is closed in the Zariski topology on
$\mathfrak{g}$ and invariant under the commuting actions of $G$ and
$\mathbb{C}^{\times}$. The $G$-action on $\mathcal{N}$ has finitely many
orbits.

An important player in the representation theory of $G_{\mathbb{R}}$ is the
closed subvariety

$\mathcal{N}_{\theta}:=\mathcal{N}\cap\mathfrak{p}$

This subset is invariant under $K$ and $\mathbb{C}^{\times}$ (though not under
$G$). By a result of Kostant-Rallis ([12]), the $K$-action on
$\mathcal{N}_{\theta}$ has finitely-many orbits. Each $K$-orbit on
$\mathcal{N}_{\theta}$ is a Lagrangian submanifold of its $G$-saturation,
which is a $G$-orbit on $\mathcal{N}$.

If $V$ is an irreducible admissible representation of $G_{\mathbb{R}}$, there
is an associated class in the Grothendieck group
$K\mathrm{Coh}^{K}(\mathcal{N}_{\theta})$ of $K$-equivariant coherent sheaves
on $\mathcal{N}_{\theta}$, constructed as follows. The Harish-Chandra module
of $V$ is a $\mathfrak{g}$-module $M$ with an algebraic action of $K$. These
structures are compatible in the two obvious ways: the action map

$\mathfrak{g}\otimes M\to M$

is $K$-equivariant and the $\mathfrak{g}$-action coincides on $\mathfrak{k}$
with the differentiated action of $K$. We say, in short, that $M$ is a
$(\mathfrak{g},K)$-module. $M$ is irreducible, since $V$ is.

Every finite-length $(\mathfrak{g},K)$-module $M$ admits a good filtration
$...\subset M_{-1}\subset M_{0}\subset M_{1}\subset...$ by $K$-invariant
subspaces. The associated graded $\operatorname{gr}(M)$ has the structure of a
graded, $K$-equivariant, coherent sheaf on $\mathfrak{p}$, with support
contained in $\mathcal{N}_{\theta}$. Although $\operatorname{gr}(M)$ depends
on the filtration used to define it, its class $[\operatorname{gr}(M)]$ in the
Grothendieck group $K\operatorname{Coh}^{K}(\mathcal{N}_{\theta})$ does not.
In particular, the set

$\mathrm{AV}(M):=\mathrm{Supp}(\operatorname{gr}(M))\subset\mathcal{N}_{\theta},$

called the _associated variety_ of $M$, is a well-defined invariant.151515Here
$\mathrm{AV}(M)$ is defined to be the _reduced_ support, so that the
associated variety is indeed a variety. Although $\operatorname{gr}(M)$ may
not be scheme-theoretically supported on $\mathrm{AV}(M)$ depending on the
filtration of $M$, it is always possible to interpret $[\operatorname{gr}(M)]$
as an element of $K_{0}(\mathrm{AV}(M))$, and we shall do so. The
correspondence between irreducible $G_{\mathbb{R}}$-representations and
classes in $K\mathrm{Coh}^{K}(\mathcal{N}_{\theta})$ is given by the
assignment $V\mapsto[\operatorname{gr}(M)]$:

Irred $G_{\mathbb{R}}$-repsIrred
$(\mathfrak{g},K)$-mods$K\mathrm{Coh}^{K}(\mathcal{N}_{\theta})$ Harish-
Chandra module $\mathrm{gr}$

The Arthur conjectures ([3],[4]) predict the existence of a small finite set
of irreducible representations of $G_{\mathbb{R}}$ – called _unipotent_ by
Arthur – with very interesting properties. A working definition can be found
for example in [5]. These representations have been studied extensively over
the past several decades, but the general theory remains elusive.

The unipotent representations of $G_{\mathbb{R}}$ give rise, by the
correspondence described above, to a finite set of distinguished classes in
$K\mathrm{Coh}^{K}(\mathcal{N}_{\theta})$, which we will call _unipotent
sheaves_.

Irred $G_{\mathbb{R}}$-repsIrred
$(\mathfrak{g},K)$-mods$K\mathrm{Coh}^{K}(\mathcal{N}_{\theta})$ Harish-
Chandra module $\mathrm{gr}$Unipotent $G_{\mathbb{R}}$-repsUnipotent sheaves

This discussion suggests the following natural question

###### Question 1.

Is there a purely geometric description of the unipotent sheaves?

###  6.1. Admissibility

In [15], Vogan provides a partial answer to Question 1. The key definition is
the following one, due originally to Schwartz:

###### Definition 6.1.1 ([14]).

Let $U$ be a homogeneous space for $K$. Choose $e\in U$ so that
$U=K/\operatorname{Stab}_{K}(e)$. Homogeneous sheaves of twisted differential
operators on $U$ are parameterized by characters of the Lie algebra
$\mathfrak{k}^{e}$ (see, for example, Appendix A in [10]). Let $\mathrm{tr}$
be the character of $\mathfrak{k}^{e}$ corresponding to the canonical line
bundle on $U$. This character is given by the explicit formula

$\mathrm{tr}(X)=\mathrm{Tr}(\mathrm{ad}(X)|_{\mathfrak{k}^{e}})-\mathrm{Tr}(\mathrm{ad}(X)|_{\mathfrak{k}})\qquad
X\in\mathfrak{k}^{e}$

A $K$-equivariant vector bundle $\mathcal{E}\in\mathrm{Vect}^{K}(U)$ is
_admissible_ if the restriction $\mathcal{E}|_{U}$ has the structure of a
strongly $K$-equivariant $\mathcal{D}_{U}^{\frac{1}{2}\mathrm{tr}}$-module,
where $\mathcal{D}_{U}^{\frac{1}{2}\mathrm{tr}}$ is the homogeneous sheaf of
twisted differential operators corresponding to the character
$\frac{1}{2}\mathrm{tr}$ of $\mathfrak{k}^{e}$.

Here is a more explicit formulation. The fiber $E=\mathcal{E}|_{e}$ carries a
representation $\rho$ of $\operatorname{Stab}_{K}(e)$. $\mathcal{E}$ is
admissible if $\rho$ satisfies the condition

$d\rho=\frac{1}{2}\mathrm{tr}\cdot\mathrm{Id}_{E}\in\mathrm{Rep}(\mathfrak{k}^{e})$

As is clear from the second formulation of Definition 6.1.1, admissibility is
a property of the class $[\mathcal{E}]\in K\mathrm{Vect}^{K}(U)$.

Vogan proves

###### Theorem 6.1.2 ([15], Theorem 8.7).

Suppose $M$ is the Harish-Chandra module of a unipotent representation of
$G_{\mathbb{R}}$. Let $U_{1},...,U_{m}$ be the open $K$-orbits on