# Stabilizer reduction for derived stacks and applications to sheaf-theoretic
invariants

Jeroen Hekking  Department of Mathematics, Universität Regensburg, Regensburg,
93053, Germany<EMAIL_ADDRESS>,  David Rydh  Department of Mathematics,
KTH Royal Institute of Technology, Stockholm, 114 28, Sweden<EMAIL_ADDRESS>and  Michail Savvas  Department of Mathematics, The University of Texas at
Austin, Austin, TX 78712, USA<EMAIL_ADDRESS>
###### Abstract.

We construct a canonical stabilizer reduction $\widetilde{X}$ for any derived
$1$-algebraic stack $X$ over $\mathbb{C}$ as a sequence of derived Kirwan
blow-ups, under mild natural conditions that include the existence of a good
moduli space for the classical truncation $X_{\mathrm{cl}}$.

Our construction has several desired features: it naturally generalizes
Kirwan’s classical partial desingularization algorithm to the context of
derived algebraic geometry, preserves quasi-smoothness, and is a derived
enhancement of the intrinsic stabilizer reduction constructed by Kiem, Li and
the third author. Moreover, if $X$ is $(-1)$-shifted symplectic, we show that
the semi-perfect and almost perfect obstruction theory of
$\widetilde{X}_{\mathrm{cl}}$ and the associated virtual fundamental cycle and
virtual structure sheaf, constructed by the same authors, are naturally
induced by $\widetilde{X}$ and its derived tangent complex. As corollaries, we
define virtual classes for moduli stacks of semistable sheaves on surfaces,
give a fully derived perspective on generalized Donaldson–Thomas invariants of
Calabi–Yau threefolds and define new generalized Vafa–Witten invariants for
surfaces via Kirwan blow-ups.

The first author was supported by the Göran Gustafsson foundation. The second
author was supported by the Göran Gustafsson Foundation for Research in
Natural Sciences and Medicine.

###### Contents

  1. 1 Recollections on derived algebraic geometry
  2. 2 Overview of derived blow-ups
  3. 3 Equivariant derived algebraic geometry
  4. 4 Derived loci of maximal stabilizer dimension
  5. 5 Intrinsic blow-ups as equivariant derived blow-ups
  6. 6 Derived stabilizer reduction of Artin stacks
  7. 7 Quasi-smooth stacks
  8. 8 $(-1)$-shifted symplectic stacks
  9. 9 Sheaf-theoretic invariants

## Introduction

### Some brief history

It has long been understood that in order to effectively study geometric
objects and their families, one needs to enlarge the category of algebraic
varieties and consider the more general notion of algebraic stacks. While the
theory of algebraic stacks now enjoys a vast literature and features
prominently within algebraic geometry and beyond, the presence of
automorphisms of objects, also called stabilizers, which is typically a
desirable feature, can sometimes be problematic regarding the use of standard
techniques that apply to algebraic varieties, e.g., (co)homological methods
like integration.

A very useful way to associate an algebraic space to an Artin stack is to
consider its moduli space, whenever this makes sense. However, depending on
the scope of one’s considerations, this might not be adequate, given that it
is not necessarily a well-behaved operation. For example, a smooth stack need
not have a smooth moduli space.

Stabilizer reduction refers to the process of resolving the stackiness of an
algebraic stack to produce a canonical Deligne–Mumford stack, whose
stabilizers are all finite. In classical algebraic geometry, Kirwan [Kir85]
was the first to carry this out for smooth quotient stacks
$X=[Q^{\mathrm{ss}}/G]$ obtained by Geometric Invariant Theory [MFK94], namely
by an action of a reductive group $G$ on a smooth projective variety $Q$
together with a linearization.

More precisely (assuming that $Q^{s}\neq\emptyset$), Kirwan produced a
sequence of GIT quotient stacks

$X_{0}=X=[Q^{\mathrm{ss}}/G],\ X_{1}=[Q_{1}^{\mathrm{ss}}/G],\ \dots\ ,\
\widetilde{X}\coloneqq X_{n}=[\widetilde{Q}^{\mathrm{ss}}/G]$

by iteratively blowing up $X_{i}$ along the (smooth) locus
$X_{i}^{{\mathrm{max}}}\subseteq X_{i}$ of points with maximal dimensional
stabilizer and then deleting unstable points by a careful study of stability
on the blow-up. The final result
$\widetilde{X}=[\widetilde{Q}^{\mathrm{ss}}/G]$ is a GIT quotient
Deligne–Mumford stack satisfying
$\widetilde{Q}^{\mathrm{ss}}=\widetilde{Q}^{s}$. At the level of GIT
quotients, each morphism $X_{i+1}\to X_{i}$ fits into a natural commutative
square

(0.5)

where the lower horizontal arrow is a birational blow-up map. Thus, since
$\widetilde{Y}$ only has finite quotient singularities and admits a
projective, birational map $\widetilde{Y}\to Y$, it is a partial
desingularization of $Y$. Kirwan’s construction has had a wealth of
applications, see for example [Kir86, KL04, Kie07].

Subsequently, Kirwan’s construction has been independently generalized by
Edidin–Rydh [ER21], and Kiem–Li and the third author [KLS17, Sav20], to apply
to (possibly singular) Artin stacks $X$ with a good moduli space $q\colon X\to
Y$ in the sense of Alper [Alp13]. Such stacks are a natural generalization of
GIT quotient stacks $X=[Q^{\mathrm{ss}}/G]$ with good moduli space morphism
$q\colon[Q^{\mathrm{ss}}/G]\to Q/\\!\\!/G$. In fact, by deep results of
Alper–Hall–Rydh [AHR19, AHR20], the GIT picture is the étale local model for
this general class of stacks.

The two constructions, while related, differ in the notion of blow-up being
applied at each iterated step: Edidin–Rydh introduce a saturated blow-up by
first blowing up $X$ along $X^{{\mathrm{max}}}$ and then carefully deleting a
closed substack of unstable points, whereas Kiem–Li and the third author use a
so-called Kirwan blow-up by first taking an intrinsic blow-up of $X$ along
$X^{{\mathrm{max}}}$ and then deleting unstable points. The intrinsic blow-up
was constructed using local embeddings into smooth quotient stacks and
preserved Kuranishi models. It was thus expected to be the classical shadow of
a derived blow-up.

### Derived stabilizer reduction: statement of results

In this paper, we generalize Kirwan’s algorithm to the setting of derived
algebraic stacks $X$ whose classical truncation $X_{\mathrm{cl}}$ admits a
good moduli space $q\colon X_{\mathrm{cl}}\to Y$. By the hidden smoothness
principle, derived stacks should behave in analogy with smooth classical
stacks, and hence it is natural to expect that a derived stabilizer reduction
$\widetilde{X}$ should be the result of repeatedly blowing up $X$ along
$X^{{\mathrm{max}}}$ and then deleting unstable points. This is exactly what
we do.

Our first result proves the existence of $X^{{\mathrm{max}}}$ in the derived
setting, under mild finiteness assumptions which we leave implicit here and in
the rest of the introduction. Uniqueness refers to uniqueness up to a
contractible space of homotopies.

######  Theorem (Theorem 4.14).

Let $X$ be a derived Artin stack whose classical truncation admits a good
moduli space $X_{\mathrm{cl}}\to Y$. Let $d$ be the maximal dimension of
stabilizers of points of $X$.

Then there exists a unique, closed immersion $X^{{\mathrm{max}}}\to X$ of
derived algebraic stacks such that for any étale morphism $[U/G]\to X$, with
$U$ an affine derived scheme and $G$ a reductive group of dimension $d$, whose
classical truncation fits in a Cartesian diagram

(0.10)

there exists a Cartesian diagram of derived stacks

where $G^{0}\subseteq G$ is the identity component of $G$. The classical
truncation of $X^{{\mathrm{max}}}$ is naturally isomorphic to
$(X_{\mathrm{cl}})^{{\mathrm{max}}}$.

For a classical stack, a precise treatment of $X^{{\mathrm{max}}}$ was given
in the appendix of [ER21]. Our definition gives a derived enhancement,
directly inspired by the classical construction. To prove Theorem 4.14 and to
compare the derived blow-up of $X$ along the locus $X^{{\mathrm{max}}}$ to the
intrinsic blow-up, we use properties of derived fixed loci, which are defined
via Weil restrictions. A substantial part of the paper is devoted to a
thorough treatment of elements of derived equivariant geometry for this
reason.

With $X^{{\mathrm{max}}}$ at hand, the recent theory of derived blow-ups in
arbitrary derived centers—generalized by the first author in [Hek21] from the
quasi-smooth case treated in [KR18], and further developed in [HKR22]—together
with its natural extension to (equivariant) derived stacks, allows us to
define the derived intrinsic blow-up
$X^{\mathrm{intr}}\coloneqq\operatorname{Bl}_{X^{{\mathrm{max}}}}X$ of $X$ in
the center $X^{{\mathrm{max}}}$. Our second result shows that its classical
truncation is the intrinsic blow-up $(X_{\mathrm{cl}})^{\mathrm{intr}}$ of
$X_{\mathrm{cl}}$ defined by Kiem–Li and the third author. We may then delete
unstable points as in [KLS17] to obtain an open substack $\widehat{X}\subseteq
X^{\mathrm{intr}}$, which we define to be the derived Kirwan blow-up of $X$.

We summarize these results in the following theorem.

######  Theorem (Theorem 5.3, Proposition 6.2, Theorem 6.7).

There exists a canonical derived Artin stack $\widehat{X}$, called the derived
Kirwan blow-up of $X$, together with a morphism $\pi\colon\widehat{X}\to X$,
such that:

  1. (i)

Its classical truncation $\widehat{X}_{\mathrm{cl}}$ admits a good moduli
space morphism $\widehat{q}\colon\widehat{X}_{\mathrm{cl}}\to\widehat{Y}$.

  2. (ii)

The maximum stabilizer dimension of points in $\widehat{X}$ is strictly
smaller than that of $X$.

  3. (iii)

For any affine, étale, stabilizer-preserving morphism $[U/G]\to X$ whose
classical truncation fits in a Cartesian diagram (0.10), the base change
$\widehat{X}\times_{X}[U/G]$ is naturally isomorphic to the derived Kirwan
blow-up of $[U/G]$.

  4. (iv)

$\pi|_{\pi^{-1}(X^{\mathrm{s}})}$ is an isomorphism over the open locus
$X^{\mathrm{s}}$ of stable points.

$\widehat{X}$ is the semi-stable locus
$(X^{{\mathrm{intr}}})^{\mathrm{ss}}\subseteq X^{{\mathrm{intr}}}$, an open
substack of the Artin stack
$X^{{\mathrm{intr}}}=\operatorname{Bl}_{X^{{\mathrm{max}}}}X$, called the
derived intrinsic blow-up of $X$.

The classical truncations of $X^{{\mathrm{intr}}}$ and $\widehat{X}$ are the
classical intrinsic and Kirwan blow-ups of the classical truncation
$X_{\mathrm{cl}}$ respectively.

The main ingredient in the proof is an explicit, non-trivial computation of
the classical truncation of the equivariant blow-up of $\operatorname{Spec}A$
along $(\operatorname{Spec}A)^{G^{0}}$, using several properties of derived
blow-ups and fixed loci. Since
$(\operatorname{Bl}_{X^{{\mathrm{max}}}}X)_{\mathrm{cl}}$ is in general not
the blow-up of $X_{\mathrm{cl}}$ along $X_{\mathrm{cl}}^{{\mathrm{max}}}$,
this theorem explains the difference between the two constructions [ER21] and
[KLS17]. When $X$ is smooth, these do coincide and both blow-ups are the same.

As an immediate consequence of this theorem we get the desired derived
stabilizer reduction $\widetilde{X}\to X$ of $X$ by repeatedly taking derived
Kirwan blow-ups until the maximal stabilizer dimension of points drops to
zero. $\widetilde{X}$ is a derived Deligne–Mumford stack, which by
construction is a natural derived enhancement of the intrinsic stabilizer
reduction $\widetilde{X}_{\mathrm{cl}}$ of the classical truncation
$X_{\mathrm{cl}}$ constructed in [Sav20] (this in particular also applies to
the case where $X=X_{\mathrm{cl}}$ is classical to begin with).

With $\widetilde{X}$ at our disposal, we investigate its properties for quasi-
smooth and $(-1)$-shifted symplectic stacks and obtain associated virtual
fundamental classes. These constructions have natural applications towards
enumerative invariants of sheaves on surfaces and threefolds.

### Quasi-smoothness and invariants of Donaldson-type

Using the properties of derived blow-ups and the definition of derived
intrinsic and Kirwan blow-ups, we show that they preserve the property of
being quasi-smooth (see Proposition 7.4). We thus obtain the following
theorem.

######  Theorem (Theorem 7.1).

The derived stabilizer reduction $\widetilde{X}$ of a quasi-smooth derived
Artin stack $X$ is quasi-smooth. In particular, the intrinsic stabilizer
reduction $\widetilde{X}_{\mathrm{cl}}$ admits a natural virtual fundamental
class $[\widetilde{X}_{\mathrm{cl}}]^{\mathrm{vir}}\in
A_{\ast}(\widetilde{X}_{\mathrm{cl}})$ and virtual structure sheaf
$[\mathcal{O}_{\widetilde{X}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{X}_{\mathrm{cl}})$.

Quasi-smooth derived Artin stacks naturally appear as derived moduli stacks
${\mathcal{M}}$ of semistable sheaves or perfect complexes on smooth,
projective surfaces. There are several variations of how one can obtain such
stacks, e.g., by considering reduced versions when the genus of the surface is
positive, or fixing the determinant. For more details, the interested reader
can consult [Moc09] or [Joy21].

Let $\mathcal{M}$ be a quasi-smooth moduli stack of this type such that the
good moduli space $M_{\mathrm{cl}}$ of $\mathcal{M}_{\mathrm{cl}}$ is proper.
Then $\widetilde{\mathcal{M}}_{\mathrm{cl}}$ is a proper Deligne–Mumford stack
and integrals
$\int_{[\widetilde{\mathcal{M}}_{\mathrm{cl}}]^{\mathrm{vir}}}\gamma$ for
cohomology classes $\gamma$ on $\widetilde{\mathcal{M}}_{\mathrm{cl}}$ may be
viewed as Donaldson-type invariants (see Definition 9.2).

###  $(-1)$-shifted symplectic geometry and Donaldson–Thomas invariants

The main motivation for the introduction of the intrinsic stabilizer reduction
$\widetilde{X}_{\mathrm{cl}}$ in [KLS17, Sav20] was the construction of
intersection-theoretic generalized Donaldson–Thomas invariants.

Namely, it was shown by local computation that if $X$ is a $(-1)$-shifted
symplectic derived Artin stack, then the intrinsic stabilizer reduction
$\widetilde{X}_{\mathrm{cl}}$ admits a canonically induced semi-perfect
obstruction theory (cf. [CL11]) of virtual dimension zero, and later in [KS22]
that in fact this can be promoted to an almost perfect obstruction theory. In
particular, there are an associated virtual fundamental cycle and virtual
structure sheaf

(0.11) $\displaystyle[\widetilde{X}_{\mathrm{cl}}]^{\mathrm{vir}}\in
A_{0}(\widetilde{X}_{\mathrm{cl}}),\quad[\mathcal{O}_{\widetilde{X}_{\mathrm{cl}}}^{\mathrm{vir}}]\in
K_{0}(\widetilde{X}_{\mathrm{cl}}),$

and one can define

$\mathrm{DTK}(X)=\int_{[\widetilde{X}_{\mathrm{cl}}]^{\mathrm{vir}}}1\in\mathbb{Q}$

as the numerical generalized Donaldson–Thomas invariant via Kirwan blow-ups
(DTK invariant) associated to $X$, along with $K$-theoretic invariants. Since
derived moduli stacks of semi-stable sheaves and perfect complexes on
Calabi–Yau threefolds are $(-1)$-shifted symplectic by [PTVV13], this gives
rise to DTK invariants that act as virtual counts of these objects (after
rigidifying scaling automorphisms).

By suitable local computation, we further show that our derived stabilizer
reduction procedure gives a derived enhancement of
$\widetilde{X}_{\mathrm{cl}}$ which completely recovers the above obstruction
theory.

######  Theorem (Theorem 8.15).

Let $X$ be a $(-1)$-shifted symplectic derived Artin stack and $\widetilde{X}$
its derived stabilizer reduction. Then:

  1. (i)

The $[0,1]$-truncation
$E^{\bullet}=\tau^{[0,1]}\mathbb{T}_{\widetilde{X}}|_{\widetilde{X}_{\mathrm{cl}}}$
(with cohomological indexing notation) is a perfect complex.

  2. (ii)

Its dual $E_{\bullet}=(E^{\bullet})^{\vee}$ together with the derived
structure of $\widetilde{X}$ naturally recover the data of the semi-perfect
obstruction theory of [KLS17, Sav20] and almost perfect obstruction theory of
[KS22] and hence the virtual fundamental cycle
$[\widetilde{X}_{\mathrm{cl}}]^{\mathrm{vir}}$ and virtual structure sheaf
$[\mathcal{O}_{\widetilde{X}_{\mathrm{cl}}}^{\mathrm{vir}}]$ in (0.11).

This gives a fully derived perspective on generalized Donaldson–Thomas
invariants via Kirwan blow-ups. As a corollary, we may combine the DTK
formalism with virtual torus localization (cf. [GP99]) in order to define new,
generalized Vafa–Witten invariants enumerating Gieseker semi-stable Higgs
pairs $(E,\phi\colon E\to E\otimes K_{S})$ on $S$ with fixed positive rank and
Chern classes, fixed determinant $\det E$ and zero trace $\mathrm{tr}\ \phi=0$
(see Definition 9.6).

### Future directions

Given Kirwan’s original algorithm and its generalization in [ER21], it is
natural to wonder whether there is a theory of derived good moduli spaces for
derived algebraic stacks, such that, in particular, our derived stabilizer
reduction procedure gives a sequence of diagrams

where the vertical maps are derived good moduli space morphisms, and which are
thus derived enhancements of the diagrams (0.10). In fact, using the derived
étale slice theorem and local-to-global methods, one can indeed construct
derived good moduli spaces whose existence is preserved by the operation of
derived Kirwan blow-up. The details will appear elsewhere to keep the length
of this paper under control.

Another interesting direction of inquiry would be to investigate the
properties and degeneracy of the pullback of an $n$-shifted symplectic form to
the derived stabilizer reduction and possibly develop a theory of $n$-shifted
quasi-symplectic derived stacks, of which derived stabilizer reductions are a
particular case.

Other natural extensions of this work would be to generalize the stabilizer
reduction algorithm to derived stacks over a base of positive or even mixed
characteristic. Since our main arguments ultimately depend on the classical
situation and on the formal properties of Weil restrictions, we in fact expect
a generalized stabilizer reduction algorithm to exist in any nonconnective
algebraic geometry. Such geometries are extensions of derived algebraic
geometry—of which derived analytic geometry in the sense of [BK17], [BM22],
[BKK] is an example—and are proposed by Ben-Bassat and the first author in
[BH23], together with a generalization of derived blow-ups to this setting.

Finally, there are several purely enumerative potential ramifications for
sheaf-theoretic invariants that remain to be explored.

We intend to come back to these questions in the future.

### Layout of the paper

In §1 we review the requisite background and language of derived algebraic
geometry that we use. In §2, we recall derived blow-ups and discuss and prove
statements that are necessary in subsequent parts of the paper. §3 is devoted
to a careful study of derived equivariant geometry with a focus on fixed loci
of derived group actions. §4 gives the construction of the locus
$X^{{\mathrm{max}}}$ of points of maximal stabilizers of a derived stack $X$.
In §5 we establish the relationship between intrinsic blow-ups and equivariant
derived blow-ups, and in §6 we construct the derived stabilizer reduction of a
derived Artin stack. §7 deals with the case of quasi-smooth stacks, while §8
with the case of $(-1)$-shifted symplectic stacks. Finally, §9 discusses
applications to generalized Donaldson–Thomas and Vafa–Witten invariants.

### Acknowledgements

The first author would like to thank Adeel Khan for valuable conversations
about derived algebraic stacks.

The third author would like to thank Jarod Alper and Daniel Halpern-Leistner
for helpful answers to questions related to the étale slice theorem and good
moduli spaces and Hyeonjun Park for discussions related to fixed loci.

### Notation and conventions

Throughout, we work over $\mathbb{C}$. All rings and algebras are assumed to
be commutative. We will use both simplicial/derived rings and commutative
differential graded algebras interchangeably, assuming their usual equivalence
under the Dold–Kan correspondence. The particular model we are working with
will be clear from context.

All derived stacks and all good moduli space morphisms are assumed to have
affine diagonal. By a derived algebraic stack we mean a derived $1$-algebraic
stack.

The classical truncation of $X$ is denoted by $X_{\mathrm{cl}}$ and the
topological space of points of $X$ by $\lvert X\rvert$. If $x\in\lvert
X\rvert$, then $G_{x}$ denotes the automorphism group/stabilizer of $x\in
X_{\mathrm{cl}}$. In practice, we will mostly work with stabilizers of closed
points $x\in\lvert X\rvert$. These will be reductive for most stacks of
interest in this paper, namely stacks whose classical truncation admits a good
moduli space.

For a morphism $\rho\colon X\to Y$ and a sheaf or complex $\mathcal{E}$ on
$Y$, we often use $\mathcal{E}|_{X}$ to denote $\rho^{*}\mathcal{E}$,
suppressing the pullback from the notation. We also use the phrases “closed
embedding” and “closed immersion” interchangeably, likewise for “Artin stack”
and “algebraic stack”.

$G$, $H$ denote complex reductive groups throughout. Since we are working over
$\mathbb{C}$, being reductive is equivalent to being linearly reductive and
hence all finite-dimensional and rational $G$-representations are completely
reducible in this paper (see, for example, [Bor66, Theorem 5.2]). Usually, $H$
will be a subgroup of $G$. $G^{0}$ is the identity component of $G$. $T$ is
the torus $\mathbb{G}_{m}=\mathbb{C}^{\times}$.

If $X$ is a derived stack with $G$-action, $X^{G}$ is used to denote the
derived fixed point locus of $G$ in $X$, see §3.4. For $X$ algebraic,
$X^{\mathrm{intr}},\widehat{X}$ are used to denote the derived intrinsic and
Kirwan blow-ups of $X$ respectively, see §6.2. $\widetilde{X}$ denotes the
derived stabilizer reduction of a stack $X$, see §6.3.

Quasi-coherent modules and algebras are written as
$\mathcal{M},\mathcal{N},\dots$ and $\mathcal{A},\mathcal{B},\dots$ to
distinguish them from their affine counterparts $M,N,\dots$ and $A,B,\dots$.

We will use homological indexing notation, unless otherwise stated.

We will mostly ignore set-theoretic issues to avoid distractions. These can be
resolved by any of the standard methods, such as assuming the existence of
Grothendieck universes.

##  1\. Recollections on derived algebraic geometry

We review the language of derived algebraic geometry which we will be using.
Much of this material can be found in the standard literature on the topic,
like [SAG], [HAGII], [GR17]. For convenience, we collect some constructions
and results here, but refer the reader to these sources for more details.

###  1.1. Background on $\infty$-categories

Recall that an $\infty$-category is a generalization of a $1$-category that
allows us to talk about $n$-morphisms for all $n\in\mathbb{N}_{\geq 1}$, where
all morphisms above $n=1$ are invertible (in other words, these are
$(\infty,1)$-categories). We thus have, between any two objects $x,y$ in a
given $\infty$-category $\mathsf{C}$, a mapping space $\mathsf{C}(x,y)$ which
has the set of equivalence classes of $1$-morphisms $x\to y$ as connected
components. Here, a space is considered as a homotopy type, which can be
modelled for example on simplicial sets.

We use similar shorthands for $\infty$-categories as is customary for
$1$-categories, by expressing the existence of a homotopy as a property. In
particular, when we say that a diagram is commutative, we mean that there is a
homotopy that makes the diagram commutative. Likewise, by uniqueness we will
mean uniqueness up to a contractible space of choices.

Write $\mathsf{Cat}$ for the $\infty$-category of $\infty$-categories, and
$\mathsf{Spc}$ for the $\infty$-category of spaces. Although we will
explicitly distinguish $\infty$-categories from 1-categories throughout, we do
adopt the common language of $\infty$-categories, where limits, colimits, Kan
extensions, etc., are always taken in their $\infty$-categorical meaning.
Recall, if one chooses a point-set model, then these become homotopy limits,
colimits, Kan extensions, etc..

######  Definition 1.1.

Let $\mathsf{C}$ be an $\infty$-category. Define the cocompletion of
$\mathsf{C}$ under sifted colimits, written $\mathsf{P}_{\Sigma}(\mathsf{C})$,
as the full subcategory of the $\infty$-category $\mathsf{P}(\mathsf{C})$ of
presheaves on $\mathsf{C}$ spanned by those presheaves that send finite
coproducts in $\mathsf{C}$ to products in $\mathsf{Spc}$.

######  Remark 1.2.

The $\infty$-category $\mathsf{P}_{\Sigma}(\mathsf{C})$ is the universal way
of adding sifted colimits to $\mathsf{C}$. It is also called the nonabelian
derived category of $\mathsf{C}$. See [HTT, §5.5.8] for details.

###  1.2. Derived rings and modules

Let $\mathsf{Mod}$ be the symmetric monoidal $\infty$-category of modules over
$\mathbb{C}$ in the stably homotopic sense, i.e., of spectra endowed with a
$\mathbb{C}$-action. Endow $\mathsf{Mod}$ with the standard $t$-structure
(with homological indexing convention). Recall that $\mathsf{Mod}$ is also the
$\infty$-category associated to the unbounded derived category of
$\mathbb{C}$-vector spaces.

Write $\mathsf{Mod}^{0}$ for the full subcategory of $\mathsf{Mod}$ spanned by
the discrete, finitely generated $\mathbb{C}$-modules. Then $\mathsf{Mod}^{0}$
generates $\mathsf{Mod}_{\geq 0}$ under sifted colimits.

######  Definition 1.3.

Let $\mathsf{Alg}$ be the $\infty$-category of connective
$\mathbb{E}_{\infty}$-algebras in $\mathsf{Mod}$.

Let $\mathsf{Poly}$ be the category of finitely generated polynomial rings
over $\mathbb{C}$. Since we are working over a ring of characteristic 0,
$\mathsf{Alg}$ is equivalent to the $\infty$-category
$\mathsf{P}_{\Sigma}(\mathsf{Poly})$, hence also to the $\infty$-category
associated to the model category of simplicial $\mathbb{C}$-algebras, endowed
with the projective model structure. It is also equivalent to the
$\infty$-category of cdgas over $\mathbb{C}$ concentrated in homological
degree $\geq 0$.

An object $R$ of $\mathsf{Alg}$ will be called an algebra. We can associate to
$R$ a 1-categorical model $R^{\prime}$, either incarnated as a simplicial ring
or as a cdga. Of course, any $R^{\prime\prime}$ which is weakly equivalent to
$R^{\prime}$ will then also be a 1-categorical model of $R$. To keep the
indexing convention consistent, we will write our cdgas as chain complexes in
degree $\geq 0$.

The forgetful functor $\mathsf{Alg}\to\mathsf{Mod}_{\geq 0}$ has a left
adjoint, written $\mathrm{LSym}$. For $R\in\mathsf{Alg}$, we write
$\mathsf{Alg}_{R}$ for the slice-$\infty$-category $\mathsf{Alg}_{R/}$.
Objects in $\mathsf{Alg}_{R}$ are called $R$-algebras. Likewise, we write
$\mathsf{Mod}_{R}$ for the $\infty$-category of $R$-modules in $\mathsf{Mod}$.
The forgetful functor $\mathsf{Alg}_{R}\to(\mathsf{Mod}_{R})_{\geq 0}$ again
has a left adjoint, written $\mathrm{LSym}_{R}$.

Let $\mathsf{Mod}^{0}_{R}$ be the full subcategory of $\mathsf{Mod}_{R}$
spanned by finitely generated free $R$-modules, and let
$\mathsf{Poly}_{R}\subset\mathsf{Alg}_{R}$ be the essential image of
$\mathrm{LSym}_{R}$ on $\mathsf{Mod}^{0}_{R}$. Then $\mathsf{Mod}^{0}_{R}$
again generates $(\mathsf{Mod}_{R})_{\geq 0}$ under sifted colimits, and the
canonical map $\mathsf{P}_{\Sigma}(\mathsf{Poly}_{R})\to\mathsf{Alg}_{R}$ is
an equivalence. Moreover, $\mathsf{Alg}_{R}$ is the $\infty$-category of
connective $\mathbb{E}_{\infty}$-algebras in $\mathsf{Mod}_{R}$.

The functor $\pi_{0}\colon\mathsf{Spc}\to\mathsf{Set}$ induces adjunctions

$\displaystyle\pi_{0}:\mathsf{Mod}_{\geq
0}\leftrightarrows\mathsf{Mod}^{\heartsuit}:i$
$\displaystyle\pi_{0}:\mathsf{Alg}\leftrightarrows\mathsf{Alg}^{\heartsuit}:i$

where $\mathsf{Mod}^{\heartsuit}$ is the heart of $\mathsf{Mod}$ with respect
to the given $t$-structure, i.e., the 1-category of $\mathbb{C}$-vector
spaces, and $\mathsf{Alg}^{\heartsuit}$ is the 1-category of discrete
$\mathbb{C}$-algebras.

For $R\in\mathsf{Alg}$ and $M\in\mathsf{Mod}_{R}$, the homotopy groups
$\pi_{n}(M)$ are canonically $\pi_{0}(R)$-modules, for all $n\in\mathbb{Z}$.
In particular, $\pi_{n}(R)$ is a $\pi_{0}(R)$-module for all $n\geq 0$.

###  1.3. Cell attachments

Let $R\in\mathsf{Alg}$. For $n\geq 0$, we adjoin a free variable in
homological degree $n$ to $R$ by writing $R[u]=R[S^{n}]$, which has the
universal property that the space of $R$-algebra maps $R[u]\to B$ is
equivalent to the space of maps $(S^{n},*)\to(B,0)$ of pointed spaces, i.e.,
to $\mathsf{Mod}_{R}(R[n],B)$.

Let a sequence $\underline{\sigma}=(\sigma_{1},\dots,\sigma_{k})$ of elements
$\sigma_{i}\in\pi_{n_{i}}(R)$ be given. Then we define the finite quotient of
$R$ by $\sigma$, written
$R/(\underline{\sigma})=R/(\sigma_{1},\dots,\sigma_{k})$, via the pushout

${{R[u_{1},\dots,u_{k}]}}$${R}$${R}$${R/(\sigma_{1},\dots,\sigma_{k})}$$\scriptstyle{z}$$\scriptstyle{s}$

where the $u_{i}$ are free in homological degree $n_{i}$, the map $s$ is
induced by lifts $S^{n_{i}}\to R$ of $\sigma_{i}$, and $z$ sends each $u_{i}$
to zero. When $k=1$, we also say that $R/(\underline{\sigma})$ is obtained
from $R$ by attaching an $(n+1)$-cell. A zero cell attachment is the map $R\to
R[t]$, where $t$ is in homological degree 0. Note that the input of attaching
several $(n+1)$-cells is equivalent to a morphism $\sigma\colon M[n]\to R$ of
$R$-modules, where $M$ is free of finite rank. Giving this datum, we write the
quotient also as $R/(\sigma)$.

A finite quotient of the form $R\to R/(f_{1},\dots,f_{k})$, where each $f_{i}$
is in homological degree $0$, is called quasi-smooth.

###  1.4. Standard forms

For the purpose of performing computations, it will be useful to work with
certain $1$-categorical models.

Recall that a morphism $A\to B$ of algebras (and the corresponding map on
spectra) is _locally of finite presentation_ if $B$ is a compact object in
$\mathsf{Alg}_{A}$, i.e., the $\mathsf{Spc}$-valued functor
$\mathsf{Alg}_{A}(B,-)$ preserves filtered colimits. It is _finitely
presented_ if $B$ can be obtained from $A$ by a finite number of cell
attachments. Then the following are equivalent: (1) $B$ is locally of finite
presentation over $A$; (2) the map
$\operatorname{Spec}B\to\operatorname{Spec}A$ is, Zariski-locally on
$\operatorname{Spec}B$, finitely presented; and (3) $B$ is a retract of a
finitely presented $A$-algebra.

######  Definition 1.4.

Let $A$ be an algebra. A cdga model $R$ for $A$ is _in standard form_ if
$R_{0}$ is smooth with $\Omega_{R_{0}}$ free, and the underlying graded-
commutative ring of $R$ is freely generated over $R_{0}$ by a finite number of
generators.

######  Remark 1.5.

Observe, if a cdga $R$ is in standard form, then it can be obtained from
$R_{0}$ by a finite number of cell-attachments, say $R(0)\to R(1)\to\dots\to
R(n)$, in such a way that when we terminate the process at $k\leq n$, then
$R(k)$ is also in standard form. In fact, the underlying graded-commutative
ring of $R(k)$ is the subring of the underlying graded-commutative ring of $R$
generated by elements of homogeneous degree $\leq k$, and
$R(k+1)=R(k)/(\sigma)$ where $\sigma\colon M_{k+1}[k]\to R(k)$ and $M_{k+1}$
is a free module of finite rank over $R(k)$. Moreover
$R_{k+1}=R(k)_{k+1}\oplus(M_{k+1})_{0}$. We say that $R$ is _generated_ by the
$M_{k+1}$.

If $R$ is a model of standard form for $A$, then the cotangent complex of
$\operatorname{Spec}A$ is given by the Kähler differentials $\Omega_{R}$, see
[BBJ19, §2.3].

######  Definition 1.6.

Let $x\in\operatorname{Spec}A$ with $A\in\mathsf{Alg}$ be given. Then a model
$R$ in standard form for $A$ is _minimal at $x$_ if the differentials of
$\Omega_{R}|_{x}$ are all zero.

By the following lemma, locally finitely presented derived schemes can
Zariski-locally be described by models in standard form.

######  Lemma 1.7.

Let $A\in\mathsf{Alg}$ be locally of finite presentation and
$x\in\operatorname{Spec}A$. Then, up to Zariski localizing around $x$, we can
find a model $R$ in standard form for $A$ such that $R_{0}$ is a finitely
generated polynomial ring, or such that it is minimal at $x$.

###### Proof.

This follows for example from [BBJ19, Theorem 4.1]. ∎

######  Remark 1.8.

In general, the two conditions cannot be met simultaneously, see [BBJ19,
Remark 3.2].

###  1.5. Graded derived rings

Derived blow-ups in arbitrary centers are defined along similar lines as the
classical construction—namely, through a derived version of the Rees algebra,
of which one then takes the projective spectrum. To explain this, let us first
review graded algebras in the derived setting as defined in [Hek21].

Let $\mathbb{M}$ be a commutative monoid, considered as a discrete symmetric
monoidal category in the obvious way. Let $\mathsf{Mod}^{\mathbb{M}}$ be the
functor $\infty$-category $\mathrm{Fun}(\mathbb{M},\mathsf{Mod})$, endowed
with symmetric monoidal structure via Day convolution and with $t$-structure
inherited from $\mathsf{Mod}$. An object $M$ of $\mathsf{Mod}^{\mathbb{M}}$ is
called an $\mathbb{M}$-graded module, written as
$\bigoplus_{d\in\mathbb{M}}M_{d}$, where $M_{d}$ is $M$ evaluated at $d$.

For $d\in\mathbb{M}$ and $N\in\mathsf{Mod}$, we write $N(d)$ for the
$\mathbb{M}$-graded module which is $N$ concentrated in homogeneous degree
$-d$. Let then $(\mathsf{Mod}^{\mathbb{M}})^{0}$ be the full subcategory of
$\mathsf{Mod}^{\mathbb{M}}$ generated by finite coproducts of modules of the
form $N(d)$, where $N\in\mathsf{Mod}^{0}$ and $d\in\mathbb{M}$. Then
$(\mathsf{Mod}^{\mathbb{M}})^{0}$ generates $(\mathsf{Mod}^{\mathbb{M}})_{\geq
0}$ under sifted colimits.

Define the $\infty$-category of $\mathbb{M}$-graded algebras, written
$\mathsf{Alg}^{\mathbb{M}}$, as the $\infty$-category of connective
$\mathbb{E}_{\infty}$-algebras in $\mathsf{Mod}^{\mathbb{M}}$. As in the
ungraded case, the forgetful functor
$\mathsf{Alg}^{\mathbb{M}}\to(\mathsf{Mod}^{\mathbb{M}})_{\geq 0}$ has a left
adjoint, written $\mathrm{LSym}^{\mathbb{M}}$. Let
$\mathsf{Poly}^{\mathbb{M}}$ be the essential image of
$\mathrm{LSym}^{\mathbb{M}}$ evaluated on $(\mathsf{Mod}^{\mathbb{M}})^{0}$.
Then the canonical map
$\mathsf{P}_{\Sigma}(\mathsf{Poly}^{\mathbb{M}})\to\mathsf{Alg}^{\mathbb{M}}$
is an equivalence.

A key result from [Hek21] is that, as expected, the $\infty$-category
$\mathsf{Alg}^{\mathbb{Z}}$ is (contravariantly) equivalent to the
$\infty$-category of affine derived schemes with $\mathbb{G}_{m}$-action. The
general theory of $\infty$-actions is reviewed in §2.1, and we give an in-
depth analysis of $G$ actions on derived stacks for reductive algebraic groups
$G$ in §3.

For $B\in\mathsf{Alg}^{\mathbb{M}}$, write $\mathsf{Mod}^{\mathbb{M}}_{B}$ for
the $\infty$-category of $\mathbb{M}$-graded $B$-modules, which by definition
are $B$-modules in $\mathsf{Mod}^{\mathbb{M}}$. As in the ungraded case,
$\mathsf{Mod}^{\mathbb{M}}_{B}$ is stable and symmetric monoidal, and
$(\mathsf{Mod}^{\mathbb{M}}_{B})_{\geq 0}$ is generated under sifted colimits
by the $\infty$-category of finitely generated, free, $\mathbb{M}$-graded
$B$-modules. Once again,
$\mathsf{Alg}^{\mathbb{M}}_{B}\coloneqq(\mathsf{Alg}^{\mathbb{M}})_{B/}$ is
generated under sifted colimits by the finitely generated, $\mathbb{M}$-graded
polynomial $B$-algebras, which are defined similarly as in the ungraded case.

We are primarily interested in $\mathbb{N}$-graded and $\mathbb{Z}$-graded
rings. These exhibit familiar behavior: to a $\mathbb{Z}$-graded ring $B$ we
can associate an $\mathbb{N}$-graded ring $B_{\geq 0}$ by discarding the
pieces in negative homogeneous degrees, and an $\mathbb{N}$-graded ring
$B^{\prime}$ can be promoted to a $\mathbb{Z}$-graded ring by putting $0$ in
negative homogeneous degrees. We can also take the homogeneous degree-zero
part $B_{0}$ of a $\mathbb{Z}$-graded ring, which gives us morphisms $B_{0}\to
B_{\geq 0}\to B$ of $\mathbb{Z}$-graded rings (where we think of
$B_{0},B_{\geq 0}$ as $\mathbb{Z}$-graded by the procedure just described,
which we suppress from notation). If $B$ is $\mathbb{N}$-graded, we also have
a morphism $B\to B_{0}$ such that the composition $B_{0}\to B\to B_{0}$ is
homotopic to the identity.

Let $\mathbb{M}$ be either $\mathbb{N}$ or $\mathbb{Z}$, and let
$B\in\mathsf{Alg}^{\mathbb{M}}$. As in the ungraded case, we can adjoin a free
variable to $B$ in homological degree $n\in\mathbb{N}$ and homogeneous degree
$d\in\mathbb{M}$, which produces an $\mathbb{M}$-graded $B$-algebra $B[u]$
with the universal property that the space of $\mathbb{M}$-graded $B$-algebra
maps $B[u]\to C$ is equivalent to the space of maps $(S^{n},*)\to(C_{d},0)$,
i.e., to $\mathsf{Mod}^{\mathbb{M}}(B[n]_{d},C)$, where $B[n]_{d}$ is the
homogeneous degree-$d$ part of the module $B[n]$.

The $\mathbb{M}$-grading on $B$ induces an $\mathbb{M}$-grading on each
$\pi_{n}(B)$. We can thus define a finite quotient of $B$ by a sequence
$\underline{\sigma}=(\sigma_{1},\dots,\sigma_{k})$ of elements
$\sigma_{i}\in\pi_{n_{i}}(B)_{d_{i}}$, with $n_{i}\in\mathbb{N}$ and
$d_{i}\in\mathbb{M}$, using the graded free variables construction just
described and performing cell attachments in the same way as in the ungraded
case. The result is again written as $B/(\underline{\sigma})$.

###  1.6. Derived stacks

Let $\mathbf{P}$ be one of the following properties: étale, a (Zariski) open
immersion, smooth. A morphism $A\to B$ of derived rings is called $\mathbf{P}$
if $\pi_{0}A\to\pi_{0}B$ is $\mathbf{P}$, and the morphism
$(\pi_{n}A)\otimes_{\pi_{0}A}(\pi_{0}B)\to\pi_{n}B$ is an isomorphism, for any
$n\geq 0$. Define the $\infty$-category of affine derived schemes as
$\mathsf{Aff}\coloneqq\mathsf{Alg}^{\mathrm{op}}$, and write
$\operatorname{Spec}(-)\colon\mathsf{Alg}^{\mathrm{op}}\to\mathsf{Aff}$ for
the obvious functor. Declare morphisms in $\mathsf{Aff}$ to be $\mathbf{P}$ if
the corresponding map in $\mathsf{Alg}$ is $\mathbf{P}$. For
$U=\operatorname{Spec}A$ in $\mathsf{Aff}$, the underlying classical scheme is
$U_{\mathrm{cl}}\coloneqq\operatorname{Spec}\pi_{0}A$.

We endow $\mathsf{Aff}$ with the étale topology by declaring a family
$\\{U_{i}\to U\\}_{i}$ in $\mathsf{Aff}$ to be a cover if all $U_{i}\to U$ are
étale, and the induced maps on classical schemes $(U_{i})_{\mathrm{cl}}\to
U_{\mathrm{cl}}$ are jointly surjective.

A derived prestack is a functor
$X\colon\mathsf{Aff}^{\mathrm{op}}\to\mathsf{Spc}$. A derived stack is a
derived prestack which is a sheaf (in the $\infty$-categorical sense) for the
étale topology. We write $\mathsf{Stk}$ for the $\infty$-category of derived
stacks. For $X\in\mathsf{Stk}$, we write $\mathsf{Stk}_{X}$ for the
$\infty$-category $\mathsf{Stk}_{/X}$ of derived stacks over $X$. The Yoneda
embedding induces an embedding
$\operatorname{Spec}(-)\colon\mathsf{Aff}\to\mathsf{Stk}$.

Recall that a space $K$ is called _$n$ -truncated_ if $\pi_{m}(K,x)$ is
trivial for all points $x\in K$ and all $m>n$. A stack $X$ is called _$n$
-truncated_ if $X(T)$ is $n$-truncated for all classical affine schemes $T$. A
morphism $X\to Y$ of derived stacks is affine if
$X\times_{Y}(\operatorname{Spec}A)$ is affine, for all
$\operatorname{Spec}A\to Y$. Then a derived stack $X$ has affine diagonal if
the canonical morphism $\Delta\colon X\to X\times X$ is affine. Observe that
$X$ has affine diagonal if and only if the intersection $U\times_{X}V$ is
affine, for any pair of affine derived schemes $U,V$ over $X$.

######  Assumption 1.9.

From here on, assume that all derived stacks are 1-truncated and have affine
diagonal, unless otherwise stated.

Let ${\mathrm{cl}\mathsf{Stk}}$ be category of classical stacks. Composition
with the inclusion $\mathsf{Alg}^{\heartsuit}\to\mathsf{Alg}$ gives a functor
$(-)_{\mathrm{cl}}\colon\mathsf{Stk}\to{\mathrm{cl}\mathsf{Stk}}$ which sends
$X\in\mathsf{Stk}$ to the underlying classical stack $X_{\mathrm{cl}}$. This
functor has a fully faithful left adjoint $\iota$, and both functors
$\iota,(-)_{\mathrm{cl}}$ preserve affines. A morphism $X\to Y$ of derived
stacks is quasi-compact/a closed immersion if the morphism $X_{\mathrm{cl}}\to
Y_{\mathrm{cl}}$ is quasi-compact/a closed immersion.

###  1.7. Derived schemes

In this and the next paragraph, let $\mathbf{P}$ be one of the properties:
étale, smooth, an open immersion. Although we could add here any property
which is étale-local in an appropriate sense, we restrict ourselves to what we
need.

Let $X$ be a derived stack. A family $\\{U_{i}\to X\\}_{i}$ of morphisms in
$\mathsf{Stk}$ is a cover if the canonical map
$\operatorname{{colim}}\check{C}(f)\to X$ is an equivalence, where
$\check{C}(f)$ is the Čech nerve of $f\colon\bigsqcup U_{i}\to X$, and the
colimit is taken in $\mathsf{Stk}$. A morphism $T\to X$ in $\mathsf{Stk}$ with
affine source is $\mathbf{P}$ if $S\times_{X}T\to S$ is $\mathbf{P}$ for all
affine derived scheme $S$ over $X$. The latter is well-defined, since we
assume that all derived stacks have affine diagonal.

A derived stack $U$ is called a derived scheme if there is cover $\\{U_{i}\to
U\\}$ of open immersions $U_{i}\to U$ by affine derived schemes $U_{i}$.
Observe, the adjunction
$\iota:{\mathrm{cl}\mathsf{Stk}}\leftrightarrows\mathsf{Stk}:(-)_{\mathrm{cl}}$
restricts to an adjunction between classical schemes (which we assume to also
have affine diagonal) and derived schemes. The $\infty$-category
$\mathsf{Sch}$ of derived schemes is closed under fiber products in
$\mathsf{Stk}$.

A morphism $U\to X$ of derived stacks with $U$ a derived scheme is
$\mathbf{P}$ if $U\times_{X}V\to V$ is $\mathbf{P}$, for any derived scheme
$V$ over $X$.

###  1.8. Derived algebraic spaces and stacks

A derived algebraic space is a $0$-truncated derived stack $Y$ for which there
is an étale cover $U\to Y$ by a derived scheme $U$. A morphism $X^{\prime}\to
X$ of derived stacks is representable if for all derived schemes $U$ over $X$
the fiber product $U\times_{X}X^{\prime}$ is a derived algebraic space.
Observe that any morphism $Y\to X$ from a derived algebraic space to a derived
stack is representable. Again, this is because $X$ has affine diagonal, and
since for any affine morphism $X^{\prime}\to V$ in $\mathsf{Stk}$ with $V$ a
derived scheme it holds that $X^{\prime}$ is a derived scheme as well.

A morphism $Y^{\prime}\to Y$ of algebraic spaces is $\mathbf{P}$ if there is
an étale cover $V\to Y^{\prime}$ by a derived scheme $V$ such that the
composition $V\to Y$ is $\mathbf{P}$. A representable morphism $X^{\prime}\to
X$ of derived stacks is $\mathbf{P}$ if for any derived scheme $U$ over $X$,
the map $U\times_{X}X^{\prime}\to U$ is $\mathbf{P}$.

A derived algebraic stack is a derived stack $X$ such that there is a smooth
cover $Y\to X$ with $Y$ a derived algebraic space. Observe, the adjunction
$\iota:{\mathrm{cl}\mathsf{Stk}}\leftrightarrows\mathsf{Stk}:(-)_{\mathrm{cl}}$
restricts to an adjunction between classical algebraic stacks and derived
algebraic stacks, and also between classical algebraic spaces and derived
algebraic spaces.

Let $X^{\prime}\to X$ be a morphism of algebraic stacks. Then $X^{\prime}\to
X$ is smooth if there is a smooth cover $Y^{\prime}\to X^{\prime}$ by a
derived algebraic space $Y^{\prime}$ such that the composition $Y^{\prime}\to
X$ is smooth. Likewise, $X^{\prime}\to X$ is étale if, smooth-locally on $X$,
there is an étale cover $Y^{\prime}\to X^{\prime}$ by a derived algebraic
space $Y^{\prime}$, such that the composition $Y^{\prime}\to X$ is étale.
Finally, $X^{\prime}\to X$ is an open immersion if it is representable, étale
and a monomophism.

###  1.9. Quasi-coherent modules and algebras

Let $f\colon\operatorname{Spec}B\to\operatorname{Spec}A$ be a morphism of
affine derived schemes. Then we have an adjunction
$f^{*}:\mathsf{Mod}_{A}\leftrightarrows\mathsf{Mod}_{B}:f_{*}$, where $f^{*}$
sends $M\in\mathsf{Mod}_{A}$ to $M\otimes_{A}B$. This gives us a functor
$\mathsf{Aff}^{\mathrm{op}}\to\mathsf{Cat}$ that sends $\operatorname{Spec}R$
to $\mathsf{Mod}_{R}$. Taking the right Kan extension yields the functor

$\displaystyle\mathsf{Stk}^{\mathrm{op}}\to\mathsf{Cat}\colon
X\mapsto\operatorname{QCoh}(X)$

which sends $X$ to the $\infty$-category of quasi-coherent
$\mathcal{O}_{X}$-modules. Concretely, for $X\in\mathsf{Stk}$, we can write
$\operatorname{QCoh}(X)$ as the limit of the diagram
$\operatorname{Spec}R\mapsto\mathsf{Mod}_{R}$ over the comma $\infty$-category
$\mathsf{Aff}^{\mathrm{op}}_{/X}$ consisting of derived affine schemes over
$X$. This description tells us that we can think of a quasi-cohorent
$\mathcal{O}_{X}$-module $\mathcal{F}$ as a homotopy coherent diagram
$\\{\mathcal{F}_{A}\\}_{\operatorname{Spec}A\to X}$, where each
$\mathcal{F}_{A}$ is in $\mathsf{Mod}_{A}$. We also write
$\mathcal{F}(A)\coloneqq\mathcal{F}(\operatorname{Spec}A)\coloneqq\mathcal{F}_{A}$.

For $X\in\mathsf{Stk}$ algebraic, we can restrict the indexing category in the
formula $\operatorname{QCoh}(X)\simeq\lim_{\operatorname{Spec}R\to
X}\mathsf{Mod}_{R}$ to derived affine schemes which are smooth over $X$, and
still get the same result. Then $\operatorname{QCoh}(X)$ is naturally a
stable, presentable, symmetric monoidal $\infty$-category for which the tensor
product commutes with colimits in each variable separately, and moreover
$\operatorname{QCoh}(X)$ carries a $t$-structure such that
$\mathcal{F}\in\operatorname{QCoh}(X)$ is connective if and only if
$\mathcal{F}_{A}$ is, for all $\operatorname{Spec}A\to X$. For a morphism
$f\colon X\to Y$ of derived algebraic stacks, by the adjoint functor theorem
we have an adjunction
$f^{*}:\operatorname{QCoh}(Y)\leftrightarrows\operatorname{QCoh}(X):f_{*}$,
where $f^{*}$ is symmetric monoidal and $f_{*}$ is right-lax symmetric
monoidal.

Following the same procedure, we have a functor
$\mathsf{Aff}^{\mathrm{op}}\to\mathsf{Cat}$ which sends $\operatorname{Spec}R$
to $\mathsf{Alg}_{R}$. Right-Kan extending this yields a functor
$\mathsf{Stk}^{\mathrm{op}}\to\mathsf{Cat}$, which we write as
$X\mapsto\mathrm{QAlg}(X)$. Objects in $\mathrm{QAlg}(X)$ are called quasi-
coherent $\mathcal{O}_{X}$-algebras. When restricted to derived algebraic
stacks, this functor lands in the $\infty$-category of presentable, symmetric
monoidal $\infty$-categories, and symmetric monoidal colimit preserving
functors between them. Since we are in characteristic zero, we can also
describe $\mathrm{QAlg}(X)$ as the $\infty$-category of
$\mathbb{E}_{\infty}$-algebras in $\operatorname{QCoh}(X)_{\geq 0}$. We
furthermore have a left adjoint to the forgetful functor
$\mathrm{QAlg}(X)\to\operatorname{QCoh}(X)_{\geq 0}$, written
$\mathrm{LSym}_{X}$.

######  Example 1.10.

For $X\in\mathsf{Stk}$, let $\mathcal{O}_{X}\in\mathrm{QAlg}(X)$ be such that
$\mathcal{O}_{X}(A)=A$, for all $\operatorname{Spec}A\to X$, which is a unit
in $\mathrm{QAlg}(X)$. We also write $\mathcal{O}_{X}$ for the underlying
quasi-coherent module.

Let $X\in\mathsf{Stk}$ be algebraic. For $\mathcal{A}\in\mathrm{QAlg}(X)$, the
(relative) spectrum of $\mathcal{A}$ is the derived algebraic stack
$\operatorname{Spec}\mathcal{A}$ over $X$ such that for $f\colon T\to X$, the
space $(\operatorname{Spec}\mathcal{A})(T)$ is the space of quasi-coherent
$\mathcal{O}_{T}$-algebra maps $f^{*}\mathcal{A}\to\mathcal{O}_{T}$.

Write $\mathsf{Aff}_{X}$ for the full subcategory of $\mathsf{Stk}_{X}$
spanned by derived stacks which are affine over $X$, i.e., for which the
structure map is affine. For $\mathcal{A}\in\mathrm{QAlg}(X)$, it holds that
$\operatorname{Spec}\mathcal{A}$ is affine over $X$. In fact, the functor
$\mathcal{A}\mapsto\operatorname{Spec}\mathcal{A}$ yields an equivalence
$\mathrm{QAlg}(X)^{\mathrm{op}}\simeq\mathsf{Aff}_{X}$.

Let $X\in\mathsf{Stk}$ be algebraic, and let
$\mathcal{E}\in\operatorname{QCoh}(X)_{\geq(-1)}$ be given. Define
$\mathbb{V}(\mathcal{E})$ as the derived stack over $X$ with the universal
property that for $f\colon T\to X$ it holds that $\mathbb{V}(\mathcal{E})$ is
the space of $\mathcal{O}_{T}$-module maps
$f^{*}\mathcal{E}\to\mathcal{O}_{T}$. If $\mathcal{E}$ is connective, then
$\mathbb{V}(\mathcal{E})\simeq\operatorname{Spec}\mathrm{LSym}_{X}(\mathcal{E})$.

###  1.10. The cotangent complex and the normal bundle

Let $X$ be a derived algebraic stack. A quasi-coherent
$\mathcal{O}_{X}$-module $\mathcal{F}$ is called almost connective if for all
$\operatorname{Spec}A\to X$ there is some $n\in\mathbb{Z}$ such that
$\mathcal{F}_{A}\in(\mathsf{Mod}_{A})_{\geq n}$.

Let $T=\operatorname{Spec}A$ be an affine derived scheme, and $M$ a connective
$A$-module. Then there is a derived version of the square-zero extension of
$A$ by $M$, written $A\oplus M$, which is an $A$-algebra over $A$. We write
$T[M]$ for $\operatorname{Spec}(A\oplus M)$.

Let $f\colon X\to Y$ be a morphism of derived algebraic stacks. The cotangent
complex of $f$ is the $(-1)$-connective, quasi-coherent
$\mathcal{O}_{X}$-module ${\mathbb{L}}_{X/Y}$ with the following universal
property. For any $f\colon T\to X$ with $T$ affine, and any connective
$\mathcal{O}_{T}$-module $M$, the space of maps $f^{*}{\mathbb{L}}_{X/Y}\to M$
of $\mathcal{O}_{T}$-modules is equivalent to the space of diagram fillers

${T}$${X}$${T[M]}$${Y}$$\scriptstyle{f}$

Let $X\xrightarrow{f}Y\to Z$ be a morphism of derived algebraic stacks. Then
we have an exact sequence
$f^{*}{\mathbb{L}}_{Y/Z}\to{\mathbb{L}}_{X/Z}\to{\mathbb{L}}_{X/Y}$ in
$\operatorname{QCoh}(X)$. Moreover, for $X^{\prime}\to Y^{\prime}$ obtained by
pulling back a derived algebraic stack $Y^{\prime}$ over $Y$ along $f$, it
holds that
$h^{*}{\mathbb{L}}_{X/Y}\simeq{\mathbb{L}}_{X^{\prime}/Y^{\prime}}$, where $h$
is the structure map $X^{\prime}\to X$.

For $f\colon X\to Y$ a morphism of derived algebraic stacks, we let $N_{X/Y}$
be the shifted cotangent complex ${\mathbb{L}}_{X/Y}[-1]$, and define the
normal bundle of $f$ as $\mathbb{N}_{X/Y}\coloneqq\mathbb{V}(N_{X/Y})$. Note
that $\mathbb{V}(N_{X/Y})$ will in general be 2-truncated, contrary to our
standing assumption. This does not play a role in what follows.

##  2\. Overview of derived blow-ups

We continue with developing the background for our main results, but specify
to the recent theory on derived blow-ups and derived Rees algebras as
developed in [KR18], [Hek21], [HKR22] and [BH23]. We also establish a few
statements about derived blow-ups and their local models that will be very
useful in subsequent sections.

###  2.1. $\infty$-group actions

Although the construction of the derived blow-up only uses $G$-actions on
derived stacks for reductive algebraic groups $G$—which we analyse in-depth in
§3—we will use actions by $\infty$-groups in $\mathsf{Stk}$ in §4 for the
construction of $X^{\mathrm{max}}$. In this paragraph, we therefore review the
parts of this theory that we will be using from [NSS15], [HTT]. We omit proofs
since we only deviate slightly from the literature in terminology and
notation.

Throughout, let $\mathsf{H}$ be an $\infty$-topos. Although we will only use
the case $\mathsf{H}=\mathsf{Stk}$, presenting the theory in full generality
highlights the formal character of the story.

######  Definition 2.1.

Let $X\colon\Delta^{\mathrm{op}}\to\mathsf{H}$ be a simplicial diagram in
$\mathsf{H}$. Then $X$ satisfies the Segal condition if, for all $n\geq 0$,
the natural map

$\displaystyle\varphi_{n}\colon X_{n}\to
X_{1}\times_{X_{0}}X_{1}\times_{X_{0}}\cdots\times_{X_{0}}X_{1}$

is an equivalence, where the morphisms $\varphi_{n}$ come about from the spine
inclusions $(0\to 1\to\dots\to n)\to\Delta[n]$. If this is the case, then we
say that morphisms are invertible in $X$ if the map $X_{2}\to
X_{1}\times_{X_{0}}X_{1}$ induced by the horn inclusion
$\Lambda^{2}[2]\to\Delta[2]$ is an equivalence.

$X$ is called an $\infty$-groupoid (in $\mathsf{H}$) if it satisfies the Segal
condition such that morphisms are invertible. If moreover $X_{0}\simeq*$, then
$X$ is called an $\infty$-group (in $\mathsf{H}$). Write
$\operatorname{Grp}(\mathsf{H})$ for the full subcategory of
$\mathsf{H}^{\Delta^{\mathrm{op}}}$ spanned by $\infty$-groups in
$\mathsf{H}$.

For $G$ an $\infty$-group, we think of $G_{1}$ as the underlying object of
$G$, and often conflate the two in our notation. We write $B(*,G)$ to
specifically mean the full simplicial diagram $[n]\mapsto G_{n}\simeq
G^{\times n}$, and call it the _bar construction_.

######  Example 2.2.

An $\infty$-group(oid) in $\mathsf{Set}$ is a group(oid) in the classical
sense. An $\infty$-group in $\mathsf{Spc}$ is a loop space, which one can
think of as a topological group where the group structure is relaxed in a
homotopy coherent way. If $\mathsf{H}$ is the category of derived stacks on an
$\infty$-site $\mathsf{C}$, then $\operatorname{Grp}(\mathsf{H})$ is
equivalent to $\operatorname{Grp}(\mathsf{Spc})$-valued stacks on
$\mathsf{C}$. In particular, classical group stacks are $\infty$-groups in
$\mathsf{Stk}$.

For an $\infty$-category $\mathsf{C}$, write $\operatorname{Arr}(\mathsf{C})$
for the arrow category $\mathrm{Fun}(\Delta[1],\mathsf{C})$, and let
$\operatorname{Epi}(\mathsf{C})$ be the full subcategory of
$\operatorname{Arr}(\mathsf{C})$ spanned by the effective epimorphisms. For an
object $T$ in $\mathsf{C}$, write $\operatorname{Epi}_{T/}(\mathsf{C})$ for
the full subcategory of $\operatorname{Epi}(\mathsf{C})$ spanned by morphisms
with source $T$.

######  Proposition 2.3.

$\infty$-groupoids in $\mathsf{H}$ are effective, meaning that the Čech nerve
construction gives an adjoint equivalence of $\infty$-categories
$\check{C}\colon\operatorname{Epi}(\mathsf{H})\simeq\operatorname{Grpd}(\mathsf{H})$,
with left adjoint induced by taking the colimit. Restricting to
$\operatorname{Grp}(\mathsf{H})$ gives an adjoint equivalence

$\displaystyle
B:\operatorname{Grp}(\mathsf{H})\rightleftarrows\operatorname{Epi}_{*/}(\mathsf{H}):\Omega$

where $\Omega(*\to X)$ is the derived based loop stack $*\times_{X}*$ (with
suppressed $\infty$-group action).

Let $G$ be an $\infty$-group in $\mathsf{H}$ and $X$ an object in
$\mathsf{H}$. Then a $G$-object in $\mathsf{H}$ with underlying object $X$ is
an $\infty$-groupoid $B(X,G)$ over $B(*,G)$ of the form $[n]\mapsto
G^{n}\times X$ such that $d_{1}\colon G\times X\to X$ is the projection.
Informally, we say that $X$ is endowed with a $G$-action $\sigma\colon G\times
X\to X$ if there is a $G$-object $B(X,G)$ such that $d_{0}\simeq\sigma$, and
refer to $B(X,G)$ as the bar-construction of $\sigma$.

The $\infty$-category of $G$-objects is the full subcategory of
$\mathrm{Fun}(\Delta^{\mathrm{op}},\mathsf{H})_{B(*,G)}$ spanned by
$G$-objects. Informally, if $X,Y$ are endowed with a $G$-action, then a
$G$-equivariant morphism $f\colon X\to Y$ is a morphism of the corresponding
$G$-objects $f_{\bullet}\colon B(X,G)\to B(Y,G)$ such that $f_{1}\simeq f$.
For a $G$-object $B(X,G)$, we write the colimit of the diagram $[n]\mapsto
G^{n}\times X$ as $[X/G]$. If $X=*$, we simply write $BG$. Then the
$\infty$-category of $G$-objects is equivalent to the $\infty$-category
$\mathsf{H}_{/BG}$.

Let $f\colon T\to[X/G]$ be a morphism. Pulling back along the projection map,
we obtain a $G$-object $P$ and a $G$-equivariant morphism $P\to X$, such that
$[P/G]\to[X/G]$ is equivalent to $f$. A morphism $P\to T$ obtained this way is
a $G$-torsor or $G$-bundle.

For an object $V$ in $\mathsf{H}$, let $\underline{\mathrm{Aut}}(V)$ be the
internal automorphism $\infty$-group of $V$. This is constructed as follows.
Up to size-issues which we will ignore, there is an object classifier
$\widehat{\mathrm{Ob}(\mathsf{H})}\to\mathrm{Ob}(H)$, meaning that any object
$X$ of $\mathsf{H}$ fits in a Cartesian diagram

${X}$${\widehat{\mathrm{Ob}(\mathsf{H})}}$${*}$${\mathrm{Ob}(\mathsf{H}).}$

Then $B\underline{\mathrm{Aut}}(V)$ is the $\infty$-image of the morphism
$*\to\mathrm{Ob}(\mathsf{H})$ which classifies $V\to*$, where the
$\infty$-image is by definition the colimit of the Čech nerve of
$*\to\mathrm{Ob}(\mathsf{H})$. In particular, $*\to
B\underline{\mathrm{Aut}}(V)$ is an effective epimorphism, and
$\underline{\mathrm{Aut}}(V)$ is by definition the associated $\infty$-group.

For $G$ an $\infty$-group in $\mathsf{H}$, the adjoint action of $G$ on itself
is defined as the $G$-object associated to the projection $\mathscr{L}(BG)\to
BG$. Here, $\mathscr{L}(BG)$ is the derived (free) loop stack, which is the
self-intersection of the diagonal of $BG$, see §4.1. We write
$G_{\mathrm{ad}}$ for this $G$-object, which has $G$ as underlying object. By
definition, this means that $[G_{\mathrm{ad}}/G]\simeq\mathscr{L}(BG)$.

######  Definition 2.4.

For $\mathsf{H}=\mathsf{Stk}$, we introduce the following terminology. An
$\infty$-group object, whose underlying object in $\mathsf{Stk}$ is a derived
stack/algebraic stack/scheme, is called a derived group stack/algebraic
stack/scheme. For a derived group scheme $G$, a derived closed subgroup scheme
is a morphism $H\to G$ of derived group schemes which is a closed immersion on
underlying derived schemes. Beware that, despite the name, a derived closed
subgroup scheme $H\to G$ is in general not a monomorphism.

###  2.2. Derived projective spectra

Let $X\in\mathsf{Stk}$ be algebraic, let $\mathbb{M}$ be either $\mathbb{N}$
or $\mathbb{Z}$. As in the ungraded case, we have an $\infty$-category
$\mathrm{QAlg}^{\mathbb{M}}(X)$ of $\mathbb{M}$-graded, quasi-coherent
$\mathcal{O}_{X}$-algebras. The operations in the affine picture on
$\mathbb{N}$-graded and $\mathbb{Z}$-graded algebras mentioned in §1.5 carry
over to the global picture, for which we use the same notation.

Let $\mathcal{B}\in\mathrm{QAlg}^{\mathbb{N}}(X)$ be given. The irrelevant
ideal $\mathcal{B}_{+}$ of $\mathcal{B}$ is the fiber of the map
$\mathcal{B}\to\mathcal{B}_{0}$. Write
$V(\mathcal{B}_{+})\to\operatorname{Spec}\mathcal{B}$ for the closed immersion
$\operatorname{Spec}(\mathcal{B}_{0})\to\operatorname{Spec}\mathcal{B}$. The
$\mathbb{N}$-grading on $\mathcal{B}$ induces a $\mathbb{G}_{m}$-action on
$\operatorname{Spec}\mathcal{B}$, which restricts to a $\mathbb{G}_{m}$-action
on $(\operatorname{Spec}\mathcal{B})\setminus V(\mathcal{B}_{+})$. Define the
(relative) projective spectrum of $\mathcal{B}$ as the quotient

$\displaystyle\operatorname{Proj}\mathcal{B}\coloneqq[((\operatorname{Spec}\mathcal{B})\setminus
V(\mathcal{B}_{+}))/\mathbb{G}_{m}]$

Projective spectra commute with base-change in the obvious way. It follows
that $\operatorname{Proj}\mathcal{B}$ is always algebraic. Moreover, if
$\mathcal{B}$ is generated in homogeneous degree 1 over $\mathcal{O}_{X}$,
then for all derived schemes $f\colon T\to X$, the derived stack
$(\operatorname{Proj}\mathcal{B})\times_{X}T\simeq\operatorname{Proj}(f^{*}\mathcal{B})$
is a derived scheme as well. Here and in what follows, an $A$-algebra $B$ is
generated by a submodule $B_{0}\subset B$ if the natural map
$\mathrm{LSym}_{A}(B_{0})\to B$ is surjective (on $\pi_{0}$).

###  2.3. Weil restrictions

Let $f\colon X\to Y$ be a morphism of stacks. Consider the pullback functor
$f^{*}\colon\mathsf{Stk}_{Y}\to\mathsf{Stk}_{X}$. Suppose that $f^{*}$ has a
right adjoint $f_{*}$. Then, for $Z\to X$, we call $f_{*}Z$ the Weil
restriction of $Z$ along $f$, written $\mathrm{Res}_{f}(Z)$.

######  Remark 2.5.

Although in general the existence of the Weil restriction
$\mathrm{Res}_{f}(-)$ is only a set-theoretic issue, we remark that it exists
without any such issues by [SAG, Con. 19.1.2.3] when $f$ is representable.
Although Lurie’s result is more general, and there are more general statements
still, this version covers all cases that we will encounter.

Since colimits are universal in any $\infty$-topos (up to size issues), the
$\infty$-category $\mathsf{Stk}$ is Cartesian closed. For a morphism of
derived stacks $X\to Y$, the right adjoint of
$X\times_{Y}(-)\colon\mathsf{Stk}_{Y}\to\mathsf{Stk}_{Y}$ is written
$\underline{\operatorname{Map}}_{/Y}(X,-)$, and is called the derived mapping
stack. Then $\mathrm{Res}_{f}(Z)$ is alternatively given by the stack

(2.1)
$\displaystyle\mathrm{Res}_{f}(Z)=\underline{\operatorname{Map}}_{/Y}(X,Z)\times_{\underline{\operatorname{Map}}_{/Y}(X,X)}Y$

Under certain finiteness conditions, the derived stack $\mathrm{Res}_{f}Z$
will be algebraic, see [HLP19, Thm. 5.1.1], [HKR22]. In terms of higher
derived stacks, the algebraicity of $\mathrm{Res}_{f}Z$ is controlled by the
algebraicity of $Z\to X$ and the Tor-amplitude of $X\to Y$. Instead of giving
a general statement, we will indicate the algebraicity of the Weil restriction
in particular cases when relevant.

###  2.4. Derived Rees algebras

Endow $\mathbb{A}^{1}=\operatorname{Spec}\mathbb{C}[t^{-1}]$ with the
$\mathbb{G}_{m}$-action which corresponds to declaring $t^{-1}$ to be of
homogeneous degree $-1$. For $X\in\mathsf{Stk}$ algebraic, consider the map
$\zeta_{X}\colon B\mathbb{G}_{m}\times
X\to[\mathbb{A}^{1}/\mathbb{G}_{m}]\times X$ induced by the zero section. Then
we can Weil-restrict along $\zeta_{X}$, i.e., we have a right adjoint
$\mathrm{Res}_{\zeta}\colon\mathsf{Stk}_{B\mathbb{G}_{m}\times
X}\to\mathsf{Stk}_{[\mathbb{A}^{1}/\mathbb{G}_{m}]\times X}$ to the pullback
functor.

Let now $Z\to X$ be a morphism of derived stacks, with $X$ still algebraic.
Define the deformation space of $Z\to X$ as the Weil restriction
$\mathscr{D}_{Z/X}\coloneqq\mathrm{Res}_{\zeta_{X}}(Z\times B\mathbb{G}_{m})$
of $Z\times B\mathbb{G}_{m}$ along $\zeta_{X}$. Write $D_{Z/X}$ for the
pullback of $\mathscr{D}_{Z/X}$ along $\mathbb{A}^{1}\times
X\to[\mathbb{A}^{1}/\mathbb{G}_{m}]\times X$. In [HKR22] it is shown that the
deformation space is not only functorial in $Z$, but also in the morphism
$Z\to X$. The resulting functor $\mathscr{D}_{(-)/(-)}$ is a right adjoint,
hence in particular is stable under base-change. Consequently, the same is
true for $D_{(-)/(-)}$.

From here on, let $Z\to X$ be a closed immersion of derived algebraic stacks.
Then $D_{Z/X}\to\mathbb{A}^{1}\times X$ is affine by [Hek21], and $D_{Z/X}$
carries a $\mathbb{G}_{m}$-action such that
$[D_{Z/X}/\mathbb{G}_{m}]\simeq\mathscr{D}_{Z/X}$. We define the (derived)
extended Rees algebra of $Z$ over $X$ as the $\mathbb{Z}$-graded, quasi-
coherent $\mathcal{O}_{X}[t^{-1}]$-algebra $\mathcal{R}_{Z/X}^{\mathrm{ext}}$
such that $D_{Z/X}=\operatorname{Spec}\mathcal{R}_{Z/X}^{\mathrm{ext}}$. We
let $\mathcal{R}_{Z/X}\coloneqq(\mathcal{R}_{Z/X}^{\mathrm{ext}})_{\geq 0}$ be
the Rees algebra of $Z$ over $X$. In the affine case, say with
$Z=\operatorname{Spec}B$ and $X=\operatorname{Spec}A$, we write
$R_{B/A}^{\mathrm{ext}},R_{B/A}$ for the algebras corresponding to the quasi-
coherent algebras $\mathcal{R}_{Z/X}^{\mathrm{ext}},\mathcal{R}_{Z/X}$.

The main properties of this construction are as follows. The (extended) Rees
algebra is stable under base-change in the obvious way. Since $Z\to X$ is
affine, we can write $Z=\operatorname{Spec}\mathcal{B}$ for some quasi-
coherent $\mathcal{O}_{X}$-algebra $\mathcal{B}$. We can then recover
$\mathcal{B}$ from the Rees algebra as
$(\mathcal{R}^{\mathrm{ext}}_{Z/X}/(t^{-1}))_{0}\simeq\mathcal{B}$. In fact,
the pullback of $D_{Z/X}\to\mathbb{A}^{1}\times X$ to $\\{0\\}\times X$ gives
the normal bundle $\mathbb{N}_{Z/X}$, living over $\\{0\\}\times X$ via the
map $Z\to\\{0\\}\times X$.

######  Example 2.6.

Let $A\in\mathsf{Alg}$, let $\sigma_{1},\dots,\sigma_{k}$ be cycles
$\sigma_{i}\in\pi_{n_{i}}(A)$, and let $B$ be the finite quotient
$A/(\sigma_{1},\dots,\sigma_{k})$. Then

$\displaystyle
R^{\mathrm{ext}}_{B/A}\simeq\frac{A[t^{-1},u_{1},\dots,u_{n}]}{(u_{1}t^{-1}-\sigma_{1},\dots,u_{n}t^{-1}-\sigma_{n})}$

where the $u_{i}$ are free in homogeneous degree 1 and homological degree
$n_{i}$.

###  2.5. Derived blow-ups and virtual Cartier divisors

We continue with the closed immersion $Z\to X$ of derived algebraic stacks. We
define the derived blow-up of $X$ in $Z$ as the projective spectrum of
$\mathcal{R}_{Z/X}$ over $X$, written $\operatorname{Bl}_{Z}X$.

The basic properties here are as follows. The construction is stable under
arbitrary base-change in the obvious way, and if $Z,X$ are derived schemes,
then so is $\operatorname{Bl}_{Z}X$. It follows that, in general,
$\operatorname{Bl}_{Z}X$ is a derived algebraic stack. The structure map
$\operatorname{Bl}_{Z}X\to X$ is always an equivalence outside $Z$. Over $Z$,
we have the exceptional divisor $E_{Z}X$, which is the projective normal
bundle $\mathbb{P}(N_{Z/X})=[(\mathbb{N}_{Z/X}\setminus 0)/\mathbb{G}_{m}]$,
and lives over $Z\to X$ as a virtual Cartier divisor (see below). We recover
the classical blow-up of $X_{\mathrm{cl}}$ in $Z_{\mathrm{cl}}$ as the
schematic closure of
$(\operatorname{Bl}_{Z}X)_{\mathrm{cl}}\setminus(E_{Z}X)_{\mathrm{cl}}$ inside
$(\operatorname{Bl}_{Z}X)_{\mathrm{cl}}$.

Let $j\colon D\to T$ be a closed immersion of derived algebraic stacks. Then
$j$ is called quasi-smooth of virtual codimension $n$ if, for any derived
scheme $T^{\prime}\to T$, there are locally defined sections
$f_{1},\dots,f_{n}\in\mathcal{O}_{T^{\prime}}$ such that, Zariski locally on
$T^{\prime}$, the pullback $D\times_{T}T^{\prime}$ is the derived vanishing
locus
$V(f_{1},\dots,f_{n})=\operatorname{Spec}(\mathcal{O}_{T^{\prime}}/(f_{1},\dots,f_{n}))$.
A virtual Cartier divisor is a quasi-smooth closed immersion of virtual
codimension 1.

######  Example 2.7.

The map $\zeta\colon B\mathbb{G}_{m}\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ is a
virtual Cartier divisor. In fact, this is the universal example: for any
virtual Cartier divisor $j\colon D\to T$ there is a map $g\colon
T\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ which classifies $j$ in the sense that
$j$ is the pullback of $\zeta$ along $g$.

######  Definition 2.8.

Let $Z\to X$ be a morphism of derived algebraic stacks, and $T$ a derived
algebraic stack. A _virtual Cartier divisor over $Z\to X$ (on $T$)_ is a
$T$-point of $\mathscr{D}_{Z/X}$.

The universality of $\zeta$ implies that a virtual Cartier divisor over $Z\to
X$ is a commutative diagram of the form

(2.2) ${D}$${T}$${Z}$${X}$$\scriptstyle{g}$

where $D\to T$ is a virtual Cartier divisor.

######  Definition 2.9.

If $Z\to X$ is a closed immersion, then a virtual Cartier divisor over $Z\to
X$ classified by the diagram 2.2 is _strict_ when the underlying classical
diagram is Cartesian, and the induced homomorphism $g^{\ast}N_{Z/X}\to
N_{D/T}$ is surjective on $\pi_{0}$. (This terminology deviates from [KR18],
but agrees with [HKR22]).

The derived blow-up represents strict virtual Cartier divisors over $Z\to X$
in the sense that $\operatorname{Bl}_{Z}X(T)$ is equivalent to the space of
strict virtual Cartier divisors over $Z\to X$ on $T$, with the exceptional
divisor as the universal example.

###  2.6. Properties and formulas of derived blow-ups

We now prove and collect a few additional properties of derived blow-ups,
which we will use in Section 5.

As a consequence of the functorial description of the derived blow-up, we have
the following proposition, which shows that the classical truncation
$(\operatorname{Bl}_{Z}U)_{\mathrm{cl}}$ depends on $Z_{\leq
1}\coloneqq\pi_{\leq 1}Z$, hence in particular is invariant under attaching
$n$-cells to $\mathcal{O}_{Z}$ for $n\geq 2$.

######  Proposition 2.10.

Suppose that $Z^{\prime}\to Z$ is a closed embedding such that
$Z^{\prime}_{\leq 1}\to Z_{\leq 1}$ is an equivalence. Then the natural
morphism $\operatorname{Bl}_{Z^{\prime}}U\to\operatorname{Bl}_{Z}U$ induces an
isomorphism on classical truncations.

###### Proof.

The classical truncations $(\operatorname{Bl}_{Z^{\prime}}U)_{\mathrm{cl}}$
and $(\operatorname{Bl}_{Z}U)_{\mathrm{cl}}$ are determined by their
$T$-points, where $T$ is a classical affine scheme, and a virtual Cartier
divisor $D\to T$ is locally (on $T$) the vanishing locus of a section
$f\in\mathcal{O}_{T}$. In particular, since $T$ is classical, the truncation
$D_{\leq 1}\to D$ is an equivalence. Since also $Z^{\prime}_{\leq 1}\simeq
Z_{\leq 1}$, any map $D\to Z$ factors uniquely through $Z^{\prime}$. ∎

We also have the following simple proposition regarding classical truncations
of derived blow-ups and pullback squares.

######  Proposition 2.11.

Suppose we are given a Cartesian diagram

${Z^{\prime}}$${U^{\prime}}$${Z}$${U}$

of derived schemes, such that $U^{\prime}_{\mathrm{cl}}\to U_{\mathrm{cl}}$ is
an isomorphism. Then the canonical map
$(D_{Z^{\prime}/U^{\prime}})_{\mathrm{cl}}\longrightarrow(D_{Z/U})_{\mathrm{cl}}$
is an isomorphism.

###### Proof.

Since the deformation space is stable under pullbacks, we have

$\displaystyle(D_{Z^{\prime}/U^{\prime}})_{\mathrm{cl}}\simeq(D_{Z/U}\times_{U}U^{\prime})_{\mathrm{cl}}\simeq(D_{Z/U})_{\mathrm{cl}}$

where the last equivalence follows since $(-)_{\mathrm{cl}}$ commutes with
pullbacks, and since $U_{\mathrm{cl}}\simeq U^{\prime}_{\mathrm{cl}}$ by
assumption. ∎

In order to perform explicit computations, we will also need the following
lemma. Note that $\mathscr{D}_{U/U}\simeq
Y\times[\mathbb{A}^{1}/\mathbb{G}_{m}]$, hence $D_{U/U}\simeq
Y\times\mathbb{A}^{1}$.

######  Lemma 2.12.

Let a sequence $Z\to U\to V$ of derived stacks be given. Then the natural map

$\displaystyle\mathscr{D}_{Z/U}\to\mathscr{D}_{U/U}\times_{\mathscr{D}_{U/V}}\mathscr{D}_{Z/V}$

induced by naturality of $\mathscr{D}_{(-)/(-)}$, is an equivalence.
Consequently, the same holds true if we replace $\mathscr{D}_{(-)/(-)}$ by
$D_{(-)/(-)}$ and $\mathsf{Stk}_{[\mathbb{A}^{1}/\mathbb{G}_{m}]}$ by
$\mathsf{Stk}_{\mathbb{A}^{1}}^{\mathbb{G}_{m}}$.

###### Proof.

This will appear in [HKR22]. ∎

Suppose now that $A$ is any algebra and we are given a sequence of closed
embeddings $Z\to W\to U\to V=\operatorname{Spec}A$, corresponding to

$\displaystyle A\to B\to C\to D$

where $B=A/(f_{1},...,f_{n})$ for certain $f_{i}\in\pi_{0}A$, and
$C=B/(\sigma_{1},...,\sigma_{k})$ for certain $\sigma_{i}\in\pi_{d_{i}}B$, and
$D=A/(g_{1},...,g_{m})$ for certain $g_{i}\in\pi_{0}A$.

We use multi-index notation, so that $B=A/(\underline{f})$ and so on. From the
theory of Rees algebras, we have a diagram

${(R^{\mathrm{ext}}_{U/V})_{1}}$${A}$${B}$${(R^{\mathrm{ext}}_{Z/V})_{1}}$${A}$${D,}$

where the rows are fiber sequences. Write
$R^{\mathrm{ext}}_{U/V}=A[t^{-1},\underline{v}]/(t^{-1}\underline{v}-\underline{f})$,
and
$R^{\mathrm{ext}}_{Z/V}=A[t^{-1},\underline{w}]/(t^{-1}\underline{w}-\underline{g})$,
and $p$ for the composition
$\pi_{0}A[t^{-1},\underline{v}]\to\pi_{0}R^{\mathrm{ext}}_{U/V}\to\pi_{0}R^{\mathrm{ext}}_{Z/V}$.
Here, $\underline{v}=(v_{1},\dots,v_{n})$ and
$\underline{w}=(w_{1},\dots,w_{m})$ are free in homological degree 0 and
homogeneous degree 1.

For each $i$, let $v_{i}^{\prime}$ be the image of $v_{i}$ under $p$. Since
$p$ sends $t^{-1}v_{i}-f_{i}$ to zero, we get
$\lambda_{ji}\in\pi_{0}A[t^{-1},\underline{w}]$ such that

$\displaystyle
t^{-1}v_{i}^{\prime}-f_{i}=\sum\nolimits_{j}\lambda_{ji}(t^{-1}w_{j}-g_{j})$

holds in $\pi_{0}A[t^{-1},\underline{w}]$. Putting $t^{-1}=0$ gives us that
$f_{i}=\sum_{j}\lambda_{ji}g_{j}$, in particular that
$\lambda_{ji}\in\pi_{0}A$. Putting $t^{-1}=1$ then gives
$v_{i}^{\prime}=\sum_{j}\lambda_{ji}w_{j}$. It follows that
$R_{U/V}^{\mathrm{ext}}\to R_{Z/V}^{\mathrm{ext}}$ sends $v_{i}$ to
$\sum_{j}\lambda_{ij}g_{j}$, in the sense that precomposing with
$A[t^{-1},\underline{v}]\to R_{U/V}^{\mathrm{ext}}$ gives the unique map
$A[t^{-1},\underline{v}]\to R_{Z/V}^{\mathrm{ext}}$ of $A[t^{-1}]$-algebras
that sends $v_{i}$ to $\sum_{j}\lambda_{ij}g_{j}$.

Write
$R^{\mathrm{ext}}_{W/U}=B[t^{-1}\underline{u}]/(t^{-1}\underline{u}-\underline{\sigma})$
where $\underline{u}=(u_{1},\dots,u_{k})$ are free in homogeneous degree 1 and
homological degrees $d_{1},\dots,d_{k}$. Then, with the same argument as
before, we get elements $(\tau_{1},\dots,\tau_{k})$ in
$\pi_{*}R_{Z/U}^{\mathrm{ext}}$, with $\tau_{i}$ in homological degree $d_{i}$
and homogeneous degree 1 such that
$\pi_{*}R^{\mathrm{ext}}_{W/U}\to\pi_{*}R^{\mathrm{ext}}_{Z/U}$ sends $u_{i}$
to $\tau_{i}$.

######  Proposition 2.13.

We have

(2.3) $\displaystyle
R^{\mathrm{ext}}_{Z/U}\simeq\frac{A[t^{-1},\underline{w}]}{(t^{-1}\underline{w}-\underline{g},\sum_{j}\lambda_{1j}w_{j},\dots,\sum_{j}\lambda_{nj}w_{j})}$

where $\underline{w}=(w_{1},\dots,w_{m})$ are in homological degree 0 and
homogeneous degree 1. Likewise, we have

(2.4) $\displaystyle R^{\mathrm{ext}}_{Z/W}\simeq
R^{\mathrm{ext}}_{Z/U}/(\underline{\tau})$

where $\underline{\tau}=(\tau_{1},\dots,\tau_{k})$ are the elements with
$\tau_{i}$ in homological degree $d_{i}$ and homogeneous degree 1 described
above.

######  Remark 2.14.

As the proof will make clear, there is only a contractible space of choices
involved here, since the equivalence comes about from a description of
canonically given maps.

###### Proof.

By Lemma 2.12, we have $R^{\mathrm{ext}}_{Z/U}\simeq
R^{\mathrm{ext}}_{U/U}\otimes_{R^{\mathrm{ext}}_{U/V}}R^{\mathrm{ext}}_{Z/V}$.
We thus get the following diagram, where the transition maps in the bottom
right corner are the canonical maps induced by functoriality of
$R^{\mathrm{ext}}_{(-)/(-)}$.

$\textstyle{\mathbb{Z}[\underline{F}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{F}\mapsto
0}$$\scriptstyle{\underline{F}\mapsto
t^{-1}\underline{v}-\underline{f}}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A[t^{-1},\underline{v}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{v}\mapsto
0}$$\textstyle{\frac{A[t^{-1},\underline{v}]}{(t^{-1}\underline{v}-\underline{f})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{i}\mapsto\sum_{j}\lambda_{ij}w_{j}}$$\textstyle{\frac{A[t^{-1},\underline{w}]}{(t^{-1}\underline{w}-\underline{g})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A[t^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B[t^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{Z/U}^{\mathrm{ext}}}$

where all squares are pushouts, and $\underline{v}=(v_{1},\dots,v_{n})$ are
free in homogeneous degree 1 and homological degree 0, and likewise for
$\underline{F}=(F_{1},\dots,F_{n})$. The first claim follows.

For the second claim, we use $R^{\mathrm{ext}}_{Z/W}\simeq
R^{\mathrm{ext}}_{W/W}\otimes_{R^{\mathrm{ext}}_{W/U}}R^{\mathrm{ext}}_{Z/U}$
to get the diagram

$\textstyle{\mathbb{Z}[\underline{U}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{U}\mapsto\underline{u}t^{-1}-\underline{\sigma}}$$\scriptstyle{\underline{U}\mapsto
0}$$\textstyle{B[t^{-1},\underline{u}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{u}\mapsto
0}$$\textstyle{B[t^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{B[t^{-1},\underline{u}]}{(\underline{u}t^{-1}-\underline{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{u}\mapsto\underline{\tau}}$$\textstyle{C[t^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R^{\mathrm{ext}}_{Z/U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{Z/W}^{\mathrm{ext}}}$

where $\underline{U}=(U_{1},\dots,U_{k})$ are free in homogeneous degree 1 and
homological degrees $d_{1},\dots,d_{k}$. Since all squares are pushouts, the
second claim follows. ∎

##  3\. Equivariant derived algebraic geometry

A main ingredient in the canonical reduction of stabilizers algorithm for
algebraic stacks from [ER21] is the locus $X^{{\mathrm{max}}}$ of points with
maximal-dimensional stabilizers, for a given algebraic stack $X$. This locus
is a closed substack of $X$ in the case that $X$ admits a good moduli space,
as shown in [ER21, Prop. B.4].

In the next section, §4, we give a derived analogue of $X^{{\mathrm{max}}}$ in
the case that $X$ is a derived algebraic stack for which $X_{\mathrm{cl}}$
admits a good moduli space. In order to do this, in this section we give a
treatment of several elements of equivariant geometry and study fixed loci in
the derived setting. This allows us to obtain equivariant versions of the
results of the previous section, including equivariant derived blow-ups. We
note that there is some overlap between what we present here and what is
presented in [AKL+22, §A.3].

The statements of this section are derived analogues of familiar statements
from classical equivariant geometry, and the arguments follow similar lines to
arguments for derived analogues of ordinary (i.e., non-equivariant) classical
geometry.

###  3.1. Equivariant rings, modules and stacks

Fix a reductive algebraic group $G$ over $\mathbb{C}$. By definition, $G$ is a
linear algebraic group, hence a smooth closed subscheme of some
$\operatorname{GL}_{n}$ (smooth by Cartier’s theorem). A classical $G$-module
$V$ will be an algebraic representation of $G$ on a vector space $V$, i.e., a
morphism of group schemes $G\to\operatorname{GL}(V)$. Write $\mathsf{Rep}^{G}$
for the category of finite dimensional $G$-modules $V$. Recall, since
algebraic representations are rational, every $M\in\mathsf{Rep}^{G}$ splits
into a direct sum of simple modules.

######  Definition 3.1.

Let the symmetric monoidal $\infty$-category of $G$-modules, written
$\mathsf{Mod}^{G}$, be the stabilization of
$\mathsf{P}_{\Sigma}(\mathsf{Rep}_{G})$ with symmetric monoidal structure
induced from $\mathsf{Rep}_{G}$, and let $\mathsf{Alg}^{G}$ be the
$\infty$-category of connective $\mathbb{E}_{\infty}$-algebras in
$\mathsf{Mod}^{G}$.

Recall that the heart $\operatorname{QCoh}(BG)^{\heartsuit}$ is the category
of classical $G$-modules. In particular, we have a canonical inclusion
$\mathsf{Rep}_{R}\to\operatorname{QCoh}(BG)$. Since
$\operatorname{QCoh}(BG)_{\geq 0}$ is generated under colimits by the image of
$\mathsf{Rep}_{G}\to\operatorname{QCoh}(BG)$, the canonical map
$\mathsf{P}_{\Sigma}(\mathsf{Rep}_{G})\to\operatorname{QCoh}(BG)_{\geq 0}$ is
an equivalence, hence $\mathsf{Mod}^{G}\simeq\operatorname{QCoh}(BG)$. We
endow $\mathsf{Mod}^{G}$ with the canonical $t$-structure induced from
$\operatorname{QCoh}(BG)$.

Since the $\infty$-category $\mathsf{Mod}^{G}_{\geq 0}$ of connective
$G$-modules is presentable and the tensor product of $\mathsf{Mod}^{G}$
commutes with colimits in each variable separately, the forgetful functor
$\mathsf{Alg}^{G}\to\mathsf{Mod}^{G}_{\geq 0}$ admits a left adjoint, written
as

$\displaystyle\mathrm{LSym}^{G}\colon\mathsf{Mod}^{G}_{\geq
0}\to\mathsf{Alg}^{G}.$

Let $\mathsf{Poly}^{G}$ be the full subcategory of $\mathsf{Alg}^{G}$ spanned
by objects of the form $\mathrm{LSym}^{G}(M)$, for $M\in\mathsf{Rep}^{G}$.
Then, since we are in characteristic zero and the $G$-actions we consider are
rational, $\mathsf{Alg}^{G}$ is freely generated under sifted colimits by
$\mathsf{Poly}^{G}$.

The forgetful functor $\mathsf{Mod}^{G}\to\mathsf{Mod}$ is symmetric monoidal,
which gives us a canonical forgetful functor
$\mathsf{Alg}^{G}\to\mathsf{Alg}$. Also, the functor
$\mathsf{Mod}\to\mathsf{Mod}^{G}$ which endows a module with trivial
$G$-action preserves colimits. We write the right adjoint of this functor as
$(-)^{\mathrm{fix}}$. Note that $(-)^{\mathrm{fix}}$ corresponds to taking the
classical fixed part on $\mathsf{Rep}^{G}$.

######  Definition 3.2.

Write $\mathsf{Aff}^{G}$ for the $\infty$-category
$(\mathsf{Alg}^{G})^{\mathrm{op}}$. Endow $\mathsf{Aff}^{G}$ with the étale
topology by declaring a family $\\{U_{i}\to X\\}$ in $\mathsf{Aff}^{G}$ to be
a cover if the underlying family of derived affine schemes is an étale cover.
Write $\mathsf{Stk}^{G}$ for the $\infty$-category of derived stacks on
$\mathsf{Aff}^{G}$ with respect to the étale topology, and
$\operatorname{Spec}^{G}\colon(\mathsf{Alg}^{G})^{\mathrm{op}}\to\mathsf{Stk}^{G}$
for the Yoneda embedding.

For a derived stack $X$, write $\mathsf{Aff}(X)$ for the $\infty$-category of
stacks which are affine over $X$, and $\mathrm{QAlg}(X)$ for the
$\infty$-category of quasi-coherent $\mathcal{O}_{X}$-algebras, so that we
have an equivalence

$\displaystyle\operatorname{Spec}(-)\colon\mathrm{QAlg}(X)^{\mathrm{op}}\to\mathsf{Aff}(X).$

Since we are in characteristic zero, $\mathrm{QAlg}(X)$ is also the
$\infty$-category of connective $\mathbb{E}_{\infty}$-algebras in
$\operatorname{QCoh}(X)$. The case $X=BG$ therefore gives us

$\displaystyle\mathsf{Aff}^{G}\simeq\mathsf{Aff}(BG)$

Recall that pulling back along the quotient map $\varphi\colon*\to BG$ induces
an equivalence between $\mathsf{Stk}_{BG}$ and the $\infty$-category of stacks
with a $G$-action. Since $\varphi$ is affine, this implies that
$\mathsf{Aff}^{G}$ is also the $\infty$-category of affine derived schemes
with a $G$-action.

######  Proposition 3.3.

The $\infty$-category $\mathsf{Stk}^{G}$ is equivalent to $\mathsf{Stk}_{BG}$.

###### Proof.

Let $T$ be any derived stack. Since $T$ has affine diagonal, the
$\infty$-category $\mathsf{Aff}_{/T}$ of derived affine schemes over $T$ is a
full subcategory of $\mathsf{Aff}(T)$. Composition with the inclusion
$f\colon\mathsf{Aff}_{/T}\to\mathsf{Aff}(T)$ and left Kan extension gives an
adjunction

$\displaystyle
f_{!}:\mathsf{Stk}(\mathsf{Aff}_{/T})\rightleftarrows\mathsf{Stk}(\mathsf{Aff}(T)):f^{*}$

Since $f$ is fully faithful, so is $f_{!}$. Moreover, since
$\mathsf{Stk}_{T}\simeq\mathsf{Stk}(\mathsf{Aff}_{/T})$, the image of $f_{!}$
contains all stacks over $T$. Since $\mathsf{Stk}(\mathsf{Aff}(T))$ is
generated under colimits by stacks affine over $T$, it follows that $f_{!}$ is
essentially surjective. Hence
$\mathsf{Stk}_{/T}\simeq\mathsf{Stk}(\mathsf{Aff}(T))$, and putting $T=BG$
gives us what we want. ∎

Although we will use the equivalence $\mathsf{Stk}^{G}\simeq\mathsf{Stk}_{BG}$
seamlessly, we will not conflate a derived stack $X$ over $BG$ with the
corresponding derived $G$-stack $P\coloneqq X\times_{BG}\\{*\\}$. In
particular, if $X$ is endowed with a $G$-action as a derived stack over $BG$,
then this contains no information regarding the action of $P$.

######  Notation 3.4.

Let $G,G^{\prime}$ be reductive groups. The equivalences
$\mathsf{Stk}^{G}\simeq\mathsf{Stk}_{BG}$ and
$\mathsf{Stk}^{G^{\prime}}\simeq\mathsf{Stk}_{BG^{\prime}}$ induce
equivalences on functor-$\infty$-categories, which we write as

$\displaystyle
B:\mathrm{Fun}(\mathsf{Stk}^{G},\mathsf{Stk}^{G^{\prime}})\leftrightarrows\mathrm{Fun}(\mathsf{Stk}_{BG},\mathsf{Stk}_{BG^{\prime}}):\Omega$

######  Remark 3.5.

Consider the map $\psi\colon BG\to*$. Then the functor
$\mathsf{Stk}\to\mathsf{Stk}_{BG}$ given by pulling back along $\psi$ is right
adjoint to the functor $\mathsf{Stk}_{BG}\to\mathsf{Stk}$ given by composing
with $\psi$. Under the equivalence $\mathsf{Stk}_{BG}\simeq\mathsf{Stk}^{G}$,
this corresponds to the adjunction

$\displaystyle[-/G]:\mathsf{Stk}^{G}\rightleftarrows\mathsf{Stk}:\iota$

where $\iota$ endows a derived stack with the trivial action.

We also have an adjunction

$\displaystyle(-)\times G:\mathsf{Stk}\leftrightarrows\mathsf{Stk}^{G}:U$

induced by $\varphi\colon*\to BG$, where $U$ is the forgetful functor, and the
$G$-action on $Y\times G$ is given by translation, for $Y\in\mathsf{Stk}$.
Essentially, this is because any $G$-equivariant map $Y\times G\to X$ is
determined by its restriction to $Y\times\\{e\\}\to X$.

Let $B\in\mathsf{Alg}^{G}$. Define the $\infty$-category
$\mathsf{Mod}^{G}_{B}$ of $(G,B)$-modules as the $\infty$-category of
$B$-modules in $\mathsf{Mod}^{G}$, with canonical $t$-structure. Write
$\mathsf{Alg}^{G}_{B}$ for the slice category $(\mathsf{Alg}^{G})_{B/}$. As in
the absolute case, we have an adjunction

$\displaystyle\mathrm{LSym}^{G}_{B}:\mathsf{Mod}^{G}_{B}\leftrightarrows\mathsf{Alg}^{G}_{B}:U$

where $U$ is the forgetful functor. Let $\mathsf{Rep}^{G}_{B}$ be the
full-$\infty$-subcategory of $\mathsf{Mod}^{G}_{B}$ spanned by objects of the
form $B\otimes M$, where $M\in\mathsf{Rep}^{G}$. As shown in [BH23],
$\mathsf{Mod}_{B}^{G}$ is again a derived algebraic context, and
$\mathsf{Rep}^{G}_{B}$ generates $(\mathsf{Mod}^{G}_{B})_{\geq 0}$ under
sifted colimits.

As in the non-equivariant case, we can globalize this picture via a right Kan
extension to a functor

$\displaystyle\operatorname{QCoh}^{G}(-)\colon(\mathsf{Stk}^{G})^{\mathrm{op}}\to\mathsf{Cat}$

such that $\operatorname{QCoh}^{G}(X)$ is stable, presentable and symmetric
monoidal for $X$ algebraic. Moreover, for $f\colon X\to Y$ an equivariant map
of derived algebraic stacks, we have an equivariant pullback functor

(3.1)
$\displaystyle(f^{G})^{*}\colon\operatorname{QCoh}^{G}(Y)\to\operatorname{QCoh}^{G}(X)$

which is symmetric monoidal and a left adjoint.

With the same procedure, we get a functor $\mathrm{QAlg}^{G}(-)$. Now for
$X\in\mathsf{Stk}^{G}$ and $\mathcal{A}\in\mathrm{QAlg}^{G}(X)$, it holds that
$\mathcal{A}$ is an $\mathbb{E}_{\infty}$-algebra in
$\operatorname{QCoh}^{G}(X)$. We can thus define the $\infty$-category
$\operatorname{QCoh}_{\mathcal{A}}^{G}(X)$ of $(G,\mathcal{A})$-modules as the
$\infty$-category of $\mathcal{A}$-modules in $\operatorname{QCoh}^{G}(X)$.
The subcategory of $\operatorname{QCoh}_{\mathcal{A}}^{G}(X)$ spanned by
modules with trivial $G$-action is written
$\operatorname{QCoh}_{\mathcal{A}}(X)$, which is equivalent to the
$\infty$-category of $\mathcal{A}$-modules in $\operatorname{QCoh}(X)$.

###  3.2. Restriction and co-induction

Suppose that a reductive algebraic group $G$ acts trivially on
$A\in\mathsf{Alg}$. Then $\mathsf{Mod}^{G}_{A}$ is equivalent to the
$\infty$-category of $A$-modules endowed with a $G$-action, i.e., we have an
equivalence

(3.2)
$\displaystyle\mathsf{Mod}^{G}_{A}\simeq\mathsf{Mod}^{G}\otimes_{\mathsf{Mod}}\mathsf{Mod}_{A}\simeq\mathrm{Fun}_{\sigma}((\mathsf{Rep}^{G})^{\mathrm{op}},\mathsf{Mod}_{A})$

where
$\mathrm{Fun}_{\sigma}((\mathsf{Rep}^{G})^{\mathrm{op}},\mathsf{Mod}_{A})$ is
the $\infty$-category of functors
$(\mathsf{Rep}^{G})^{\mathrm{op}}\to\mathsf{Mod}_{A}$ that send finite
coproducts in $\mathsf{Rep}^{G}$ to products in $\mathsf{Mod}_{A}$, and
$(-)\otimes(-)$ is the Lurie tensor product, see [Rak20, Rem. 2.2.8].

Let now $f\colon G\to G^{\prime}$ be a homomorphism of reductive algebraic
groups, and let $G^{\prime}$ also act trivially on $A$. Write the induced
morphism on stacks as $\varphi\colon\operatorname{Spec}A\times
BG\to\operatorname{Spec}A\times BG^{\prime}$, and the adjunction
$\operatorname{QCoh}(\operatorname{Spec}A\times
BG^{\prime})\leftrightarrows\operatorname{QCoh}(\operatorname{Spec}A\times
BG)$ induced by $\varphi$ as

$\displaystyle\mathrm{Res}_{\varphi}:\mathsf{Mod}^{G^{\prime}}_{A}\leftrightarrows\mathsf{Mod}^{G}_{A}:\mathrm{cInd}_{\varphi}$

We call $\mathrm{Res}_{\varphi}$ the restriction functor, and
$\mathrm{cInd}_{\varphi}$ the co-induction functor.

Recall (see, e.g., [GR17, §3.2]), that $\mathrm{Res}_{\varphi}$ is symmetric
monoidal and that $\mathrm{cInd}_{\varphi}$ is right-lax symmetric monoidal.
Since $BG^{\prime}$ has affine and flat diagonal, $BG\to BG^{\prime}$ is flat
and cohomologically affine. It follows that the functors
$\mathrm{Res}_{\varphi}$ and $\mathrm{cInd}_{\varphi}$ are $t$-exact. The
adjunction therefore lifts to the level of algebras, which we again write as

$\displaystyle\mathrm{Res}_{\varphi}:\mathsf{Alg}^{G^{\prime}}_{A}\leftrightarrows\mathsf{Alg}^{G}_{A}:\mathrm{cInd}_{\varphi}$

######  Notation 3.6.

Consider the obvious maps
$\varphi\colon\operatorname{Spec}A\to\operatorname{Spec}A\times BG^{\prime}$
and $\psi\colon\operatorname{Spec}A\times BG\to\operatorname{Spec}A$. Then the
functor
$\mathrm{Res}_{\varphi}\colon\mathsf{Mod}^{G^{\prime}}_{A}\to\mathsf{Mod}_{A}$
is forgetting the $G^{\prime}$-action, which we will write as $(-)^{u}$; the
functor $\mathrm{Res}_{\psi}\colon\mathsf{Mod}_{A}\to\mathsf{Mod}^{G}_{A}$
endows an $A$-module with trivial $G$-action, which we will write as
$(-)^{\mathrm{tv}}$; and
$\mathrm{cInd}_{\psi}\colon\mathsf{Mod}^{G}_{A}\to\mathsf{Mod}_{A}$ sends an
$A$-module with $G$-action $M$ to the $G$-fixed points $M^{\mathrm{fix}}$.
Again, we use the same notation for the induced functors on algebras.

Globalizing $(-)^{\mathrm{tv}}\dashv(-)^{\mathrm{fix}}$ via right Kan
extensions gives us an adjunction

(3.3) $\displaystyle
i:\operatorname{QCoh}(X)\leftrightarrows\operatorname{QCoh}^{G}(X):(-)^{\mathrm{fix}}$

for any stack $X$ with trivial $G$-action.

######  Definition 3.7.

Let $X\in\mathsf{Stk}$ algebraic with trivial $G$-action and
$\mathcal{F}\in\operatorname{QCoh}^{G}(X)$ be given. Consider the counit
$\mathcal{F}^{\mathrm{fix}}\to\mathcal{F}$ of the adjunction
$i\dashv(-)^{\mathrm{fix}}$. The moving part of $\mathcal{F}$ is the cofiber
of the counit, written $\mathcal{F}\to\mathcal{F}^{\mathrm{mv}}$. We write
$\operatorname{QCoh}^{\mathrm{mv}}(X)$ for the essential image of the functor
$(-)^{\mathrm{mv}}\colon\operatorname{QCoh}^{G}(X)\to\operatorname{QCoh}^{G}(X)$.

Recall that a sequence of colimit-preserving functors

$\displaystyle\mathsf{C}\xrightarrow{i}\mathsf{D}\xrightarrow{g}\mathsf{E}$

between presentable stable $\infty$-categories is exact if $gi\simeq 0$, $i$
is fully faithful, and $\mathsf{E}$ is the Verdier quotient
$\mathsf{D}/\mathsf{C}$, i.e., it is the localization of $\mathsf{D}$ at the
morphisms for which the cofiber is in $\mathsf{C}$. Such a sequence is split-
exact if the right adjoints $i\dashv f$ and $g\dashv j$, which exist by the
adjoint functor theorem, are such that $fi\simeq\mathrm{id}_{\mathsf{C}}$ and
$gj\simeq\mathrm{id}_{\mathsf{E}}$.

######  Remark 3.8.

The proof of the third point of Proposition 3.9 is inspired by [BR19, App. A],
and uses the following facts on (split) exact sequences that can be found in
the detailed exposition [CDH+20, App. A] on the matter.

A sequence $\mathsf{C}\xrightarrow{i}\mathsf{D}\xrightarrow{g}\mathsf{E}$ with
adjoints $j,f$ as above is exact if and only if $i$ is a fully faithful
functor with essential image spanned by the objects $M\in\mathsf{D}$ such that
$gM\simeq 0$. If the sequence is moreover split, then the sequence
$\mathsf{E}\xrightarrow{j}\mathsf{D}\xrightarrow{f}\mathsf{C}$ is also a split
exact sequence. In this case, the square

${M}$${jg(M)}$${if^{\prime}(M)}$${if^{\prime}jg(M)}$

induced by the units of $g\dashv j$ and $f^{\prime}\dashv i$, is Cartesian,
for any $M\in\mathsf{D}$.

######  Proposition 3.9.

Let $X$ be a derived algebraic stack with a $G$-action and $\varphi_{X}\colon
X\to[X/G]$ the projection map. Consider the equivariant pullback functor

$\displaystyle(\varphi_{X}^{G})^{*}\colon\operatorname{QCoh}^{G}([X/G])\to\operatorname{QCoh}^{G}(X).$

as displayed in 3.1, where the $G$-action on $[X/G]$ is trivial.

  1. (i)

Composing $(\varphi_{X}^{G})^{*}$ with
$i_{[X/G]}\colon\operatorname{QCoh}([X/G])\to\operatorname{QCoh}^{G}([X/G])$
from (3.3) induces an equivalence

$\operatorname{QCoh}([X/G])\simeq\operatorname{QCoh}^{G}(X).$

  2. (ii)

Suppose that the $G$-action on $X$ is trivial and let $\psi_{X}\colon[X/G]\to
X$ be the structure map. Composing the adjunction
$\psi_{X}^{*}\dashv{\psi_{X}}_{*}$ with the equivalence from (i) yields the
adjunction

$\displaystyle
i:\operatorname{QCoh}(X)\leftrightarrows\operatorname{QCoh}^{G}(X):(-)^{\mathrm{fix}}$

from (3.3).

  3. (iii)

Suppose that the $G$-action on $X$ is trivial. Then the sequence

$\displaystyle\operatorname{QCoh}(X)\xrightarrow{\quad
i\quad}\operatorname{QCoh}^{G}(X)\xrightarrow{(-)^{\mathrm{mv}}}\operatorname{QCoh}^{\mathrm{mv}}(X)$

is split exact.

###### Proof.

(i). By descent, this is a local question, so we may assume that $X$ is
affine, say $X=\operatorname{Spec}A$. Let $\mathcal{A}\in\mathrm{QAlg}(BG)$
correspond to $A\in\mathsf{Alg}^{G}$ under the equivalence
$\mathsf{Alg}^{G}\simeq\mathrm{QAlg}(BG)$, so that
$[X/G]=\operatorname{Spec}\mathcal{A}$ and
$\operatorname{QCoh}([X/G])\simeq\operatorname{QCoh}_{\mathcal{A}}(BG)$. Now
the $G$-actions on $BG$, on $[X/G]$, and on $\mathcal{A}$ are all trivial.
Hence, with the same argument as in (3.2), we also have

$\displaystyle\operatorname{QCoh}^{G}([X/G])\simeq\operatorname{QCoh}_{\mathcal{A}}^{G}(BG)\simeq\operatorname{QCoh}^{G}(BG)\otimes_{\operatorname{QCoh}(BG)}\operatorname{QCoh}_{\mathcal{A}}(BG)$

Moreover,
$\operatorname{QCoh}^{G}(X)\simeq\mathsf{Mod}^{G}_{A}\simeq\operatorname{QCoh}_{\mathcal{A}}(BG)$
holds by definition of $\mathsf{Mod}^{G}_{A}$. The composition
$(\varphi_{X}^{G})^{*}\circ i_{[X/G]}$ is then equivalent to the composition

$\displaystyle\operatorname{QCoh}_{\mathcal{A}}(BG)\to\operatorname{QCoh}^{G}(BG)\otimes_{\operatorname{QCoh}(BG)}\operatorname{QCoh}_{\mathcal{A}}(BG)\to\operatorname{QCoh}_{\mathcal{A}}(BG)$

where the first arrow sends $M\in\operatorname{QCoh}_{\mathcal{A}}(BG)$ to the
external tensor product $\mathcal{O}_{BG}\boxtimes M$, where
$\mathcal{O}_{BG}$ has trivial $G$-action, and the second arrow sends
$N\boxtimes
K\in\operatorname{QCoh}^{G}(BG)\otimes_{\operatorname{QCoh}(BG)}\operatorname{QCoh}_{\mathcal{A}}(BG)$
to $N^{u}\otimes K$. The claim follows.

(ii). It suffices to show that $i\simeq(\varphi^{G}_{X})^{*}\circ
i_{[X/G]}\circ\psi_{X}^{*}$. Since the action on $X$ is now trivial, we have
$\operatorname{QCoh}([X/G])\simeq\operatorname{QCoh}(BG)\otimes_{\mathsf{Mod}}\operatorname{QCoh}(X)$
by [GR17, Prop. 3.3.5], and $\psi_{X}^{*}$ is equivalent to the functor

$\displaystyle\mathcal{O}_{BG}\boxtimes(-)\colon\operatorname{QCoh}(X)\to\operatorname{QCoh}(BG)\otimes_{\mathsf{Mod}}\operatorname{QCoh}(X)$

Since $(\varphi^{G}_{X})^{*}\circ i_{[X/G]}$ is equivalent to the functor
$N\boxtimes K\to N\otimes K$ from
$\operatorname{QCoh}(BG)\otimes_{\mathsf{Mod}}\operatorname{QCoh}(X)$ to
$\operatorname{QCoh}^{G}(X)$, the claim follows.

(iii). The functor
$(-)^{\mathrm{mv}}\colon\operatorname{QCoh}^{G}(X)\to\operatorname{QCoh}^{\mathrm{mv}}(X)$
preserves colimits, hence has a right adjoint, say
$j\colon\operatorname{QCoh}^{\mathrm{mv}}(X)\to\operatorname{QCoh}^{G}(X)$.
For $M\in\operatorname{QCoh}^{\mathrm{mv}}(X)$, take
$N\in\operatorname{QCoh}^{G}(X)$ such that $N^{\mathrm{mv}}\simeq M$. Consider
the following commutative diagram

${N^{\mathrm{fix}}}$${N^{\mathrm{fix}}}$${(jM)^{\mathrm{fix}}}$${N^{\mathrm{fix}}}$${N}$${jM}$${0}$${N^{\mathrm{mv}}}$${(jM)^{\mathrm{mv}}}$

The middle row is exact, hence so is the top one by exactness and idempotency
of $(-)^{\mathrm{fix}}$, and the right column is exact by definition. It
follows that $(jM)^{\mathrm{fix}}=0$, hence $jM\simeq(jM)^{\mathrm{mv}}$. The
diagram also shows that $(-)^{\mathrm{mv}}\circ(-)^{\mathrm{fix}}\simeq 0$.

By construction, for any $K\in\operatorname{QCoh}^{G}(X)$, it holds that
$K^{\mathrm{mv}}\simeq 0$ if and only if $K^{\mathrm{fix}}\to K$ is an
equivalence. Since $i$ is fully faithful, the sequence

(3.4)
$\displaystyle\operatorname{QCoh}(X)\xrightarrow{i}\operatorname{QCoh}^{G}(X)\xrightarrow{(-)^{\mathrm{mv}}}\operatorname{QCoh}^{\mathrm{mv}}(X)$

is thus split exact, by Remark 3.8. ∎

######  Corollary 3.10.

Let $G$ act trivially on a derived algebraic stack $X$. Then for each
$\mathcal{F}\in\operatorname{QCoh}^{G}(X)$ the canonical sequence

(3.5)
$\displaystyle\mathcal{F}^{\mathrm{fix}}\to\mathcal{F}\to\mathcal{F}^{\mathrm{mv}}$

is split. Consequently, $(-)^{\mathrm{fix}}$ is also a left adjoint of $i$.

###### Proof.

Since the sequence (3.4) is split, we have a left adjont $f^{\prime}\dashv i$,
and for any $\mathcal{F}\in\operatorname{QCoh}^{G}(X)$ a Cartesian diagram

${\mathcal{F}}$${j\mathcal{F}^{\mathrm{mv}}}$${if^{\prime}\mathcal{F}}$${if^{\prime}j\mathcal{F}^{\mathrm{mv}}}$

For both claims, it suffices to show that
$if^{\prime}j\mathcal{F}^{\mathrm{mv}}\simeq 0$. Since $f^{\prime},i$ are both
left adjoint, this reduced to the classical case, which is know. ∎

###  3.3. Relative fixed loci of derived stacks

Let $f\colon G\to G^{\prime}$ be a surjective homomorphism of reductive
algebraic groups, and $\varphi\colon BG\to BG^{\prime}$ the induced map on
stacks.

######  Definition 3.11.

Let $f_{*}\colon\mathsf{Stk}^{G}\to\mathsf{Stk}^{G^{\prime}}$ be the functor
that sends a derived stack $X\colon\mathsf{Alg}^{G}\to\mathsf{Spc}$ to

$\displaystyle\mathsf{Alg}^{G^{\prime}}\xrightarrow{\mathrm{Res}_{\varphi}}\mathsf{Alg}^{G}\xrightarrow{X}\mathsf{Spc}$

Let $B,\Omega$ be as in Notation 3.4. Write $\varphi_{*}$ for the functor
$Bf_{*}\colon\mathsf{Stk}_{BG}\to\mathsf{Stk}_{BG^{\prime}}$. Conversely, let
$\varphi^{*}\colon\mathsf{Stk}_{BG^{\prime}}\to\mathsf{Stk}_{BG}$ be the
pullback functor along $\varphi$, and let $f^{*}$ be the functor
$\Omega\varphi^{*}\colon\mathsf{Stk}^{G^{\prime}}\to\mathsf{Stk}^{G}$.

######  Lemma 3.12.

The functor
$\mathrm{Res}_{\varphi}\colon\mathsf{Alg}^{G^{\prime}}\to\mathsf{Alg}^{G}$ is
fully faithful.

###### Proof.

Since
$\mathrm{Res}_{\varphi}\colon\mathsf{Mod}^{G^{\prime}}\to\mathsf{Mod}^{G}$ is
$t$-exact, we can restrict it to a functor between connective objects, written
$H\colon\mathsf{Mod}^{G^{\prime}}_{\geq 0}\to\mathsf{Mod}^{G}_{\geq 0}$. It
suffices to show that $H$ is fully faithful. Let
$h\colon\mathsf{Rep}^{G^{\prime}}\to\mathsf{Rep}^{G}$ be the classical
restriction functor.

Since $H$ preserves colimits, it is determined by its restriction to
$\mathsf{Rep}^{G^{\prime}}$. Since $H$ restricted to
$\mathsf{Rep}^{G^{\prime}}$ is $h$, we see that $H$ is the left derived
functor of $h$ by [HTT, Prop. 5.5.8.15]. By [HTT, Prop. 5.5.8.22], it
therefore suffices to show that
$H\colon\mathsf{Rep}^{G^{\prime}}\to\mathsf{Mod}^{G}_{\geq 0}$ is fully
faithful and with essential image contained in the compact projective objects
of $\mathsf{Mod}^{G}_{\geq 0}$.

We already know that the essential image of $H$ on $\mathsf{Rep}^{G^{\prime}}$
consists of compact projective objects, since $H=h$ on
$\mathsf{Rep}^{G^{\prime}}$. What remains to show is that $h$ is fully
faithful, which follows from the fact that $f$ is surjective. ∎

######  Proposition 3.13.

The functors $f^{*},f_{*}$ fit into an adjunction

$\displaystyle
f^{*}:\mathsf{Stk}^{G^{\prime}}\leftrightarrows\mathsf{Stk}^{G}:f_{*}$

where $f^{*}$ is fully faithful. Consequently, the functor $\varphi_{*}$ is
the Weil restriction along $\varphi\colon BG\to BG^{\prime}$.

###### Proof.

Take a left adjoint $F$ to $f_{*}$ via left Kan extensions. Since
$\mathrm{Res}_{\varphi}$ is fully faithful by Lemma 3.12, it holds that $F$
and $f^{*}$ agree when restricted to $\mathsf{Aff}^{G^{\prime}}$. Since both
functors commute with colimits and $\mathsf{Aff}^{G^{\prime}}$ generates
$\mathsf{Stk}^{G^{\prime}}$ under colimits, $F$ and $f^{*}$ therefore agree on
all of $\mathsf{Stk}^{G^{\prime}}$. ∎

###  3.4. Absolute fixed loci of derived stacks

Consider now, for a reductive algebraic group $G$, the adjunction induced by
the canonical map $G\to*$ via Proposition 3.13, written as

$\displaystyle\iota:\mathsf{Stk}\leftrightarrows\mathsf{Stk}^{G}:(-)^{G}$

######  Remark 3.14.

Let $U\in\mathsf{Stk}^{G}$ and put $X\coloneqq[U/G]$. Observe that the derived
mapping stack $\underline{\operatorname{Map}}(BG,X)$ exists without any set-
theoretic issues. Define $\underline{\operatorname{Map}}^{G}(\ast,X)$ via the
Cartesian diagram

Proposition 3.13 and the formula (2.1) for the Weil restriction imply that
$\underline{\operatorname{Map}}^{G}(\ast,X)$ is the Weil restriction
$\psi_{*}X$ along $\psi\colon BG\to*$. Moreover, it holds

$\displaystyle\mathsf{Stk}(T,\psi_{*}X)\simeq\mathsf{Stk}_{BG}(T\times
BG,X)\simeq\mathsf{Stk}^{G}(T,U)\simeq\mathsf{Stk}(T,U^{G})$

giving another description of $U^{G}$ in terms of mapping stacks.

Assume now that $X$ is algebraic and locally of finite presentation. Observe
that $\psi\colon BG\to\ast$ is formally proper by [HLP19, Prop. 4.3.4], that
$\psi_{*}\mathcal{O}_{BG}\simeq\psi_{*}\psi^{*}\mathbb{C}\simeq\mathbb{C}$
since $\psi^{*}$ is the fully faithful functor
$(-)^{\mathrm{tv}}\colon\mathsf{Mod}_{\mathbb{C}}\to\mathsf{Mod}^{G}_{\mathbb{C}}$,
and that $\psi_{*}$ is of Tor-amplitude $\geq 0$ by the previous point.
Therefore, because $X$ is locally of finite presentation and all stacks are
assumed to have affine diagonal, [HLP19, Thm. 5.1.1] implies that all mapping
stacks in the above diagram are algebraic, hence that $U^{G}$ is as well.

Write $\mathrm{cl}\mathsf{Stk}$ for the category of classical stacks, and
$\mathrm{cl}\mathsf{Stk}^{G}$ for the category of classical stacks with
$G$-action. We then have an adjunction

$\displaystyle\iota_{\mathrm{cl}}:\mathrm{cl}\mathsf{Stk}\rightleftarrows\mathrm{cl}\mathsf{Stk}^{G}:(-)^{G,\mathrm{cl}}$

where $\iota_{\mathrm{cl}}$ endows a stack with trivial $G$-action, and
$(-)^{G,\mathrm{cl}}$ takes the classical fixed points. We thus have a diagram

(3.6)
${\mathsf{Stk}}$${\mathsf{Stk}^{G}}$${\mathsf{Stk}}$${\mathrm{cl}\mathsf{Stk}}$${{\mathrm{cl}\mathsf{Stk}}^{G}}$${{\mathrm{cl}\mathsf{Stk}}}$$\scriptstyle{\iota}$$\scriptstyle{(-)_{\mathrm{cl}}}$$\scriptstyle{(-)^{G}}$$\scriptstyle{(-)_{\mathrm{cl},G}}$$\scriptstyle{(-)_{\mathrm{cl}}}$$\scriptstyle{\iota_{\mathrm{cl}}}$$\scriptstyle{(-)^{G,\mathrm{cl}}}$

where $(-)_{\mathrm{cl},G}$ is taking the underlying classical $G$-stack. We
will often write $(-)_{\mathrm{cl},G}$ resp. $(-)^{G,\mathrm{cl}}$ simply as
$(-)_{\mathrm{cl}}$ resp. as $(-)^{G}$, which is justified by the first point
of the following result.

######  Proposition 3.15.

  1. (i)

The diagram (3.6) is commutative.

  2. (ii)

Suppose that $X\in\mathsf{Stk}^{G}$ is a derived scheme such that
$X_{\mathrm{cl}}$ is separated. Then $X^{G}$ is a derived scheme, and the unit
$\iota X^{G}\to X$ of the adjunction $\iota\dashv(-)^{G}$ is a closed
immersion. In particular, for $B\in\mathsf{Alg}^{G}$, it holds that
$(\operatorname{Spec}^{G}B)^{G}$ is affine.

###### Proof.

(i) Clearly, it holds
$\iota_{\mathrm{cl}}\circ(-)_{\mathrm{cl}}\simeq(-)_{\mathrm{cl},G}\circ\iota$.
Let now $j\colon\mathrm{cl}\mathsf{Stk}\to\mathsf{Stk}$ and
$j^{G}\colon\mathrm{cl}\mathsf{Stk}^{G}\to\mathsf{Stk}^{G}$ be the inclusions.
Then also $\iota\circ j=j^{G}\circ\iota_{\mathrm{cl}}$. By composing
adjunctions, we have

$\displaystyle\iota\circ j\dashv(-)_{\mathrm{cl}}\circ(-)^{G}$
$\displaystyle\&$ $\displaystyle
j^{G}\circ\iota_{\mathrm{cl}}\dashv(-)^{G,\mathrm{cl}}\circ(-)_{\mathrm{cl},G}$

and therefore
$(-)_{\mathrm{cl}}\circ(-)^{G}\simeq(-)^{G,\mathrm{cl}}\circ(-)_{\mathrm{cl},G}$.

(ii) Let $X\in\mathsf{Stk}^{G}$ be a derived scheme such that
$X_{\mathrm{cl}}$ is separated. By [CGP15, Prop. A.8.10] it holds that
$X_{\mathrm{cl}}^{G}$ is a closed subscheme of $X_{\mathrm{cl}}$, hence that
$X^{G}$ is a closed subscheme of $X$. ∎

We can now give an explicit description of
$\mathcal{O}_{(\operatorname{Spec}B)^{G}}$ for $B\in\mathsf{Poly}^{G}$. By
Proposition 3.15, the adjunction $\iota\dashv(-)^{G}$ restricts to affine
objects. On the algebra side, this gives us an adjunction

$\displaystyle(-)_{G}:\mathsf{Alg}^{G}\leftrightarrows\mathsf{Alg}:(-)^{\mathrm{tv}}$

where $(-)^{\mathrm{tv}}$ is the functor which endows an algebra with trivial
$G$-action.

######  Definition 3.16.

Let $A\in\mathsf{Alg}^{G}$ and $(M\in\mathsf{Mod}^{G}_{A})_{\geq 0}$. Define

$\displaystyle A[M]\coloneqq\mathrm{LSym}^{G}_{A}(M)$

Let now $B\in\mathsf{Alg}^{G}_{A}$ and $\sigma\colon M\to B$ a map of
$(G,A)$-modules be given. Define $B/(\sigma)$ as the pushout

${A[M]}$${A}$${B}$${B/(\sigma)}$$\scriptstyle{\sigma}$$\scriptstyle{p}$

where $p$ is induced by the map $M\to 0$.

Suppose that $M\in\mathsf{Rep}^{G}$, say $\dim M=d$. Then
$\sigma\colon\mathbb{C}[M[k]]\to B$ induces elements
$v_{1},\dots,v_{d}\in\pi_{k}(B)$, and the underlying algebra of $B/(\sigma)$
is equivalent to the derived quotient $B/(v_{1},\dots,v_{d})$. This follows
because forgetting the action is a left adjoint, see Notation 3.6.

######  Proposition 3.17.

For $M\in\mathsf{Mod}^{G}_{\geq 0}$ it holds
$\mathbb{C}[M]_{G}\simeq\mathbb{C}[M^{\mathrm{fix}}]$.

###### Proof.

Let $R\in\mathsf{Alg}$. By Corollary 3.10, we have that $(-)^{\mathrm{fix}}$
is a left adjoint of $(-)^{\mathrm{tv}}$. We thus have

$\displaystyle\mathsf{Alg}(\mathbb{C}[M]_{G},R)\simeq\mathsf{Alg}^{G}(\mathbb{C}[M],R^{\mathrm{tv}})\simeq\mathsf{Mod}^{G}(M,R^{\mathrm{tv}})\simeq\mathsf{Mod}(M^{\mathrm{fix}},R)$

from which the claim follows. ∎

The same methods also imply the following statement about relative fixed loci.

######  Proposition 3.18.

Let $G\to G^{\prime}$ be a surjective homomorphism of reductive groups with
kernel $H$ and $X\in\mathsf{Stk}^{G}$. Then $H$ is reductive and the Weil
restriction of $[X/G]$ along the morphism $BG\to BG^{\prime}$ is given by the
$H$-fixed locus $[X^{H}/G^{\prime}]$.

###  3.5. The cotangent complex of the fixed locus

Let $X$ be a derived algebraic stack with a $G$-action, and write
$\varphi:X\to[X/G]$ for the projection map. By Proposition 3.9, the pullback
$\varphi^{*}$ is the forgetful functor
$\operatorname{QCoh}^{G}(X)\to\operatorname{QCoh}(X)$. Since
${\mathbb{L}}_{X}\simeq\varphi^{*}{\mathbb{L}}_{[X/G]/BG}$, it follows that
${\mathbb{L}}_{X}$ carries a canonical $G$-action, with which we tacitly endow
it throughout.

######  Example 3.19.

Recall that the Lie algebra $\mathfrak{g}$ is the tangent bundle at the
identity $e\colon*\to G$, hence $\mathfrak{g}^{\vee}\simeq
e^{*}{\mathbb{L}}_{G}\simeq{\mathbb{L}}_{*/BG}$. We thus have an exact
sequence

$\displaystyle\varphi^{*}{\mathbb{L}}_{[X/G]}\to{\mathbb{L}}_{X}\to\mathfrak{g}^{\vee}\otimes\mathcal{O}_{X}$

In particular, when ${\mathbb{L}}_{X}$ is connective, then we can think of
$\varphi^{*}{\mathbb{L}}_{[X/G]}$ pictorially as the chain complex
${\mathbb{L}}_{X}\to\mathfrak{g}^{\vee}\otimes\mathcal{O}_{X}$.

######  Proposition 3.20.

Let $X$ be a derived algebraic $G$-stack, write $i\colon X^{G}\to X$ for the
canonical map. Then the exact sequence

$\displaystyle
i^{*}{\mathbb{L}}_{X}\to{\mathbb{L}}_{X^{G}}\to{\mathbb{L}}_{X^{G}/X}$

is equivalent to the exact sequence

$\displaystyle
i^{*}{\mathbb{L}}_{X}\to(i^{*}{\mathbb{L}}_{X})^{\mathrm{fix}}\to(i^{*}{\mathbb{L}}_{X})^{\mathrm{mv}}[1]$

induced by the splitting
$i^{*}{\mathbb{L}}_{X}\simeq(i^{*}{\mathbb{L}}_{X})^{\mathrm{fix}}\oplus(i^{*}{\mathbb{L}}_{X})^{\mathrm{mv}}$
from Corollary 3.10.

###### Proof.

By construction, the $G$-action on $X^{G}$ is trivial. Let
$j\colon[X^{G}/G]\to[X/G]$ and $p\colon[X^{G}/G]\to X^{G}$ be the canonical
maps. By Proposition 3.9, the functor $j^{*}$ is equivalent to the
$G$-equivariant pullback $(i^{G})^{*}$, and the functor $p_{*}$ is equivalent
to $(-)^{\mathrm{fix}}$. It follows that
$p_{*}j^{*}{\mathbb{L}}_{[X/G]/BG}\simeq(i^{*}{\mathbb{L}}_{X})^{\mathrm{fix}}$.

Recall from Remark 3.14 that the Weil restriction $\psi_{*}[X/G]$ along
$\psi\colon BG\to*$ is $X^{G}$. Then the pullback of $\psi$ is the map $p$,
and $j$ is induced by the counit of the adjunction $\psi^{*}\dashv\psi_{*}$.
Moreover, by Proposition 3.9, it holds that $p_{*}$ is also a left adjoint of
$p^{*}$. Therefore, by the formula for the cotangent complex of Weil
restrictions given in [HKR22], it also holds
$p_{*}j^{*}{\mathbb{L}}_{[X/G]/BG}\simeq{\mathbb{L}}_{X^{G}}$. The claim
follows. ∎

###  3.6. Base-change of fixed loci

We have the following étale base change lemma. Let $G$ be a reductive
algebraic group as usual.

######  Lemma 3.21.

Let $U\to V$ be a $G$-equivariant morphism of derived schemes, write
$f\colon[U/G]\to[V/G]$ for the induced map, and consider the natural
commutative square

${{[U^{G^{0}}/G]}}$${{[V^{G^{0}}/G]}}$${{[U/G]}}$${{[V/G]},}$$\scriptstyle{f}$

where $G^{0}\subseteq G$ denotes the identity component of $G$. If $f$ is
étale and $f_{\mathrm{cl}}$ is separated, then the square is Cartesian.

###### Proof.

Write $X=[U/G]$ and $Y=[V/G]$. By naturality of $(-)^{G^{0}}$, if
$V^{G^{0}}=\emptyset$, then $U^{G^{0}}=\emptyset$, and there is nothing to
prove. So we assume that $V^{G^{0}}\not=\emptyset$. Now
$U^{G^{0}}\not=\emptyset$ if and only if $X$ and $Y$ have the same maximal
stabilizer dimension. Hence, this is true for the underlying classical
truncations, since we have a natural isomorphism
$[U_{\mathrm{cl}}^{G^{0}}/G]=(X_{\mathrm{cl}})^{{\mathrm{max}}}\to(Y_{\mathrm{cl}})^{{\mathrm{max}}}\times_{Y_{\mathrm{cl}}}X_{\mathrm{cl}}=[V_{\mathrm{cl}}^{G^{0}}/G]\times_{Y_{\mathrm{cl}}}X_{\mathrm{cl}}$
by the discussion following the statement of Proposition C.5 in [ER21].

By $G$-equivariance and étaleness of the morphism $X\to Y$, we have a
commutative diagram

where the vertical arrows are equivalences and the splittings are with respect
to the $G^{0}$-action.

But
${\mathbb{L}}_{X^{G^{0}}}\simeq({\mathbb{L}}_{X}|_{X^{G^{0}}})^{\mathrm{fix}}$
and
${\mathbb{L}}_{Y^{G^{0}}}\simeq({\mathbb{L}}_{Y}|_{Y^{G^{0}}})^{\mathrm{fix}}$
and it thus follows that ${\mathbb{L}}_{X^{G^{0}}/Y^{G^{0}}}\simeq 0$ and
$X^{G^{0}}\to Y^{G^{0}}$ is étale.

Now $X^{G^{0}}\to Y^{G^{0}}$ factors through the étale morphism
$X\times_{Y}Y^{G^{0}}\to Y^{G^{0}}$, so the map $X^{G^{0}}\to
X\times_{Y}Y^{G^{0}}$ must also be étale and hence an isomorphism, as it
induces an isomorphism on classical truncations. ∎

###  3.7. Equivariant standard forms

Our discussion so far allows us to give natural equivariant extensions of
standard forms and their existence.

To this end, let $\mathsf{mod}^{G}$ be the 1-category of presheaves on
$\mathsf{Rep}^{G}$ that send finite coproducts in $\mathsf{Rep}^{G}$ to
products of sets. Then $\mathsf{mod}^{G}$ is the category of discrete
$G$-modules $(\mathsf{Mod}^{G})^{\heartsuit}$, and $\mathsf{Mod}^{G}$ is the
unbounded derived category of $\mathsf{mod}^{G}$. A _$G$ -cdga_ is a
commutative algebra object in the category $\mathrm{Ch}_{\geq
0}(\mathsf{mod}^{G})$ of chain complexes in homological degree $\geq 0$, with
the natural symmetric monoidal structure. Similarly, we define a _graded-
commutative $G$-ring_ as a graded-commutative algebra object in
$\mathsf{mod}^{G}$.

######  Definition 3.22.

Let an algebra $A\in\mathsf{Alg}^{G}$ and $x\in\operatorname{Spec}A$ fixed by
$G$ be given. A $G$-cdga model $R$ of $A$ is said to be in standard form if
$R_{0}$ is smooth with $\Omega_{R_{0}}$ free, and the underlying graded-
commutative $G$-ring of $R$ is freely generated over $R_{0}$ on a finite
number of generators. If this holds, then we say that $R$ is _minimal at $x$_
if the underlying cdga (which is non-equivariantly in standard form) is
minimal at $x$.

Remark 1.5 also holds in the equivariant case. That is, if a $G$-cdga $R$ is
in standard form, then it can be obtained from $R_{0}$ by equivariant cell-
attachements $R(0)\to R(1)\to\dots\to R(n)$ such that, for all $k$, the
underlying graded-commutative $G$-ring of $R(k)$ is the subring of the
underlying graded-commutative $G$-ring of $R$ generated by elements in
homogeneous degree $\leq k$, and $R(k+1)=R(k)/(\sigma)$ for a $G$-equivariant
map $\sigma:M_{k+1}[k]\to R(k)$, where $M_{k+1}$ is a free $G$-module over
$R(k)$.

As in the non-equivariant case, we say that $B\in\mathsf{Alg}^{G}_{A}$ (or the
corresponding map on spectra) is locally of finite $G$-presentation if $B$ is
a compact object in $\mathsf{Alg}^{G}_{A}$. Likewise, we call $B$ _finitely
$G$-presented_ if it is contained in the smallest subcategory of
$\mathsf{Alg}^{G}_{A}$ that contains $\mathsf{Poly}^{G}_{A}$ and is closed
under finite colimits. We note that any classical, finitely presented, affine
scheme $T\in\mathsf{Stk}^{G}$ admits a $G$-equivariant closed embedding into a
smooth classical scheme $\operatorname{Spec}^{G}(\mathbb{C}[M])$ for some
$M\in\mathsf{Rep}^{G}$. Moreover, $B$ is finitely $G$-presented if and only if
$B$ is finitely presented and admits a $G$-action (one can see this by the
argument sketched in the proof of the following lemma). Then
$\operatorname{Spec}^{G}B\to\operatorname{Spec}^{G}A$ is locally of finite
$G$-presentation if and only if $\operatorname{Spec}^{G}B$ is Zariski locally
in $\mathsf{Stk}^{G}$ finitely $G$-presented over $\operatorname{Spec}^{G}A$.
We then have the following.

######  Lemma 3.23.

Let an algebra $A\in\mathsf{Alg}^{G}$ and $x\in\operatorname{Spec}A$ fixed by
$G$ be given. Suppose that $A$ is locally of finite $G$-presentation over
$\mathbb{C}$. Then, up to equivariant Zariski localizing around $x$, there is
a $G$-cdga model $R$ of $A$ in standard form. We can arrange $R$ to be such
that $R_{0}=\mathbb{C}[M]$ for some $M\in\mathsf{Rep}^{G}$. Alternatively, we
can arrange $R$ to be minimal at $x$.

In addition, in both cases, up to further equivariant Zariski shrinking, the
underlying $R(i)$-modules $M_{i+1}$ may be taken, for all $i\geq 0$, in the
form $M_{i+1}=R(i)\otimes_{\mathbb{C}}W_{i+1}$ for $G$-representations
$W_{i+1}\in\mathsf{Rep}^{G}$.

###### Proof.

The same argument that proves Lemma 1.7 applies equivariantly to establish
everything, but the last assertion.

We briefly sketch how to equivariantly adapt the steps of the proof of [BBJ19,
Theorem 4.1]. Write $X=\operatorname{Spec}^{G}A$ for brevity. Since $A$ is
locally of finite $G$-presentation over $\mathbb{C}$, we may assume, up to
$G$-equivariant Zariski open localizing around $x$, that $A$ is finitely
$G$-presented. Following [BBJ19, Theorem 4.1], we may now either choose a
$G$-equivariant closed embedding
$X_{\mathrm{cl}}\to\operatorname{Spec}^{G}(\mathbb{C}[M])$ for some
$M\in\mathsf{Rep}^{G}$ or, using the reductivity of $G$, after further
equivariant Zariski open localizing around $x\in X_{\mathrm{cl}}$, a
$G$-equivariant closed embedding
$X_{\mathrm{cl}}\to\operatorname{Spec}^{G}(R_{0})$ into a smooth scheme, that
is minimal at $x$ for some finitely presented $R_{0}\in\mathsf{Alg}^{G}$.

We now continue with the minimal case, as the case with $R_{0}=\mathbb{C}[M]$
proceeds similarly. We observe that since $[\operatorname{Spec}^{G}(R_{0})/G]$
is smooth over $BG$, the closed embedding
$[X_{\mathrm{cl}}/G]\to[\operatorname{Spec}^{G}(R_{0})/G]$ can be lifted over
$BG$ to a closed embedding $[X/G]\to[\operatorname{Spec}^{G}(R_{0})/G]$, which
by Proposition 3.3 amounts to a $G$-equivariant map $R_{0}\to A$, lifting
$R_{0}\to\pi_{0}(A)$. Let $F$ be the fiber of $R_{0}\to A$. Then, up to
possible further equivariant Zariski open shrinking around $x$, there exists a
free $G$-module $M_{1}$ over $R_{0}$ that surjects onto $\pi_{0}(F)$ and whose
fiber at $x$ is isomorphic to $\pi_{0}(F)|_{x}$. For this to be possible, we
use the reductivity of $G$ and that $x$ is fixed by $G$. The morphism
$M_{1}\to\pi_{0}(F)\to R_{0}$ determines a $G$-cdga $R(1)$, together with a
$G$-equivariant morphism $R(1)\to A$ that is an isomorphism on $\pi_{0}$. This
showcases the first step of the construction. One argues similarly to
construct the remaining $R(i)$, by following the proof of [BBJ19, Theorem
4.1].

To establish the last assertion, it clearly suffices to prove that for any
$G$-equivariant vector bundle $\mathcal{W}$ on a derived affine $G$-scheme $U$
there exists a $G$-invariant Zariski open neighbourhood $x\in U_{x}\subseteq
U$ and a $G$-equivariant isomorphism $\mathcal{W}|_{U_{x}}\cong
W\otimes_{\mathbb{C}}\mathcal{O}_{U}$, where $W$ is a $G$-representation.

Since $x$ is fixed by $G$, $G$ acts on the fiber $W\coloneqq\mathcal{W}|_{x}$
and we have a $G$-equivariant surjection $\mathcal{W}=\Gamma(U,\mathcal{W})\to
W$. Since $G$ is reductive, we may split this surjection and consider
$W\subseteq\Gamma(U,\mathcal{W})$. This induces a $G$-equivariant morphism
$W\otimes_{\mathbb{C}}\mathcal{O}_{U}\to\mathcal{W}$. For $U$ classical, the
locus where this map is an isomorphism is clearly a $G$-invariant open
subscheme of $U$ which contains $x$, and the general case reduces to the
classical case, since $\mathcal{W}$ is a vector bundle. This concludes the
proof. ∎

Now, our prior discussion of fixed loci immediately implies an explicit
description for $(\operatorname{Spec}B)^{G}$ when $B$ has a model in standard
form, given in the next propositions.

For $R$ a model in standard form for $B\in\mathsf{Alg}$, we write $B(i)$ for
the algebra in $\mathsf{Alg}$ corresponding to the cdga $R(i)$.

######  Proposition 3.24.

Suppose that $B\in\mathsf{Alg}^{G}$ has a model $R$ in standard form such that
$M_{i+1}=R(i)\otimes_{\mathbb{C}}W_{i+1}$ for some $G$-representations
$W_{i+1}$. Then $B_{G}$ also has a model $R_{G}$ in standard form with
$(R_{G})_{0}=\mathcal{O}_{(\operatorname{Spec}R_{0})^{G}}$ and generated by
the modules $N_{i+1}$ defined inductively by
$N_{i+1}=R(i)_{G}\otimes_{\mathbb{C}}W_{i+1}^{\mathrm{fix}}$, where $R(i)_{G}$
is the model in standard form for $B(i)_{G}$ generated by $N_{j}$ for $j\leq
i$.

In addition, for a point $x\in\operatorname{Spec}B$ fixed by $G$, if $R$ is
minimal at $x$, we can arrange $R_{G}$ to be minimal at $x$ as well.

###### Proof.

Recall that $R_{0}$ is smooth and classical by definition. We first show that
$(\operatorname{Spec}R_{0})^{G}$ is smooth and classical and coincides with
the classical fixed locus. To this end, write $X=\operatorname{Spec}R_{0}$.
Then, by [Dré04, Lemma 5.1], for any $G$-fixed point $x$, there is a Zariski
open $G$-invariant neighbourhood $U_{x}\subseteq X$ together with an étale
$G$-equivariant morphism $U_{x}\to T_{x}X$, where $T_{x}X$ denotes the tangent
space of $X$ at $x$. Let $U$ be the disjoint union of the $U_{x}$ as $x$
ranges over the fixed points of $X$. We obtain a $G$-equivariant étale
morphism $f\colon U\to X$. Since $f=f_{\mathrm{cl}}$ is separated, by Lemma
3.21, $U^{G}=X^{G}\times_{X}U$. Applying the same lemma to each morphism
$U_{x}\to T_{x}X$, which is étale and separated, we get
$(U_{x})^{G}=U_{x}\times_{X}(T_{x}X)^{G}$. By Proposition 3.17, since
$T_{x}X=\operatorname{Spec}(\mathbb{C}[M])$ for some $M\in\mathsf{Rep}^{G}$,
it follows that $(T_{x}X)^{G}$ is smooth and classical, and hence so is
$(U_{x})^{G}$, and therefore $U^{G}$ as well. Then, $U^{G}=X^{G}\times_{X}U$,
and the fact that $U^{G}\to X^{G}$ is surjective on points by construction,
implies that $X^{G}$ must be smooth and classical. In particular, it must
coincide with the classical fixed locus in this case.

For $i\geq 0$, we assume by induction that the cdga $R(i)_{G}$, generated by
$N_{j}$ for $j\leq i$, is a model for $B(i)_{G}$, and put $N_{i+1}\coloneqq
R(i)_{G}\otimes_{\mathbb{C}}W_{i+1}^{\mathrm{fix}}$. Then we have a (strictly)
commutative diagram of cdgas

(3.7)
${\mathbb{C}}$${\mathbb{C}[W_{i+1}[i]]}$${\mathbb{C}}$${R(i)}$${\mathrm{LSym}_{R(i)}(M_{i+1}[i])}$${R(i)}$${R(i)}$${R(i+1)}$$\scriptstyle{\alpha}$$\scriptstyle{W_{i+1}[i]\mapsto
0}$$\scriptstyle{M_{i+1}[i]\mapsto 0}$

where the square on the left is the natural one coming from the fact that
$M_{i+1}=R(i)\otimes W_{i+1}$, and the bottom square comes from the definition
of $R(i+1)$. One shows that all squares are homotopy pushouts, using that
$M_{i+1}$ is free and [BBJ19, §2.2]. Define $R(i+1)_{G}$ as the homotopy
pushout of the map $\mathbb{C}[W_{i+1}^{\mathrm{fix}}[i]]\to R(i)_{G}$
(induced by $\alpha$) along the map
$\mathbb{C}[W_{i+1}^{\mathrm{fix}}[i]]\to\mathbb{C}$ induced by the zero map.
Then, since $(-)_{G}$ commutes with pushouts, we see that $R(i+1)_{G}$ is a
model for $B(i+1)_{G}$, generated by $N_{j}$ for $j\leq i+1$. This concludes
the first claim.

Let a point $x\in\operatorname{Spec}B$ be fixed by $G$ such that $R$ is
minimal at $x$. By [BBJ19, Prop. 2.12], the cotangent complex for $R$
restricted to $x$ has underlying chain complex

$\displaystyle\cdots\to W_{i+1}|_{x}\to W_{i}|_{x}\to W_{i-1}|_{x}\to\cdots$

and the cotangent complex for $R_{G}$ restricted to $x$ has underlying chain
complex

$\displaystyle\cdots\to W_{i+1}^{\mathrm{fix}}|_{x}\to
W_{i}^{\mathrm{fix}}|_{x}\to W_{i-1}^{\mathrm{fix}}|_{x}\to\cdots$

The claim thus holds by definition of minimality. ∎

###  3.8. Equivariant blow-ups

With similar arguments as in the non-equivariant case, we can apply the
machinery of [Hek21, HKR22] to the $G$-equivariant setting to produce
$G$-equivariant Rees algebras and blow-ups. This is carried out in [BH23] in
much more generality. We summarise the results here.

For $Z\to X$ a $G$-equivariant closed immersion, we have a _$G$ -equivariant
extended Rees algebra_ $\mathcal{R}_{Z/X}^{G,\mathrm{ext}}$ with a similar
universal property as $\mathcal{R}_{Z/X}^{\mathrm{ext}}$, but now against all
$G$-equivariant $T\to X$. We define the _$G$ -equivariant blow-up_ as

$\operatorname{Bl}^{G}_{Z}X\coloneqq[((\operatorname{Spec}^{G}\mathcal{R}_{Z/X}^{G})\setminus
V((\mathcal{R}_{Z/X}^{G})_{+})]$

where
$\mathcal{R}_{Z/X}^{G}\coloneqq(\mathcal{R}_{Z/X}^{G,\mathrm{ext}})_{\geq 0}$.
The following can be shown by comparing universal properties.

######  Proposition 3.25.

It holds that
$[\operatorname{Bl}^{G}_{Z/X}/G]\simeq\operatorname{Bl}_{[Z/G]/[X/G]}$.

One can show the next result along similar lines as done in [Hek21] or as in
[BH23].

######  Proposition 3.26.

Take $A\in\mathsf{Alg}^{G}$ and let $\sigma\colon M\to A$ be a map of
connective $(G,A)$-modules. Put $B\coloneqq A/(\sigma)$. Then we can compute
the $G$-equivariant extended Rees algebra of $B$ over $A$ as

$\displaystyle
R^{G,{\mathrm{ext}}}_{B/A}\simeq\frac{A[t^{-1},M]}{(t^{-1}M-\sigma)}$

where the quotient is with respect to the map $M\to A[t^{-1},M]$ which is
induced by substracting $\sigma\colon M\to A\to A[t^{-1},M]$ from the map

$\displaystyle\mathrm{LSym}^{G}_{A[t^{-1}]}(\times
t^{-1})\colon\mathrm{LSym}^{G}_{A[t^{-1}]}(M[t^{-1}])\to\mathrm{LSym}^{G}_{A[t^{-1}]}(M[t^{-1}])$

######  Example 3.27.

Suppose that $M\in(\mathsf{Mod}_{A}^{G})_{\geq 0}$ is of the form
$M=M_{1}[k_{1}]\oplus\dots\oplus M_{n}[k_{n}]$, where each
$M_{i}\in\mathsf{Rep}_{A}^{G}$ is of the form $M_{i}=A\otimes V_{i}$, for
$V_{i}\in\mathsf{Rep}^{G}$ of dimension $d_{i}$, and
$k_{1},\dots,k_{n}\in\mathbb{N}$. For each $i$, let
$\underline{v}_{i}=(v_{i1},\dots,v_{id_{i}})$ be a basis for the underlying
$A$-module of $M_{i}$.

Let a map $\sigma\colon M\to A$ be given, and let $B$ be the quotient
$A/(\sigma)$. Then the restrictions $M_{i}[k_{i}]\to A$ of $\sigma$ induce
elements $\underline{a}_{i}=(a_{i1},\dots,a_{id_{i}})\in\pi_{k_{i}}(A)$. From
Proposition 3.26, it follows that the underlying $A[t^{-1}]$-algebra of
$R^{G,{\mathrm{ext}}}_{B/A}$ is

$\displaystyle\frac{A[t^{-1},\underline{x}_{1},\dots,\underline{x}_{n}]}{(\underline{x}_{1}t^{-1}-\underline{a}_{1},\dots,\underline{x}_{n}t^{-1}-\underline{a}_{n})}$

where each $\underline{x}_{i}$ is a sequence of free variables
$x_{i1},\dots,x_{id_{i}}$ in homogeneous degree 1 and homological degree
$k_{i}$.

######  Example 3.28.

Let $A\in\mathsf{Alg}^{G}$ be given. Take a connective $(G,A)$-module $M$ of
the form $M=M_{1}[k_{1}]\oplus\dots\oplus M_{n}[k_{n}]$, with each
$M_{i}\in\mathsf{Rep}^{G}_{A}$, say $M_{i}=A\otimes V_{i}$ for some
$V_{i}\in\mathsf{Rep}^{G}$. Let $\sigma\colon M\to A$ be a given map of
$(G,A)$-modules, and put $B\coloneqq A/(\sigma)$. We will give a formula for
$R^{G,{\mathrm{ext}}}_{B_{G}/B}$.

Observe that

$\displaystyle A[M]_{G}\simeq(\mathrm{LSym}_{A}^{G}(A\otimes
V))_{G}\simeq(\mathrm{LSym}^{G}(V)\otimes A)_{G}\simeq A_{G}[M^{\mathrm{mv}}]$

It follows that $B_{G}$ fits into the following pushout square

${A_{G}[M^{\mathrm{mv}}]}$${A_{G}}$${A_{G}}$${B_{G}}$$\scriptstyle{s}$$\scriptstyle{z}$

where $z$ is induced by the zero map and $s$ by the restriction of $\sigma$ to
$M^{\mathrm{mv}}$. It follows that $B^{\mathrm{mv}}$ is the cofibre of
$M^{\mathrm{mv}}\to A^{\mathrm{mv}}$.

Write $M$ as $M^{\prime}\oplus M^{\prime\prime}$, where $M^{\prime}$ is the
direct sum of those $M_{i}[k_{i}]\subset M$ for which the restriction
$M^{\prime}\to A$ is nonzero. Then $M^{\prime\prime}\to A$ is zero. It follows
that

$\displaystyle B^{\mathrm{mv}}\simeq N\coloneqq
A^{\mathrm{mv}}/M^{\prime\prime}\oplus M^{\prime}[1]$

hence, by Example 3.27,

$\displaystyle
R^{G,{\mathrm{ext}}}_{B_{G}/B}\simeq\frac{B[t^{-1},N]}{(t^{-1}N-\tau)}.$

##  4\. Derived loci of maximal stabilizer dimension

Let $X$ be a derived algebraic stack whose classical truncation
$X_{\mathrm{cl}}$ is Noetherian and admits a good moduli space. In [ER21], it
is shown that there is a canonical closed substack
$X_{\mathrm{cl}}^{{\mathrm{max}}}$ of $X_{\mathrm{cl}}$ which is the locus of
points of $X_{\mathrm{cl}}$ with stabilizer group of maximal dimension. Our
objective in this section is to establish one of the central results of this
paper, the existence of a derived locus of maximal-dimensional stabilizers
$X^{{\mathrm{max}}}$. Our definition of $X^{{\mathrm{max}}}$ is directly
inspired by the definition of $X_{\mathrm{cl}}^{\mathrm{max}}$ in [ER21], and
gives a derived enhancement. Along the way we will review derived loop stacks
and derived bundles twisted by inner automorphisms, and give a derived Luna
étale slice theorem.

###  4.1. Derived loop stacks

Let $X\in\mathsf{Stk}$ be a derived stack. Recall that the derived loop stack
$\mathscr{L}X$ of $X$ is the self-intersection of the diagonal
$\Delta_{X}\colon X\to X\times X$, which is algebraic if $X$ is. Let $k\colon
T\to X$ be a morphism of derived stacks. Then the stabilizer of $X$ in $T$ is
the pullback

$\displaystyle G_{k}\coloneqq G_{T}\coloneqq T\times_{X}\mathscr{L}X$

Since $(\mathscr{L}X)_{\mathrm{cl}}$ is the inertia stack of
$X_{\mathrm{cl}}$, it holds that $(G_{T})_{\mathrm{cl}}$ is the stabilizer
$I_{X_{\mathrm{cl}}}$ of $X_{\mathrm{cl}}$ at $T_{\mathrm{cl}}$. We call
$G_{T}$ connected if $G_{T}\to T$ has connected classical fibers. If
$\mathcal{O}_{T}$ is a field, then $k\colon T\to X$ corresponds to a point
$x\in X$, and we call $G_{x}\coloneqq G_{k}$ the stabilizer (at $x$). Note
that $(G_{x})_{\mathrm{cl}}$ is a classical affine group scheme.

######  Example 4.1.

The basic example is when $X$ is of the form $[U/G]$, where $U$ is a derived
scheme, for which it holds

$\displaystyle\mathscr{L}X\simeq[S_{U}/G].$

Here, $S_{U}$ is the derived stabilizer group scheme $S_{U}$, defined as the
fibre product $(G\times U)\times_{U\times U}U$ of the map
$(\sigma,\pi_{2})\colon G\times U\to U\times U$ (where $\sigma$ is the
$G$-action) with the diagonal. See [Toë14, §4.4].

Let $f\colon X\to Y$ be a morphism of derived stacks. Then we have a natural
morphism $\mathscr{L}f\colon\mathscr{L}X\to\mathscr{L}Y\times_{Y}X$ of derived
stacks over $X$. We call $f$ monodromy-free if $\mathscr{L}f$ is an
equivalence. Likewise, we call $f$ stabilizer-preserving if
$(\mathscr{L}f)_{\mathrm{cl}}$ is an equivalence. If the latter is the case,
then for any $k\colon T\to X$ with $T$ classical, the morphism
$(\mathscr{L}f)_{\mathrm{cl}}$ induces an equivalence
$(G_{k})_{\mathrm{cl}}\simeq(G_{fk})_{\mathrm{cl}}$.

We call a morphism of derived algebraic stacks $f\colon X\to Y$ separated
(proper) if $f_{\mathrm{cl}}$ is separated (proper), as in [Stcks, Tag 04YW].
If $f\colon X\to Y$ is representable, then $f$ is separated if and only if
$\Delta_{f}$ is a closed immersion [Stcks, Tag 04YS]. Hence, since in the
general case the diagonal is representable, one can also define $f$ to be
separated by asking $\Delta_{f}$ to be of finite type, universally closed, and
separated. The basic examples are: affine morphisms are separated [Stcks, Tag
06TZ], and closed immersions are proper [Stcks, Tag 0CL8].

######  Example 4.2.

Let $X$ be a derived algebraic stack. Since we assume that all stacks have
affine diagonal, in particular $X$ has separated diagonal. Therefore, for $T$
a separated derived algebraic stack, any morphism $T\to X$ is separated by
[Stcks, Tag 050M].

Suppose that we have a Cartesian diagram

${X}$${Y}$${M}$${N}$$\scriptstyle{f}$

of derived algebraic stacks, where the vertical arrows are good moduli spaces.
Then if $M$ is separated, so is $f$.

######  Remark 4.3.

Let $f\colon X\to Y$ be a morphism of derived algebraic stacks. Then we have a
Cartesian diagram

(4.1)
${\mathscr{L}X}$${X}$${f^{*}\mathscr{L}Y}$${X\times_{Y}X}$${Y}$${X}$${X\times
X}$${Y\times