# Langlands duality on the Beilinson-Drinfeld Grassmannian

Justin Campbell and Sam Raskin

###### Abstract.

We calculate various categories of equivariant sheaves on the Beilinson-
Drinfeld Grassmannian in Langlands dual terms. For one, we obtain the
factorizable derived geometric Satake theorem. More generally, we calculate
the categorical analogue of unramified vectors in the Jacquet module of
sheaves on the Grassmannian.

In all cases, our spectral categories involve factorization modules for
factorization algebras related to the Langlands dual group.

###### Contents

  1. 1 Introduction
  2. 2 Renormalizing crystals of categories
  3. 3 Preliminaries on factorization
  4. 4 A local acyclity theorem
  5. 5 Spectral Hecke categories
  6. 6 Construction of the derived Satake transform
  7. 7 The case of a torus
  8. 8 Factorization modules at a point
  9. 9 The equivalence for $P=G$
  10. 10 The equivalence for a proper parabolic

##  1\. Introduction

###  1.1. What is this paper about?

Our main objectives in this paper are as follows:

  * •

Realize a strategy of Gaitsgory-Lurie giving a natural construction of the
derived geometric Satake equivalence [BF08] (cf. the first footnote in _loc.
cit_.).

  * •

Extend derived geometric Satake from the usual affine Grassmannian to the
Beilinson-Drinfeld (or factorizable) affine Grassmannian.

  * •

Extend the Arkhipov-Bezrkuvaknikov-Ginzburg equivalence [ABG04] to the
Beilinson-Drinfeld affine Grassmannian.

In fact, we unify the latter two subjects with a mutual generalization to
parabolic subgroups $P$ of $G$; for $P=G$ we obtain factorizable derived
Satake, while for $P=B$ we obtain factorizable Arkhipov-Bezrukavnikov-
Ginzburg.

These results have been widely anticipated. In particular, Gaitsgory has
spoken and written about our main results for some time, cf. [Gai15] §4.7 and
§6.6.3, [Gai11] §10.3.3, [ABG+14] Talk V.2 Conjecture 1, [ABC+18] Program
§4.1.1, and [AG15] Remark 12.6.7 to name a few.

However, the questions we consider have previously resisted precise
formulations. In the above sources, one finds no definitions of the spectral
(or _Langlands dual_) sides of the equivalences under consideration. So
although our results have been anticipated, we can find no precisely
formulated conjecture in the literature that we can point to and say “We
proved _this_ conjecture!” In large part, our task in this paper is to
precisely formulate the problem and to do so in a way that is amenable to its
solution.

Accordingly, below we will give an extended discussion of our point of view on
the spectral side.

###  1.2. Applications

Derived Satake at a point is closely related to subtle issues in the global
geometric Langlands program; see the above references as well as [FR23] and
[FR22]. Beraldo [Ber19] has given a lovely interpretation of the meaning of
full factorizable Satake, for which we refer to _loc. cit_.

For a long time (cf. [Gai11] and [Gai15]), it has been anticipated that our
results in the parabolic case are required to understand global geometric
Langlands near reducible local systems. Forthcoming work [CGR23] of Chen-
Gaitsgory-R. uses our present work in completely settling exactly this
problem.

###  1.3. The Gaitsgory-Lurie heuristic (spherical case)

We now recall the idea of Gaitsgory and Lurie, in part because it was never
written down (as far as we are aware). For purposes of an introduction, we
give ourselves the liberty to be lax with our homological algebra and notation
around factorization structures.

We let $\operatorname{Gr}_{G}$ denote the affine Grassmannian, which is the
quotient $\mathfrak{L}G/\mathfrak{L}^{+}G$ of the algebraic loop group of $G$
by the group of jets into $G$. Beilinson-Drinfeld viewed
$\operatorname{Gr}_{G}$ as a factorization space fibered over the Ran space of
finite subsets of the curve $X$. The geometric side of derived Satake is the
category
$\mathsf{Sph}_{G}:=\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}^{+}G}$ of
D-modules on the affine Grassmannian111In the main body of this text, the
symbol $\mathsf{Sph}_{G}$ is used for a slightly modified version of
$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}^{+}G}$, which we will not
need in this introduction..

Gaitsgory-Lurie observed that $\mathsf{Sph}_{G}$ is a _factorization monoidal
category_ because of the Beilinson-Drinfeld factorization structure on the
affine Grassmannian, and as such, it acts on the factorization category
$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}$ of Whittaker
D-modules on the affine Grassmannian.

By a version of the geometric Casselman-Shalika formula of [FGV01], it is
known that

$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\simeq\mathsf{Rep}(\check{G})$
(1.3.1)

as factorization categories (see Theorem 6.5.1 below).

We use the following standard heuristics: factorization categories are
analogous to $\mathbb{E}_{2}$-categories (also known as braided monoidal
categories), while factorization monoidal categories are analogous to
$\mathbb{E}_{3}$-categories. These heuristics are made somewhat precise in the
literature, see [Lur17a] §5.5 and notes from the talk by Nick Rozenblyum in
[ABC+18]. See also [Noc23], where the $\mathbb{E}_{3}$-structure on
$\mathsf{Sph}_{G}$ is made completely rigorous.

Now recall that for an $\mathbb{E}_{n}$-category $\mathcal{C}$, there is a
certain category $Z_{\mathbb{E}_{n}}(\mathcal{C})$ called its
_$\mathbb{E}_{n}$ -center_, which is the universal $\mathbb{E}_{n+1}$-category
acting on $\mathcal{C}$ as an $\mathbb{E}_{n}$-category in a suitable sense –
we refer to [Lur17a] §5.3 for definitions and details. Explicitly, as a
monoidal category, one can calculate $Z_{\mathbb{E}_{n}}(\mathcal{C})$ as
$\mathsf{End}_{\mathcal{C}\text{\textendash}\mathsf{mod}_{\mathbb{E}_{n}}}(\mathcal{C})$,
the category of endofunctors of $\mathcal{C}$ as an $\mathbb{E}_{n}$-module
category over itself.

Following this heuristic, we imagine that $\mathsf{Sph}_{G}$ is an
$\mathbb{E}_{3}$-category (which we know it literally is, cf. [Noc23]) acting
on the $\mathbb{E}_{2}$-category $\mathsf{Rep}(\check{G})$ via the equivalence
(1.3.1). Therefore, heuristically, we obtain a canonical functor of
$\mathbb{E}_{3}$-categories

$\mathsf{Sph}_{G}\longrightarrow Z_{\mathbb{E}_{2}}(\mathsf{Rep}(\check{G})).$
(1.3.2)

On the other hand, it is well-known that:

$Z_{\mathbb{E}_{2}}(\mathsf{Rep}(\check{G}))=Z_{\mathbb{E}_{2}}(\mathsf{QCoh}(\mathbb{B}\check{G}))=\mathsf{QCoh}((\mathbb{B}\check{G})^{\mathbb{S}^{2}})$

(cf. [BZFN10] Corollary 5.12). Here the last term
$(\mathbb{B}\check{G})^{\mathbb{S}^{2}}$ is the derived stack parametrizing
maps from the homotopy type of the $2$-sphere to the stack
$\mathbb{B}\check{G}$. Writing $\mathbb{S}^{2}$ as the suspension of
$\mathbb{S}^{1}$, this stack may be calculated more explicitly as

$(\mathbb{B}\check{G})^{\mathbb{S}^{2}}=\mathbb{B}\check{G}\underset{(\mathbb{B}\check{G})^{\mathbb{S}^{1}}}{\times}\mathbb{B}\check{G}=\mathbb{B}\check{G}\underset{(\check{G}\overset{\operatorname{ad}}{/}\check{G})}{\times}\mathbb{B}\check{G}=(\Omega_{1}\check{G})/\check{G}$

where

$\Omega_{1}\check{G}:=\operatorname{Spec}(k)\underset{\check{G}}{\times}\operatorname{Spec}(k)$

is the DG group of automorphisms of the identity in $\check{G}$. Using
logarithms, one finds that

$\Omega_{1}\check{G}=\Omega_{0}\check{\mathfrak{g}}=\operatorname{Spec}(\operatorname{Sym}(\check{\mathfrak{g}}[1])),$

so we can interpret the functor (1.3.2) as a functor

$\mathsf{Sph}_{G}\longrightarrow\operatorname{Sym}(\check{\mathfrak{g}}[1]))\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{G})).$

The derived Satake theorem then says that this functor is _almost_ an
equivalence. More precisely, the derived Satake theorem of [BF08] says that
this is true _up to renormalization_ – we should use the _renormalized Satake
category_ and the category
$\mathsf{IndCoh}((\mathbb{B}\check{G})^{\mathbb{S}^{2}})$.

To summarize, their heuristic says to follow the recipe:

  1. (1)

Use the factorization action of $\mathsf{Sph}_{G}$ on
$\mathsf{Rep}(\check{G})$ to obtain a functor to a suitable analogue of the
$\mathbb{E}_{2}$-center of $\mathsf{Rep}(\check{G})$.

  2. (2)

Settle the homological algebra issues to obtain a functor that is plausibly an
equivalence.

  3. (3)

Perform an explicit calculation to prove that the functor so obtained is an
equivalence.

######  Remark 1.3.1.

In [BF08], one finds that many of the difficulties have to do with
constructing the functor. The Gaitsgory-Lurie idea is of a markedly different
nature than what Bezrukavnikov-Finkelberg considered.

###  1.4. Centers and $\mathbb{E}_{n}$-modules

We will reinterpret centers (and more generally: centralizers) in terms of
modules over rings (or near enough). This material is not necessary, but we
find it plays an important motivating role for our main results.

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal
categories.

Given an $\mathbb{E}_{n}$-monoidal functor $F:\mathcal{C}\to\mathcal{D}$, its
_$\mathbb{E}_{n}$ -centralizer_ $Z_{\mathbb{E}_{n}}(F)$ is the category

$\mathsf{Hom}_{\mathcal{C}\text{\textendash}\mathsf{mod}_{\mathbb{E}_{n}}}(\mathcal{C},\mathcal{D}),$

i.e., the $\mathbb{E}_{n}$-Hochschild homology of $\mathcal{D}$ as an
$\mathbb{E}_{n}$-module category for $\mathcal{C}$. Its key property is that
an $\mathbb{E}_{n}$-map $\mathcal{E}\to Z_{\mathbb{E}_{n}}(F)$ is equivalent
to an $\mathbb{E}_{n}$-map $\mathcal{C}\otimes\mathcal{E}\to\mathcal{D}$ whose
composition

$\mathcal{C}\xrightarrow{\operatorname{id}_{\mathcal{C}}\otimes\mathds{1}_{\mathcal{E}}}\mathcal{C}\otimes\mathcal{E}\longrightarrow\mathcal{D}$

is $F$ as an $\mathbb{E}_{n}$-functor. (We recover the $\mathbb{E}_{n}$-center
of $\mathcal{C}$ by taking the centralizer of
$\operatorname{id}_{\mathcal{C}}$.)

We now give an alternative to $\mathbb{E}_{n}$-centralizers (that coincides in
some cases). First, one may form $\mathsf{Hom}(\mathcal{C},\mathcal{D})$, the
category of continuous DG functors from $\mathcal{C}$ to $\mathcal{D}$. Using
Day convolution, this category is naturally symmetric monoidal as well.

We remind the reader that $\mathbb{E}_{n}$-algebras in
$\mathsf{Hom}(\mathcal{C},\mathcal{D})$ are the same as (left) lax
$\mathbb{E}_{n}$-monoidal (continuous DG) functors from $\mathcal{C}$ to
$\mathcal{D}$. A variant: given $F:\mathcal{C}\to\mathcal{D}$ a lax
$\mathbb{E}_{n}$-monoidal functor, thought of as an $\mathbb{E}_{n}$-algebra
in $\mathsf{Hom}(\mathcal{C},\mathcal{D})$, an $\mathbb{E}_{n}$-module for $F$
in $\mathsf{Hom}(\mathcal{C},\mathcal{D})$ is a functor
$G:\mathcal{C}\to\mathcal{D}$ of lax $\mathbb{E}_{n}$-module categories for
$\mathcal{C}$. (In fact, this variant can be put on equal footing with its
predecessor using general colored operads.)

Now suppose that $\mathcal{C}$ is _rigid_ (symmetric) monoidal. Then recall
that a _lax_ linear morphism of $\mathcal{C}$-module categories is
automatically linear – this is standard for left module categories, but holds
just the same for $\mathbb{E}_{n}$-module categories. Therefore, in the above
discussion, we have

$Z_{\mathbb{E}_{n}}(F):=\mathsf{Hom}_{\mathcal{C}\text{\textendash}\mathsf{mod}_{\mathbb{E}_{n}}}(\mathcal{C},\mathcal{D})=\mathsf{Hom}_{\mathcal{C}\text{\textendash}\mathsf{mod}_{\mathbb{E}_{n}}^{\operatorname{lax}}}(\mathcal{C},\mathcal{D})$

where
$\mathcal{C}\text{\textendash}\mathsf{mod}_{\mathbb{E}_{n}}^{\operatorname{lax}}$
is the category of _lax_ $\mathbb{E}_{n}$-module categories for $\mathcal{C}$.
In particular, we see that when $\mathcal{C}$ is rigid, we can extend the
definition of $\mathbb{E}_{n}$-centralizer to allow $F$ to be merely lax
$\mathbb{E}_{n}$-monoidal. Moreover, using the preceding paragraph, we can
think of $Z_{\mathbb{E}_{n}}(F)$ as $\mathbb{E}_{n}$-modules for $F$ in the
symmetric monoidal category $\mathsf{Hom}(\mathcal{C},\mathcal{D})$.

We now make one additional manipulation, continuing to assume $\mathcal{C}$ is
rigid. Then recall that $\mathcal{C}$ is canonically self-dual as a DG
category. In particular, we obtain

$\mathsf{Hom}(\mathcal{C},\mathcal{D})\simeq\mathcal{C}^{\vee}\otimes\mathcal{D}\simeq\mathcal{C}\otimes\mathcal{D}$

where one easily finds222For completeness, we supply the argument. Giving a
lax symmetric monoidal functor
$\mathcal{E}\to\mathsf{Hom}(\mathcal{C},\mathcal{D})$ is the same as giving a
lax symmetric monoidal functor $\mathcal{C}\otimes\mathcal{E}\to\mathcal{D}$.
For $\mathcal{E}=\mathcal{C}\otimes\mathcal{D}$, we have the lax symmetric
monoidal functor
$\mathcal{C}\otimes\mathcal{E}=\mathcal{C}\otimes\mathcal{C}\otimes\mathcal{D}\xrightarrow{m\otimes\operatorname{id}}\mathcal{C}\otimes\mathcal{D}\xrightarrow{\underline{\operatorname{Hom}}_{\mathcal{C}}(\mathds{1}_{\mathcal{C}},-)\otimes\operatorname{id}}\mathcal{D}$
(the key point being that $\mathds{1}$ is compact and
$\underline{\operatorname{Hom}}_{\mathcal{C}}(\mathds{1},-):\mathcal{C}\to\operatorname{Vect}$
is lax symmetric monoidal). This gives rise to the lax symmetric monoidal
functor
$\mathcal{C}\otimes\mathcal{D}\to\mathsf{Hom}(\mathcal{C},\mathcal{D})$. By
definition, this is the composite equivalence from before. One can check that
this lax symmetric monoidal equivalence is actually symmetric monoidal.  that
the composite equivalence is symmetric monoidal.

To summarize, we obtain the following under the rigidity assumption on
$\mathcal{C}$. Below, for $F:\mathcal{C}\to\mathcal{D}$ (possibly lax)
$\mathbb{E}_{n}$-monoidal, we let
$\mathcal{K}_{F}\in\mathcal{C}\otimes\mathcal{D}$ be the corresponding object.

  1. (1)

The $\mathbb{E}_{n}$-center of $\mathcal{C}$ is the category of
$\mathbb{E}_{n}$-modules for
$\mathcal{K}_{\operatorname{id}_{\mathcal{C}}}\in\mathbb{E}_{n}\text{\textendash}{\mathsf{alg}}(\mathcal{C}\otimes\mathcal{D})$.

  2. (2)

More generally, for $F:\mathcal{C}\to\mathcal{D}$ lax $\mathbb{E}_{n}$-linear,
the $\mathbb{E}_{n}$-centralizer of $F$ is the category of
$\mathbb{E}_{n}$-modules for
$\mathcal{K}_{F}\in\mathbb{E}_{n}\text{\textendash}{\mathsf{alg}}(\mathcal{C}\otimes\mathcal{D})$.

In particular, we have expressed $\mathbb{E}_{n}$-centers and centralizers in
terms of $\mathbb{E}_{n}$-modules.

######  Example 1.4.1.

For $\mathcal{C}=\mathsf{Rep}(\check{G})$, we find that
$Z_{\mathbb{E}_{2}}(\mathsf{Rep}(\check{G}))$ is the category of
$\mathbb{E}_{2}$-modules for the regular representation
$\mathcal{O}_{\check{G}}\in\mathsf{ComAlg}(\mathsf{Rep}(\check{G}\times\check{G}))$.

In §1.6 below, we will extend this discussion to the setting where there is a
parabolic subgroup; we remark that there, we actually do consider
$\mathbb{E}_{2}$-modules for lax monoidal functors and use centralizers rather
than centers.

######  Remark 1.4.2.

A technical remark: in [Ras21a] §6, it was observed that factorization
categories attached to _rigid_ symmetric monoidal categories have much more
favorable properties than general factorization categories. We consider the
use of rigidity in the above discussion to be related to this observation, but
here working purely in the $\mathbb{E}_{2}$-setting.

###  1.5. Chiral setting

Following standard analogies, we consider chiral (or factorization) algebras
as de Rham analogues of $\mathbb{E}_{2}$-algebras (which are Betti objects),
and similarly for $\mathbb{E}_{2}$-modules.

Recall from [Ras21a] that any symmetric monoidal category naturally gives rise
to a factorization category (analogous to thinking of an
$\mathbb{E}_{\infty}$-category as an $\mathbb{E}_{2}$-category), and any
commutative algebra in a symmetric monoidal category gives rise to a
factorization algebra in the associated factorization category (analogous to
thinking of an $\mathbb{E}_{\infty}$-algebra as an $\mathbb{E}_{2}$-algebra).

We can then form the category of factorization modules

$\mathsf{Sph}^{\operatorname{spec},\operatorname{naive}}:=\mathcal{O}_{\check{G}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(\check{G}\times\check{G})).$

In §2 and §5, we explain how to perform homological algebra corrections to
$\mathsf{Sph}^{\operatorname{spec},\operatorname{naive}}$ (analogous to
replacing $\mathsf{QCoh}$ by $\mathsf{IndCoh}$) to obtain a factorization
category $\mathsf{Sph}_{\check{G}}^{\operatorname{spec}}$.

Following Gaitsgory, we refer in this paper to this process of correcting
homological defects as _renormalization_. The specific technical issues we
face here are novel and significant. Their resolution occupies a large portion
of this work.

In §6, we explain how the Gaitsgory-Lurie paradigm plays out in the chiral
setting. Roughly speaking, their strategy for constructing the derived Satake
functor goes through without hiccups in our setting. The factorizable derived
Satake equivalence itself is stated as Theorem 6.6.1.

Our proof that the functor is an equivalence takes up §9. We do this by
reducing to the case that $G=T$ is a torus; in §7 we verify this case
directly. Our process of reduction to the torus uses the study of Jacquet
functors from [Ras21a]-[Ras16].

###  1.6. The Gaitsgory-Lurie heuristic (parabolic case)

As far as we are aware, Gaitsgory and Lurie did not observe that a similar
strategy also applies in the presence of a parabolic $P$ of $G$.

Let $N_{P}$ denote the unipotent radical of $P$ and let $M=P/N_{P}$. On the
geometric side, we consider the factorization category

$\mathsf{D}(\operatorname{Gr}_{G})_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$

where on the right hand side, we have formed the _coinvariant category_ , cf.
[Ras16]. By [Che20], we note that this category can also be thought of as

$\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N_{P^{-}}\mathfrak{L}^{+}M}$

i.e., _invariants_ for the _opposite_ parabolic.

On the spectral side, there are _two_ natural candidates to consider, attached
to two lax symmetric monoidal functors

$\operatorname{Chev}_{\Omega},\operatorname{Chev}_{\Upsilon}:\mathsf{Rep}(\check{G})\longrightarrow\mathsf{Rep}(\check{M}).$

Both functors come by first restricting a representation from $\check{G}$ to
$\check{P}$ and then applying a functor
$\mathsf{Rep}(\check{P})\to\mathsf{Rep}(\check{M})$.

For $\operatorname{Chev}_{\Omega}$, the relevant functor
$\mathsf{Rep}(\check{P})\to\mathsf{Rep}(\check{M})$ is (derived) invariants
for $\check{N}_{\check{P}}$.

The functor $\operatorname{Chev}_{\Upsilon}$ is more complicated to define.
Consider $\check{\mathfrak{n}}_{\check{P}}$ as a Lie algebra in
$\mathsf{Rep}(\check{M})$. We consider

$Z_{\mathsf{Rep}(\check{M}),\mathbb{E}_{1}}(\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M})),$

i.e., the $\mathbb{E}_{1}$-center (alias: Drinfeld center) in the sense of
$\mathsf{Rep}(\check{M})$-linear monoidal categories. Because
$\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))$
is itself symmetric monoidal, there is a natural functor

$\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))\longrightarrow
Z_{\mathsf{Rep}(\check{M}),\mathbb{E}_{1}}(\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M})).$

We remark that one may identify
$Z_{\mathsf{Rep}(\check{M}),\mathbb{E}_{1}}(\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))$
more explicitly with

$(\check{\mathfrak{n}}_{\check{P}}\otimes\operatorname{C}^{\cdot}(\mathbb{S}^{1}))\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))=U_{\mathbb{E}_{2}}(\check{\mathfrak{n}}_{\check{P}})\text{\textendash}\mathsf{mod}_{\mathbb{E}_{2}}(\mathsf{Rep}(\check{M})).$

(We highlight these later expressions are closer to those used in
[Ras21a]-[Ras16] and in the body of this paper, although they are perhaps less
conceptual.) Then $\operatorname{Chev}_{\Upsilon}$ comes by composing the
forgetful functor $\mathsf{Rep}(\check{G})\to\mathsf{Rep}(\check{P})$ with the
composite functor

$\mathsf{Rep}(\check{P})\subset\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))\longrightarrow
Z_{\mathsf{Rep}(\check{M}),\mathbb{E}_{1}}(\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))\xrightarrow{\operatorname{oblv}}\check{\mathfrak{n}}_{\check{P}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(\check{M}))\xrightarrow{\operatorname{oblv}}\mathsf{Rep}(\check{M}).$

Up to issues of completion, we expect the $\mathbb{E}_{2}$-centralizers
$Z_{\mathbb{E}_{2}}(\operatorname{Chev}_{\Omega})$ and
$Z_{\mathbb{E}_{2}}(\operatorname{Chev}_{\Upsilon})$ of these functors to be
“approximately” (i.e., up to issues of completion) equivalent by a Koszul
duality procedure.

Although $\operatorname{Chev}_{\Omega}$ is easier to define and arguably is
the better object, we use (the de Rham analogue of)
$\operatorname{Chev}_{\Upsilon}$. There are several reasons for this, which we
take a moment to record:

  * •

One should think that $\operatorname{Chev}_{\Upsilon}$ is adapted to
$\mathfrak{L}N_{P}\mathfrak{L}^{+}M$-coinvariants and
$\operatorname{Chev}_{\Omega}$ is adapted to invariants. One may use either
invariants or coinvariants due to [Che20], so we are similarly free to use
either $\operatorname{Chev}_{\Upsilon}$ or $\operatorname{Chev}_{\Omega}$.

  * •

The paper [Ras16] uses $\operatorname{Chev}_{\Upsilon}$, which makes our
references somewhat easier.

However, we remark that in the _quantum_ setting, more recent papers in the
subject, such as [CF21a], [Yan21], and [GL19] prefer
$\operatorname{Chev}_{\Omega}$.

  * •

When working factorizably (as opposed to at a single point), it appears to be
much more technically convenient to use $\operatorname{Chev}_{\Upsilon}$
rather than $\operatorname{Chev}_{\Omega}$. This was implicit in [Ras16], and
holds again in our present work, particularly as concerns renormalization, ULA
properties, and the use of t-structures.

With those remarks out of the way, we note that there is a pairing

$\mathsf{D}(\operatorname{Gr}_{G})_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\otimes\mathsf{D}(\operatorname{Gr}_{G})^{\mathfrak{L}N^{-},\psi}\longrightarrow(\mathsf{D}(\mathfrak{L}G)_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}$

by convolution. One can further $!$-restrict to:

$(\mathsf{D}(\mathfrak{L}(P\times
N_{M}^{-})_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M})^{\mathfrak{L}N^{-},\psi}=\mathsf{D}(\operatorname{Gr}_{M})^{\mathfrak{L}N_{M}^{-},\psi}$

for $N_{M}^{-}:=N^{-}\cap M$. Applying the equivalence (1.3.1) for $G$ and
$M$, we can rewrite the resulting functor as a pairing

$\mathsf{D}(\operatorname{Gr}_{G})_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\otimes\mathsf{Rep}(\check{G})\longrightarrow\mathsf{Rep}(\check{M}).$

By [Ras16] Theorem 4.15.1, the induced functor

$\mathsf{Rep}(\check{G})\longrightarrow\mathsf{Rep}(\check{M})$

obtained by pairing with the unit object in
$\mathsf{D}(\operatorname{Gr}_{G})_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}$ is
the functor $\operatorname{Chev}_{\Upsilon}$ (or rather, its de Rham
counterpart). If we imagined we were in the $\mathbb{E}_{2}$-setting, this
would yield a functor

$\mathsf{D}(\operatorname{Gr}_{G})_{\mathfrak{L}N_{P}\mathfrak{L}^{+}M}\longrightarrow
Z(\operatorname{Chev}_{\Upsilon})$

that ought to be an equivalence up to issues of renormalization.

Using the essentially the same discussion as in §1.5, we formulate and prove a
purely de Rham version of exactly this assertion, Theorem 6.12.3. As we
discuss in the paper, on underlying categories (i.e., forgetting factorization
structures), we recover (by different means) the main theorem of [ABG04].

######  Remark 1.6.1.

In truth, the references to [Ras16] above are for $P=B$. This is ultimately
due to a choice in [BG08] to work only with the Borel. Work in progress by
Færgeman-Hayash is extending [BG08] to the case of general parabolics. As
such, our results for general parabolics (besides $B$ and $G$) are conditional
on their work.

###  1.7. Notation and conventions

We work over a field $k$ of characteristic zero. We also assume that $k$ is
algebraically closed, which is not strictly necessary but simplifies some
notations.

Denote by $X$ a fixed smooth and connected (but not necessarily proper) curve
over $k$.

We fix a reductive group $G$ over $k$ and a Borel subgroup $B\subset G$. Let
$N\subset B$ denote the unipotent radical of $B$ and $T=B/N$ the Cartan.
Choose a splitting $T\to B$, so that we can consider the opposite Borel
$B^{-}$, with unipotent radical $N^{-}$. We will denote by $P$ a standard
parabolic, with unipotent radical $N_{P}$, Levi subgroup $M$, opposite
parabolic $P^{-}$, etc. The coweight lattice will be denoted by
$\Lambda=\operatorname{Hom}(\mathbb{G}_{m},T)$.

The Langlands dual group of $G$ (over $k$) will be denoted by $\check{G}$,
with corresponding standard parabolics $\check{P}$, etc.

We freely use the language of higher category theory as developed in [Lur17b]
and [Lur17a], and usefully summarized in [GR19] Chapter 1. We will use the
term _category_ to mean $(\infty,1)$_-category_ , and _DG category_ to mean
_presentable stable_ $k$_-linear category_. DG categories assemble into a
category $\mathsf{DGCat}$, which is symmetric monoidal with respect to the
Lurie tensor product. The unit for this tensor product is the DG category
$\operatorname{Vect}$ of (complexes of) $k$-vector spaces.

We work in the setting of derived algebraic geometry, i.e. our test objects
are understood to be affine DG schemes over $k$. For any prestack $Y$ over
$k$, we denote by $\mathsf{QCoh}(Y)$ the DG category of quasicoherent sheaves
on $Y$. If $Y$ is locally almost of finite type, we can also consider the DG
category $\mathsf{IndCoh}(Y)$ of ind-coherent sheaves, whose theory is
developed in [Gai13] and [GR19]. We will denote the DG category of D-modules
on such a prestack by $\mathsf{D}(Y)$.

We will make use of the theory of D-modules on certain infinite-dimensional
spaces such as loop groups, which has been developed in [Ras15b] and [Ber17].
We use the $\mathsf{D}^{*}$ version of this theory by default.

Finally, for a (possibly lax) prestack $Y$, we will denote by
$\mathsf{ShvCat}(Y)$ the category of sheaves of categories on $Y$. We refer
the reader to [Gai14] for more details on this notion, or [Ras15a] for the lax
setting.

###  1.8. Acknowledgements

As the above introduction should make clear, this paper develops an old idea
of Dennis Gaitsgory and Jacob Lurie.

We warmly thank Dennis Gaitsgory for the boundless generosity with which he
shares his ideas. We learned of his work with Lurie through conversations with
him, but also learned the major technical tools used in this paper from him.

In addition, we thank Dima Arinkin, David Ben-Zvi, Dario Beraldo, Lin Chen,
Gurbir Dhillon, Kevin Lin, Sergey Lysenko, and Nick Rozenblyum for
enlightening conversations related to this material.

S.R. was supported by NSF grant DMS-2101984 and a Sloan Research Fellowship.

##  2\. Renormalizing crystals of categories

Below $S$ will denote a smooth quasicompact scheme over $k$ unless otherwise
specified. This section lays the technical foundations for renormalization in
the presence of a crystal structure over $S$, i.e. a $\mathsf{D}(S)$-module
structure. Proposition 2.7.1 gives sufficient conditions under which the
canonical renormalization of a $\mathsf{D}(S)$-module category with compatible
t-structure admits a natural crystal structure. Definition 2.8.1 formulates
what it means for a crystal of categories with compatible t-structure to be
“almost ULA generated,” and Proposition 2.8.2 says that the canonical
renormalization of such a category is ULA generated.

### 2.1.

The forgetful functor

$\operatorname{oblv}_{S}:\mathsf{D}(S)\longrightarrow\mathsf{IndCoh}(S)$

is symmetric monoidal, and hence can be viewed as a morphism of
$\mathsf{D}(S)$-modules. Its left adjoint

$\operatorname{ind}_{S}:\mathsf{IndCoh}(S)\longrightarrow\mathsf{D}(S)$

therefore admits an oplax $\mathsf{D}(S)$-linear structure, which is easily
seen to be strict. Given a $\mathsf{D}(S)$-module category $\mathcal{C}$, we
therefore obtain adjoint functors

$\operatorname{ind}_{\mathcal{C}}:\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}\rightleftarrows\mathcal{C}:\operatorname{oblv}_{\mathcal{C}}.$
(2.1.1)

Moreover, for any $\mathsf{D}(S)$-linear functor $F:\mathcal{C}\to\mathcal{D}$
we have commutative squares

${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{D}}$${\mathcal{C}}$${\mathcal{D}}$$\scriptstyle{\operatorname{id}\otimes
F}$$\scriptstyle{F}$$\scriptstyle{\operatorname{oblv}_{\mathcal{C}}}$$\scriptstyle{\operatorname{oblv}_{\mathcal{D}}}$

and

${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{D}}$${\mathcal{C}}$${\mathcal{D}.}$$\scriptstyle{\operatorname{id}\otimes
F}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}}}$$\scriptstyle{\operatorname{ind}_{\mathcal{D}}}$$\scriptstyle{F}$

######  Proposition 2.1.1.

The adjunction (2.1.1) is both monadic and comonadic.

###### Proof.

For the monadicity it suffices to show that
$\operatorname{oblv}_{\mathcal{C}}$ is conservative, which is Lemma B.2.1 in
[Ras21a].

We claim that the monad
$\operatorname{oblv}_{\mathcal{C}}\circ\operatorname{ind}_{\mathcal{C}}$ is
_effective_ in the sense of [Ras15c] Definition 6.9.1, which will imply
comonadicity of the adjunction by Remark 6.9.2 of _loc. cit._ The convolution
action of the infinitesimal groupoid

$\widehat{(S^{2})}_{\Delta}\cong S\underset{S_{\operatorname{dR}}}{\times}S$

on $S$ gives rise to an action of
$\mathsf{IndCoh}(\widehat{(S^{2})}_{\Delta})$ on $\mathsf{IndCoh}(S)$, which
in turn induces an equivalence of monoidal categories

$\mathsf{IndCoh}(\widehat{(S^{2})}_{\Delta})\tilde{\longrightarrow}\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S)).$

The natural t-structure on $\mathsf{IndCoh}(\widehat{(S^{2})}_{\Delta})$
induces a t-structure on
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$, which we shift by
$\dim S$ so that the coconnective objects are exactly the left t-exact
functors (i.e. we normalize so that the unit
$\operatorname{id}_{\mathsf{IndCoh}(S)}$ belongs to the heart).

The monad $\operatorname{oblv}_{S}\circ\operatorname{ind}_{S}$ lifts to an
associative algebra in
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$, which corresponds to
the algebra of differential operators in
$\mathsf{IndCoh}(\widehat{(S^{2})}_{\Delta})$. According to Lemma 6.17.1 of
_loc. cit._ , to prove that the monad is effective, it suffices to show that
the following conditions are satisfied:

  1. (i)

$\operatorname{oblv}_{S}\circ\operatorname{ind}_{S}$ is coconnective in
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$;

  2. (ii)

$\operatorname{oblv}_{S}\circ\operatorname{ind}_{S}$ is right flat, i.e. the
functor

$\displaystyle\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$
$\displaystyle\longrightarrow\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$
$\displaystyle F$
$\displaystyle\mapsto\operatorname{oblv}_{S}\circ\operatorname{ind}_{S}\circ
F$

is left t-exact;

  3. (iii)

the object

$\operatorname{cofib}(\operatorname{id}_{\mathsf{IndCoh}(S)}\longrightarrow\operatorname{oblv}_{S}\circ\operatorname{ind}_{S})$

in $\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$ is right flat and
eventually connective.

All three of these conditions follow from the existence of the standard
filtration on differential operators. Namely, the corresponding filtration on
$\operatorname{oblv}_{S}\circ\operatorname{ind}_{S}$ in
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$ has associated graded
given by tensoring with the locally free sheaf
$\operatorname{Sym}_{\mathcal{O}_{S}}\operatorname{T}(S)$ (using the action of
$\mathsf{QCoh}(S)$ on $\mathsf{IndCoh}(S)$).

∎

### 2.2.

Recall that for any DG categories $\mathcal{C}$ and $\mathcal{D}$ equipped
with t-structures, their tensor product $\mathcal{C}\otimes\mathcal{D}$ admits
a canonical t-structure, whose connective subcategory
$(\mathcal{C}\otimes\mathcal{D})^{\leq 0}$ is generated under colimits by
objects $c\boxtimes d$ with $c$ and $d$ connective. If
$\mathcal{C}=\mathsf{D}(S_{1})$ and $\mathcal{D}=\mathsf{D}(S_{2})$, then this
t-structure makes the natural equivalence

$\mathsf{D}(S_{1})\otimes\mathsf{D}(S_{2})\tilde{\longrightarrow}\mathsf{D}(S_{1}\times
S_{2})$

t-exact.

We will use the following technical lemma frequently. Recall that the
t-structure on $\mathcal{C}$ is said to be _compactly generated_ if the
category $\mathcal{C}^{\leq 0}$ is compactly generated.

######  Lemma 2.2.1.

Let $\mathcal{C}_{1}$, $\mathcal{C}_{2}$, and $\mathcal{D}$ be DG categories
equipped with t-structures, and suppose we are given a functor
$F:\mathcal{C}_{1}\to\mathcal{C}_{2}$.

  1. (i)

If $F$ is right t-exact, then so is

$F\otimes\operatorname{id}_{\mathcal{D}}:\mathcal{C}_{1}\otimes\mathcal{D}\longrightarrow\mathcal{C}_{2}\otimes\mathcal{D}.$

  2. (ii)

Suppose that $F$ is left t-exact, and that either

    1. (a)

the t-structures on $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are compactly
generated, or

    2. (b)

the t-structure on $\mathcal{D}$ is compactly generated.

Then $F\otimes\operatorname{id}_{\mathcal{D}}$ is left t-exact.

###### Proof.

This is a combination of Proposition A.1, Lemma A.6, and Lemma A.9 in [YR22].

∎

###  2.3. Compatible t-structures

The Verdier self-duality equivalence

$\mathsf{D}(S)^{\vee}\cong\mathsf{D}(S)$

interchanges the symmetric monoidal structure whose multiplication is given by
$\otimes^{!}$ with the cocommutative coalgebra structure on $\mathsf{D}(S)$
whose comultiplication is given by

$\Delta_{\operatorname{dR},*}:\mathsf{D}(S)\longrightarrow\mathsf{D}(S\times
S)\cong\mathsf{D}(S)\otimes\mathsf{D}(S).$

In particular, a $(\mathsf{D}(S),\otimes^{!})$-module structure on a DG
category $\mathcal{C}$ is the same datum as a
$(\mathsf{D}(S),\Delta_{\operatorname{dR},*})$-comodule structure. In what
follows we pass freely between the two.

Similarly to the case of D-modules, the Serre self-duality equivalence

$\mathsf{IndCoh}(S)^{\vee}\cong\mathsf{IndCoh}(S)$

interchanges the symmetric monoidal structure whose multiplication is given by
$\otimes^{!}$ with the cocommutative coalgebra structure on
$\mathsf{IndCoh}(S)$ whose comultiplication is given by $\Delta_{*}$, so that
an $(\mathsf{IndCoh}(S),\otimes^{!})$-module structure is the same datum as an
$(\mathsf{IndCoh}(S),\Delta_{*})$-comodule structure.

######  Definition 2.3.1.

Given a $\mathsf{D}(S)$-module category $\mathcal{C}$, we say that a
t-structure on $\mathcal{C}$ is _compatible with the_ $\mathsf{D}(S)$_-module
structure_ if the corresponding coaction map

$\mathcal{C}\longrightarrow\mathsf{D}(S)\otimes\mathcal{C}$

is t-exact. Similarly, if $\mathcal{D}$ is an $\mathsf{IndCoh}(S)$-module
category, we say that a t-structure on $\mathcal{D}$ is _compatible with the_
$\mathsf{IndCoh}(S)$_-module structure_ if the coaction map

$\mathcal{D}\to\mathsf{IndCoh}(S)\otimes\mathcal{D}$

is t-exact.

######  Proposition 2.3.2.

If $\mathcal{C}$ is a $\mathsf{D}(S)$-module category equipped with a
compatible t-structure, then the category
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ admits a unique
t-structure such that $\operatorname{ind}_{\mathcal{C}}$ is t-exact. Moreover,
this t-structure makes $\operatorname{oblv}_{\mathcal{C}}$ t-exact and is
compatible with the $\mathsf{IndCoh}(S)$-action.

###### Proof.

We claim that the comonad
$\operatorname{ind}_{\mathcal{C}}\circ\operatorname{oblv}_{\mathcal{C}}$ is
t-exact. In view of the previous proposition, this will imply that there is a
unique t-structure on $\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$
which makes $\operatorname{ind}_{\mathcal{C}}$ t-exact (because the comonad is
left t-exact), and moreover that $\operatorname{oblv}_{\mathcal{C}}$ is
t-exact with respect to this t-structure (because the comonad is t-exact).

The cartesian square

${S}$${S\times
S_{\operatorname{dR}}}$${S_{\operatorname{dR}}}$${S_{\operatorname{dR}}\times
S_{\operatorname{dR}}.}$

gives rise to a commutative square

${\mathsf{IndCoh}(S)}$${\mathsf{IndCoh}(S)\otimes\mathsf{D}(S)}$${\mathsf{D}(S)}$${\mathsf{D}(S)\otimes\mathsf{D}(S),}$$\scriptstyle{\operatorname{ind}_{S}}$$\scriptstyle{\operatorname{ind}_{S}\otimes\operatorname{id}_{\mathsf{D}(S)}}$

where the functors are given by $\mathsf{IndCoh}$ direct image, and base
change implies that the square

${\mathsf{D}(S)}$${\mathsf{D}(S)\otimes\mathsf{D}(S)}$${\mathsf{IndCoh}(S)}$${\mathsf{IndCoh}(S)\otimes\mathsf{D}(S)}$$\scriptstyle{\operatorname{oblv}_{S}}$$\scriptstyle{\operatorname{oblv}_{S}\otimes\operatorname{id}_{\mathsf{D}(S)}}$

also commutes. Both of these squares take place in the category of
$\mathsf{D}(S)$-modules, where $\mathsf{D}(S)$ acts on the right factor of the
tensor product categories. Notice also that the horizontal functors are fully
faithful and admit $\mathsf{D}(S)$-linear right adjoints. Thus, tensoring over
$\mathsf{D}(S)$ with $\mathcal{C}$, we obtain a commutative diagram

${\mathcal{C}}$${\mathsf{D}(S)\otimes\mathcal{C}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathsf{IndCoh}(S)\otimes\mathcal{C}}$${\mathcal{C}}$${\mathsf{D}(S)\otimes\mathcal{C}.}$$\scriptstyle{\operatorname{oblv}_{\mathcal{C}}}$$\scriptstyle{\operatorname{oblv}_{S}\otimes\operatorname{id}_{\mathcal{C}}}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}}}$$\scriptstyle{\operatorname{ind}_{S}\otimes\operatorname{id}_{\mathcal{C}}}$

where the horizontal functors are fully faithful. In order to prove that
$\operatorname{ind}_{\mathcal{C}}\circ\operatorname{oblv}_{\mathcal{C}}$ is
t-exact, it therefore suffices to prove that the bottom-left circuit of the
outer square is t-exact, since the bottom horizontal functor is the coaction
functor and hence t-exact by hypothesis.

Since the upper horizontal functor is t-exact, it is enough to prove that both
right vertical functors are t-exact. The functors $\operatorname{oblv}_{S}$
and $\operatorname{ind}_{S}$ are both t-exact and the t-structures on
$\mathsf{IndCoh}(S)$ and $\mathsf{D}(S)$ are compactly generated, so this
follows from Lemma 2.2.1.

To see that the t-structure on
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ is compatible with the
$\mathsf{IndCoh}(S)$-action, consider the commutative square

${S}$${S\times S}$${S_{\operatorname{dR}}}$${S_{\operatorname{dR}}\times
S_{\operatorname{dR}}.}$

Passing to ind-coherent sheaves under direct image and tensoring over
$\mathsf{D}(S)$ with $\mathcal{C}$, we see that the coaction functors fit into
a commutative square

${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathsf{IndCoh}(S)\otimes\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathcal{C}}$${\mathsf{D}(S)\otimes\mathcal{C}.}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}}}$$\scriptstyle{\operatorname{ind}_{S}\otimes\operatorname{ind}_{\mathcal{C}}}$

Since the lower-left circuit of this square is t-exact, it suffices to show
that $\operatorname{ind}_{S}\otimes\operatorname{ind}_{\mathcal{C}}$ is
t-exact and conservative. We have seen above that

$\operatorname{ind}_{S}\otimes\operatorname{id}_{\mathcal{C}}:\mathsf{IndCoh}(S)\otimes\mathcal{C}\longrightarrow\mathsf{D}(S)\otimes\mathcal{C}$

is t-exact. Since $\operatorname{ind}_{\mathcal{C}}$ is t-exact and the
t-structure on $\mathsf{IndCoh}(S)$ is compactly generated, the functor

$\operatorname{id}_{\mathsf{IndCoh}(S)}\otimes\operatorname{ind}_{\mathcal{C}}:\mathsf{IndCoh}(S)\otimes\mathcal{C}\longrightarrow\mathsf{IndCoh}(S)\otimes\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}$

is t-exact by Lemma 2.2.1. Note that the proof of Proposition 2.1.1 shows that
$\operatorname{ind}_{S}\otimes\operatorname{id}_{\mathcal{C}}$ and
$\operatorname{id}_{\mathsf{IndCoh}(S)}\otimes\operatorname{ind}_{\mathcal{C}}$
are both comonadic and in particular conservative. Thus their composition
$\operatorname{ind}_{S}\otimes\operatorname{ind}_{\mathcal{C}}$ is t-exact and
conservative as claimed.

∎

###  2.4. Almost compact generation

This section borrows heavily from the appendix to [BZNP17]. We warn the reader
that our terminology does not agree with _loc. cit._ in some places. For
example, we use the term “coherent t-structure” for a weaker notion than
theirs.

Let $\mathcal{C}$ be a DG category equipped with a t-structure. Recall that an
object $c$ in $\mathcal{C}$ is called _almost compact_ if its truncation
$\tau^{\geq n}c$ is compact in $\mathcal{C}^{\geq n}$ for any
$n\in\mathbb{Z}$. If $c$ is eventually coconnective and almost compact, then
$c$ is called _coherent_. Denote by
$\mathcal{C}^{\operatorname{coh}}\subset\mathcal{C}$ the (non-cocomplete) full
subcategory consisting of coherent objects.

If the t-structure on $\mathcal{C}$ is right complete, then any almost compact
object is eventually connective, and hence any coherent object is
cohomologically bounded.

We will use the folllowing technical lemma repeatedly.

######  Lemma 2.4.1.

Let $\mathcal{C}$ and $\mathcal{D}$ be DG categories equipped with
t-structures which are compatible with filtered colimits and right complete.
Suppose that $F:\mathcal{C}\to\mathcal{D}$ is t-exact and admits a continuous
right adjoint, and moreover that $F$ is conservative on $\mathcal{C}^{+}$. If
$c$ belongs to $\mathcal{C}^{+}$ and $F(c)$ is almost compact, then $c$ is
almost compact (equivalently, coherent).

###### Proof.

See Step 3 in the proof of Lemma 6.11.2 in [Ras19].

∎

######  Definition 2.4.2.

We say that $\mathcal{C}$ is _coherent_ with respect to its t-structure if the
inclusion

$\mathcal{C}^{\geq 0}\longrightarrow\mathcal{C}^{\geq-1}$

preserves compact objects.

For example, if $A$ is a classical commutative algebra, then
$A\text{\textendash}\mathsf{mod}$ with its usual t-structure is coherent if
and only if $A$ is coherent, i.e. every finitely generated ideal is finitely
presented.

######  Proposition 2.4.3.

Suppose that the t-structure on $\mathcal{C}$ is compatible with filtered
colimits. Then $\mathcal{C}$ is coherent if and only if almost compact objects
in $\mathcal{C}$ are stable under truncation.

In particular, if $\mathcal{C}$ is coherent then the subcategory
$\mathcal{C}^{\operatorname{coh}}$ is stable under truncation functors, and
hence inherits a t-structure from $\mathcal{C}$. This t-structure extends
uniquely to a t-structure on the _canonical renormalization_

$\mathcal{C}^{\operatorname{ren}}:=\mathsf{Ind}(\mathcal{C}^{\operatorname{coh}})$

which is compatible with filtered colimits. The unique continuous extension

$\mathcal{C}^{\operatorname{ren}}\longrightarrow\mathcal{C}$

of the inclusion $\mathcal{C}^{\operatorname{coh}}\subset\mathcal{C}$ is
t-exact.

######  Definition 2.4.4.

We call $\mathcal{C}$ _almost compactly generated_ with respect to its
t-structure if it satisfies the following conditions:

  1. (i)

the t-structure is compatible with filtered colimits and right complete;

  2. (ii)

$\mathcal{C}$ is coherent;

  3. (iii)

$\mathcal{C}^{\geq 0}$ is compactly generated.

######  Remark 2.4.5.

Conditions (i) and (iii) are usually rather easy to verify in practice, while
coherence is more subtle. One particularly bothersome example of this is the
fact that the tensor product of coherent categories need not be coherent; for
instance, it is well-known that a ring of polynomials with coefficients in a
coherent ring need not be coherent.

The canonical renormalization behaves well for almost compactly generated
categories.

######  Proposition 2.4.6.

If $\mathcal{C}$ is almost compactly generated, then the t-exact functor
$\mathcal{C}^{\operatorname{ren}}\to\mathcal{C}$ restricts to an equivalence

$\mathcal{C}^{\operatorname{ren},+}\tilde{\longrightarrow}\mathcal{C}^{+}.$

The next lemma supplies a convenient criterion for almost compact generation.

######  Lemma 2.4.7.

Suppose that $\mathcal{C}$ is equipped with a t-structure such that

  1. (i)

the t-structure is compatible with filtered colimits and right complete;

  2. (ii)

the category $\mathcal{C}^{\heartsuit}$ is compactly generated;

  3. (iii)

any compact object in $\mathcal{C}^{\heartsuit}$ is almost compact in
$\mathcal{C}$.

Then $\mathcal{C}$ is almost compactly generated.

### 2.5.

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are DG categories equipped with
t-structures. We will write

$\operatorname{Fun}_{\operatorname{fha}}(\mathcal{C},\mathcal{D})\subset\operatorname{Fun}(\mathcal{C},\mathcal{D})$

for the full subcategory consisting of exact continuous functors of finite
homological amplitude, or equivalently those exact continuous functors which
send $\mathcal{C}^{+}$ into $\mathcal{D}^{+}$. Denote by

$\operatorname{Fun}_{+\text{-}\\!\operatorname{cts}}(\mathcal{C}^{+},\mathcal{D}^{+})\subset\operatorname{Fun}(\mathcal{C}^{+},\mathcal{D}^{+})$

the full subcategory consisting of exact functors which commute with filtered
colimits bounded uniformly from below. There is a natural restriction functor

$\operatorname{Fun}_{\operatorname{fha}}(\mathcal{C},\mathcal{D})\longrightarrow\operatorname{Fun}_{+\text{-}\\!\operatorname{cts}}(\mathcal{C}^{+},\mathcal{D}^{+}).$
(2.5.1)

######  Lemma 2.5.1.

Suppose that $\mathcal{C}$ is almost compactly generated, and that
$\mathcal{C}^{\operatorname{ren}}\tilde{\to}\mathcal{C}$. Then the functor
(2.5.1) is an equivalence.

###### Proof.

The inverse functor is given by the composite

$\operatorname{Fun}_{+\text{-}\\!\operatorname{cts}}(\mathcal{C}^{+},\mathcal{D}^{+})\longrightarrow\operatorname{Fun}(\mathcal{C}^{\operatorname{coh}},\mathcal{D})\longrightarrow\operatorname{Fun}(\mathcal{C},\mathcal{D}),$

where the first functor is restriction and the second is left Kan extension.
Namely, as observed in Example 4.4.4 of [Ras19], the fact that
$\mathcal{C}^{\operatorname{coh}}$ is stable under truncation functors implies
that any functor in
$\operatorname{Fun}_{+\text{-}\\!\operatorname{cts}}(\mathcal{C}^{+},\mathcal{D}^{+})$
is left Kan extended from $\mathcal{C}^{\operatorname{coh}}$.

∎

###  2.6. Functorial renormalization

Let us write $\mathsf{DGCat}_{\operatorname{t-str}}$ for the category of DG
categories equipped with t-structures, with $1$-morphisms given by all
continuous functors (we impose no t-exactness conditions on functors here). As
described above, the Lurie tensor product of two objects in
$\mathsf{DGCat}_{\operatorname{t-str}}$ admits a canonical t-structure, and
hence the Lurie tensor product lifts to a symmetric monoidal structure on
$\mathsf{DGCat}_{\operatorname{t-str}}$. We denote by
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}$ the associated pseudo-
tensor category.333The term “pseudo-tensor category,” which we have borrowed
from Beilinson-Drinfeld, is formally identical to “colored operad” but plays a
different conceptual role. A morphism from a colored operad to a pseudo-tensor
category should be thought of as an algebra over the colored operad in the
pseudo-tensor category. In particular, for us colored operads are “small,”
while pseudo-tensor categories tend to be “large.”

We define a $1$-full pseudo-tensor subcategory

$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}\subset(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}$

as follows. An object in $\mathsf{DGCat}_{\operatorname{t-str}}$ belongs to
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
provided that it is almost compactly generated. An $n$-ary morphism

$\mathcal{C}_{1}\otimes\cdots\otimes\mathcal{C}_{n}\longrightarrow\mathcal{D}$

belongs to
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
provided that the image of the composition

$\mathcal{C}_{1}^{+}\times\cdots\times\mathcal{C}_{n}^{+}\longrightarrow\mathcal{C}_{1}\otimes\cdots\otimes\mathcal{C}_{n}\longrightarrow\mathcal{D}$

is contained in $\mathcal{D}^{+}$.

Next, consider the symmetric monoidal category
$\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}}$ of arrows in
$\mathsf{DGCat}_{\operatorname{t-str}}$. Define a $1$-full pseudo-tensor
subcategory

$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}\subset(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}$

as follows. An object $\mathcal{C}^{\prime}\to\mathcal{C}$ in
$\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}}$ belongs to
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$
provided that the following conditions are satisfied:

  1. (i)

$\mathcal{C}^{\prime}\to\mathcal{C}$ is a t-exact functor;

  2. (ii)

$\mathcal{C}$ and $\mathcal{C}^{\prime}$ are almost compactly generated;

  3. (iii)

the category $\mathcal{C}^{\prime}$ is compactly generated by
$(\mathcal{C}^{\prime})^{\operatorname{coh}}$;

  4. (iv)

$\mathcal{C}^{\prime}\to\mathcal{C}$ induces an equivalence
$(\mathcal{C}^{\prime})^{+}\tilde{\to}\mathcal{C}^{+}$.

(Note that these conditions imply that $\mathcal{C}^{\prime}\to\mathcal{C}$
factors uniqely through a continuous t-exact equivalence
$\mathcal{C}^{\prime}\tilde{\to}\mathcal{C}^{\operatorname{ren}}$.) Given
objects

$\mathcal{C}_{1}^{\prime}\to\mathcal{C}_{1},\cdots\\!,\mathcal{C}_{n}^{\prime}\to\mathcal{C}_{n},\mathcal{D}^{\prime}\to\mathcal{D}$

in
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$,
an $n$-ary morphism

${\mathcal{C}_{1}^{\prime}\otimes\cdots\otimes\mathcal{C}_{n}^{\prime}}$${\mathcal{D}^{\prime}}$${\mathcal{C}_{1}\otimes\cdots\otimes\mathcal{C}_{n}}$${\mathcal{D},}$

in $(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}$ is an
$n$-ary morphism in
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$
provided that both horizontal functors are $n$-ary morphisms in
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$ (in
fact, it suffices to impose this condition on the lower horizontal functor).

######  Proposition 2.6.1.

The second projection

$\displaystyle\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}}$
$\displaystyle\longrightarrow\mathsf{DGCat}_{\operatorname{t-str}}$
$\displaystyle(\mathcal{C}^{\prime}\to\mathcal{C})$
$\displaystyle\mapsto\mathcal{C}$

induces an equivalence of pseudo-tensor categories

$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}\tilde{\longrightarrow}(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}.$

###### Proof.

This functor is essentially surjective on the underlying categories because
for any almost compactly generated category $\mathcal{C}$, the canonical
renormalization $\mathcal{C}^{\operatorname{ren}}\to\mathcal{C}$ is an object
of
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$
which projects to $\mathcal{C}$.

To see that this functor is fully faithful on $n$-ary morphisms, suppose that
we are given objects

$\mathcal{C}_{1}^{\prime}\to\mathcal{C}_{1},\cdots\\!,\mathcal{C}_{n}^{\prime}\to\mathcal{C}_{n},\mathcal{D}^{\prime}\to\mathcal{D}$

in
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$.
We claim that any $n$-ary morphism

$\mathcal{C}_{1}\otimes\cdots\otimes\mathcal{C}_{n}\longrightarrow\mathcal{D}$

in $(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
lifts to an $n$-ary morphism in
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$
which is unique up to contractible choice. Namely, the image of the
composition

$(\mathcal{C}_{1}^{\prime})^{\operatorname{coh}}\times\cdots\times(\mathcal{C}_{n}^{\prime})^{\operatorname{coh}}\tilde{\longrightarrow}\mathcal{C}_{1}^{\operatorname{coh}}\times\cdots\times\mathcal{C}_{n}^{\operatorname{coh}}\longrightarrow\mathcal{D}$

is contained in $\mathcal{D}^{+}$. Since this functor is exact in each
variable separately, it follows that there is a unique continuous extension to

$\mathcal{C}_{1}^{\prime}\otimes\cdots\otimes\mathcal{C}_{n}^{\prime}=\mathsf{Ind}((\mathcal{C}_{1}^{\prime})^{\operatorname{coh}})\otimes\cdots\otimes\mathsf{Ind}((\mathcal{C}_{n}^{\prime})^{\operatorname{coh}})\longrightarrow\mathcal{D}.$

Note that this extension defines an $n$-ary morphism in
$(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}$,
since any object in $(\mathcal{C}_{i}^{\prime})^{+}$ is a filtered colimit of
objects in $(\mathcal{C}_{i}^{\prime})^{\operatorname{coh}}$ bounded uniformly
from below. This completes the proof.

∎

We will make frequent use of the endomorphism of pseudo-tensor categories

$\displaystyle(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
$\displaystyle\longrightarrow(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
(2.6.1) $\displaystyle\mathcal{C}$
$\displaystyle\mapsto\mathcal{C}^{\operatorname{ren}}$

obtained as the composition

$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}\tilde{\longrightarrow}(\mathsf{DGCat}_{\operatorname{t-str}}^{\Delta^{1}})^{\otimes}_{\operatorname{ren}}\longrightarrow(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}},$

where the first functor is inverse to the equivalence of Proposition 2.6.1 and
the second functor is the first projection
$(\mathcal{C}^{\prime}\to\mathcal{C})\mapsto\mathcal{C}^{\prime}$.

###  2.7. Renormalization for crystals of categories

We continue to let $S$ denote a smooth quasicompact scheme.

Consider the pseudo-tensor category
$(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}$
whose objects are $\mathsf{D}(S)$-module categories equipped with a compatible
t-structure, with $n$-ary morphisms
$(\mathcal{C}_{1},\cdots\\!,\mathcal{C}_{n})\to\mathcal{D}$ given by arbitrary
$\mathsf{D}(S)$-linear functors

$\mathcal{C}_{1}\underset{\mathsf{D}(S)}{\otimes}\cdots\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}_{n}\longrightarrow\mathcal{D}.$

We define a pseudo-tensor subcategory

$(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}\subset(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}$

a follows. An object $\mathcal{C}$ of
$(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}$
belongs to
$(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
if $\mathcal{C}$ is almost compactly generated. An $n$-ary morphism

$\mathcal{C}_{1}\underset{\mathsf{D}(S)}{\otimes}\cdots\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}_{n}\longrightarrow\mathcal{D}$

belongs to
$(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
provided that the image of the composition

$\mathcal{C}_{1}^{+}\times\cdots\times\mathcal{C}_{n}^{+}\longrightarrow\mathcal{C}_{1}\underset{\mathsf{D}(S)}{\otimes}\cdots\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}_{n}\longrightarrow\mathcal{D}$

is contained in $\mathcal{D}^{+}$.

######  Proposition 2.7.1.

The morphism (2.6.1) lifts canonically to a morphism of pseudo-tensor
categories

$\displaystyle(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
$\displaystyle\longrightarrow(\mathsf{D}(S)\text{\textendash}\mathsf{mod}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
$\displaystyle\mathcal{C}$
$\displaystyle\mapsto\mathcal{C}^{\operatorname{ren}}.$

###### Proof.

Observe that $\mathsf{D}(S)$ is a commutative algebra object in
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
satisfying $\mathsf{D}(S)^{\operatorname{ren}}\tilde{\to}\mathsf{D}(S)$.
Moreover, for any $\mathsf{D}(S)$-module category $\mathcal{C}$, the coaction
functor

$\operatorname{coact}_{\mathcal{C}}:\mathcal{C}\longrightarrow\mathsf{D}(S)\otimes\mathcal{C}$

is left adjoint to the action functor

$\operatorname{act}_{\mathcal{C}}:\mathsf{D}(S)\otimes\mathcal{C}\longrightarrow\mathcal{C}.$

It follows that if $\mathcal{C}$ is equipped with a compatible t-structure,
then $\operatorname{act}_{\mathcal{C}}$ is left t-exact, and hence the
$\mathsf{D}(S)$-action on $\mathcal{C}$ takes place in
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$
provided that $\mathcal{C}$ is almost compactly generated. All of this implies
that (2.6.1) is functorial on $\mathsf{D}(S)$-modules; we need only check that
if the $\mathsf{D}(S)$-action on $\mathcal{C}$ is compatible with the
t-structure, then so is the induced action on
$\mathcal{C}^{\operatorname{ren}}$.

First, we claim that the functor
$\operatorname{coact}_{\mathcal{C}^{\operatorname{ren}}}$ has finite
homological amplitude. Since $(\mathcal{C}^{\operatorname{ren}})^{+}$ is
generated by compact objects of $\mathcal{C}^{\operatorname{ren}}$ under
filtered colimits bounded uniformly from below, it suffices to show that
$\operatorname{coact}_{\mathcal{C}^{\operatorname{ren}}}$ sends compact
objects to eventually coconnective objects. Observe that
$\operatorname{coact}_{\mathcal{C}^{\operatorname{ren}}}$ is left adjoint to
the continuous functor $\operatorname{act}_{\mathcal{C}^{\operatorname{ren}}}$
and hence preserves compact objects, so the claim will follow if we prove that
compact objects in $\mathsf{D}(S)\otimes\mathcal{C}^{\operatorname{ren}}$ are
eventually coconnective. The category
$\mathsf{D}(S)\otimes\mathcal{C}^{\operatorname{ren}}$ is compactly generated
by objects of the form $\mathcal{M}\boxtimes c$ where $\mathcal{M}$ and $c$
are compact in $\mathsf{D}(S)$ in $\mathcal{C}^{\operatorname{ren}}$
respectively, so we need only prove that objects of this form are eventually
coconnective, since they generate all compact objects in
$\mathsf{D}(S)\otimes\mathcal{C}^{\operatorname{ren}}$ under finite colimits
and retracts. But compact objects in $\mathsf{D}(S)$ and
$\mathcal{C}^{\operatorname{ren}}$ are eventually coconnective, which implies
that objects $\mathcal{M}\boxtimes c$ as above are eventually coconnective, by
Lemma 2.2.1.

To see that $\operatorname{coact}_{\mathcal{C}^{\operatorname{ren}}}$ is
t-exact, consider the commutative square

${\mathcal{C}^{\operatorname{ren}}}$${\mathsf{D}(S)\otimes\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}}$${\mathsf{D}(S)\otimes\mathcal{C}.}$

The left vertical and lower horizontal functors are t-exact, and the right
vertical functor is t-exact by Lemma 2.2.1. We have already shown that the
upper horizontal functor has finite homological amplitude, and hence we obtain
a commutative square

${(\mathcal{C}^{\operatorname{ren}})^{+}}$${(\mathsf{D}(S)\otimes\mathcal{C}^{\operatorname{ren}})^{+}}$${\mathcal{C}^{+}}$${(\mathsf{D}(S)\otimes\mathcal{C})^{+}.}$

Since $(\mathcal{C}^{\operatorname{ren}})^{+}\to\mathcal{C}^{+}$ is an
equivalence and in particular conservative, Lemma 4.6.2(3) in [Ras19] implies
that the right vertical functor is conservative. It follows immediately that
$\operatorname{coact}_{\mathcal{C}^{\operatorname{ren}}}$ is left t-exact. As
for right t-exactness, since the t-structure on
$\mathcal{C}^{\operatorname{ren}}$ is compactly generated, it suffices to show
that $\operatorname{coact}_{\mathcal{C}^{\operatorname{ren}}}$ sends compact
objects in $(\mathcal{C}^{\operatorname{ren}})^{\leq 0}$ into
$(\mathsf{D}(S)\otimes\mathcal{C}^{\operatorname{ren}})^{\leq 0}$. Since
compact objects in $\mathcal{C}^{\operatorname{ren}}$ are eventually
coconnective, this follows from the corresponding property of
$\operatorname{coact}_{\mathcal{C}}$ using the same commutative square.

∎

###  2.8. Almost ULA generation

Suppose that we are given a $\mathsf{D}(S)$-module category $\mathcal{C}$ with
compatible t-structure.

######  Definition 2.8.1.

An object $c$ in $\mathcal{C}$ will be called _almost ULA_ over $S$ if
$\operatorname{oblv}_{\mathcal{C}}(c)$ is almost compact in
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$. We will say that
$\mathcal{C}$ is _almost ULA generated_ if $\mathcal{C}$ is almost compactly
generated, and moreover
$(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\geq 0}$ is compactly
generated by objects of the form

$\tau^{\geq 0}(\mathcal{F}\otimes\operatorname{oblv}_{\mathcal{C}}(c))$

where $\mathcal{F}$ is a perfect complex on $S$ and $c$ is almost ULA in
$\mathcal{C}$.

As the name suggests, the renormalization of an almost ULA generated category
is ULA generated.

######  Proposition 2.8.2.

Suppose that a $\mathsf{D}(S)$-module category $\mathcal{C}$ is almost ULA
generated. Then the $\mathsf{D}(S)$-module category
$\mathcal{C}^{\operatorname{ren}}$ is ULA generated by the objects $\tau^{\geq
n}c$ in

$\mathcal{C}^{+}\cong(\mathcal{C}^{\operatorname{ren}})^{+}\subset\mathcal{C}^{\operatorname{ren}},$

where $n\in\mathbb{Z}$ and $c$ is almost ULA in $\mathcal{C}$.

This proposition is immediate from the following lemma.

######  Lemma 2.8.3.

Let $\mathcal{C}$ be a $\mathsf{D}(S)$-module category equipped with
compatible t-structure, and suppose that $\mathcal{C}$ is almost compactly
generated. Then $\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ is
coherent.

If we assume in addition that
$(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\geq 0}$ is compactly
generated (e.g. if $\mathcal{C}$ is almost ULA generated), it follows that
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ is almost compactly
generated. In that case, the functor

$\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}^{\operatorname{ren}}\longrightarrow\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}$

induced by $\mathcal{C}^{\operatorname{ren}}\to\mathcal{C}$ factors uniquely
through a t-exact equivalence

$\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}^{\operatorname{ren}}\tilde{\longrightarrow}(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C})^{\operatorname{ren}}.$

###### Proof.

For the first claim, it suffices to show that a compact object $c$ in
$(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\geq 0}$ is almost
compact (equivalently, coherent). Since $\operatorname{ind}_{\mathcal{C}}$ is
t-exact with continuous and t-exact right adjoint
$\operatorname{oblv}_{\mathcal{C}}$, the object
$\operatorname{ind}_{\mathcal{C}}(c)$ is compact in $\mathcal{C}^{\geq 0}$ and
hence almost compact in $\mathcal{C}$ by coherence of the t-structure on the
latter category. Now Lemma 2.4.1 implies that $c$ is almost compact in
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ as desired.

Next, consider the functor

$\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}^{\operatorname{ren}}\longrightarrow\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}$

induced by $\mathcal{C}^{\operatorname{ren}}\to\mathcal{C}$. This fits into
the commutative square

${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}^{\operatorname{ren}}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}^{\operatorname{ren}}}}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}}}$

and is therefore t-exact, since the vertical functors are t-exact and
conservative. For the same reason, both

$(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}^{\operatorname{ren}})^{+}\xrightarrow{\operatorname{ind}_{\mathcal{C}^{\operatorname{ren}}}}(\mathcal{C}^{\operatorname{ren}})^{+}$

and

$(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C})^{+}\xrightarrow{\operatorname{ind}_{\mathcal{C}}}\mathcal{C}^{+}$

are comonadic (cf. the Proof of Lemma 3.7.1 in [Ras19]). Since
$(\mathcal{C}^{\operatorname{ren}})^{+}\tilde{\to}\mathcal{C}^{+}$, we deduce
that the functor under consideration induces an equivalence

$(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}^{\operatorname{ren}})^{+}\tilde{\longrightarrow}(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C})^{+}.$

It folows that the functor

$(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C})^{\operatorname{ren}}\longrightarrow\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}$

factors uniquely through a t-exact continuous functor

$(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C})^{\operatorname{ren}}\longrightarrow\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}^{\operatorname{ren}},$

which we claim is an equivalence. Namely, observe that the triangles

${(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\operatorname{ren}}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}^{\operatorname{ren}}}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}}^{\operatorname{ren}}}$$\scriptstyle{\operatorname{ind}_{\mathcal{C}^{\operatorname{ren}}}}$

and

${(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\operatorname{ren}}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}^{\operatorname{ren}}}$$\scriptstyle{\operatorname{oblv}_{\mathcal{C}}^{\operatorname{ren}}}$$\scriptstyle{\operatorname{oblv}_{\mathcal{C}^{\operatorname{ren}}}}$

commute, since they commute after restricting to eventually coconnective
objects, and the categories
$(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\operatorname{ren}}$
and $\mathcal{C}^{\operatorname{ren}}$ are compactly generated by coherent
objects. Since the functor
$\operatorname{ind}_{\mathcal{C}^{\operatorname{ren}}}$ is comonadic by
Proposition 2.1.1, it suffices to show that
$\operatorname{ind}_{\mathcal{C}}^{\operatorname{ren}}$ is comonadic.

For this, observe that
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$ is almost compactly
generated with respect to the t-structure defined in the proof of Proposition
2.1.1, and that

$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))^{\operatorname{ren}}\tilde{\longrightarrow}\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S)).$

Namely, we have an equivalence

$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))\tilde{\longrightarrow}\mathsf{IndCoh}((\widehat{S^{2}})_{\Delta})$

which is t-exact up to shift, and the t-structure on the right side has these
properties. The compatibility of the $D(S)$-action with the t-structure on
$\mathcal{C}$ implies that the action of
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$ on
$\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}$ takes place in
$(\mathsf{DGCat}_{\operatorname{t-str}})^{\otimes}_{\operatorname{fha}}$, and
hence (2.6.1) induces an action on the renormalization
$(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\operatorname{ren}}$.
The monad induced by the adjunction

$\operatorname{ind}_{\mathcal{C}}^{\operatorname{ren}}:(\mathsf{IndCoh}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C})^{\operatorname{ren}}\rightleftarrows\mathcal{C}^{\operatorname{ren}}:\operatorname{oblv}_{\mathcal{C}}^{\operatorname{ren}}$

is given by the action of the associative algebra
$\operatorname{oblv}_{S}\circ\operatorname{ind}_{S}$ in
$\operatorname{End}_{\mathsf{D}(S)}(\mathsf{IndCoh}(S))$, since this is the
case on eventually coconnective objects and
$(\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C})^{\operatorname{ren}}$
is compactly generated by coherent objects. As in Proposition 2.1.1, it
follows that this monad is effective and hence that
$\operatorname{ind}_{\mathcal{C}}^{\operatorname{ren}}$ is comonadic.

∎

###  2.9. Change of base: open embedding

Suppose that $j:U\to S$ is an open embedding and $\mathcal{C}$ is a
$\mathsf{D}(S)$-module category with compatible t-structure. Put

$\mathcal{C}_{U}:=\mathsf{D}(U)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}.$

Then we have an adjunction

$j^{*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}=\mathsf{D}(S)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}\rightleftarrows\mathcal{C}_{U}:j_{*}\otimes\operatorname{id}_{\mathcal{C}}$

where the right adjoint is fully faithful, and in particular $\mathcal{C}_{U}$
admits a unique t-structure such that
$j_{*}\otimes\operatorname{id}_{\mathcal{C}}$ is left t-exact, or equivalently
such that $j^{*}\otimes\operatorname{id}_{\mathcal{C}}$ is right t-exact. Here
we are abusing notation slightly by writing $j_{*}$ rather than
$j_{\operatorname{dR},*}$, but since $j$ is an open embedding, this should not
cause any confusion.

######  Lemma 2.9.1.

The functor
$j^{*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}_{U}$ is
t-exact. If $j$ is affine, then the right adjoint
$j_{*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}_{U}\to\mathcal{C}$ is
t-exact.

###### Proof.

To see that $j^{*}\otimes\operatorname{id}_{\mathcal{C}}$ is t-exact, it
suffices to show it is left t-exact, which is equivalent to left t-exactness
of the endofunctor

$(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}\longrightarrow\mathcal{C}_{U}\longrightarrow\mathcal{C}.$

Similarly, if $(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}}$ is t-exact,
then $j_{*}\otimes\operatorname{id}_{\mathcal{C}}$ is t-exact. We have a
commutative square

${\mathcal{C}}$${\mathsf{D}(S)\otimes\mathcal{C}}$${\mathcal{C}}$${\mathsf{D}(S)\otimes\mathcal{C}}$$\scriptstyle{\operatorname{coact}_{\mathcal{C}}}$$\scriptstyle{(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}}}$$\scriptstyle{(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}}}$$\scriptstyle{\operatorname{coact}_{\mathcal{C}}}$

where the horizontal functors are fully faithful and t-exact. By Lemma 2.2.1,
the right vertical functor is left t-exact, respectively right t-exact if the
functor $j_{*}j^{*}$ is such. The lemma follows.

∎

######  Proposition 2.9.2.

If $\mathcal{C}$ is almost ULA generated over $S$, then $\mathcal{C}_{U}$ is
almost ULA generated over $U$, and the functor

$(\mathcal{C}^{\operatorname{ren}})_{U}\longrightarrow\mathcal{C}_{U}$

induced by $\mathcal{C}^{\operatorname{ren}}\to\mathcal{C}$ factors uniquely
through a t-exact equivalence

$(\mathcal{C}^{\operatorname{ren}})_{U}\tilde{\longrightarrow}(\mathcal{C}_{U})^{\operatorname{ren}}.$

###### Proof.

Suppose that $\mathcal{C}$ is almost compactly generated. Since
$j_{*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}_{U}\to\mathcal{C}$ is
left t-exact and conservative, the t-structure on $\mathcal{C}_{U}$ is
compatible with filtered colimits. For the same reason, the t-structure on
$\mathcal{C}$ is right separated and hence right complete. To see that
$(\mathcal{C}_{U})^{\geq 0}$ is compactly generated, consider the adjunction

$j^{*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}^{\geq
0}\rightleftarrows(\mathcal{C}_{U})^{\geq
0}:j_{*}\otimes\operatorname{id}_{\mathcal{C}}.$

Since $j_{*}\otimes\operatorname{id}_{\mathcal{C}}$ is continuous and fully
faithful, hence conservative, it follows that $(\mathcal{C}_{U})^{\geq 0}$ is
compactly generated.

To see that $\mathcal{C}_{U}$ is almost compactly generated, it remains to
show that the t-structure is coherent. We claim that any compact object $c$ in
$(\mathcal{C}_{U})^{\geq 0}$ is a direct summand of an object
$(j^{*}\otimes\operatorname{id}_{\mathcal{C}})(c_{0})$ where $c_{0}$ is a
compact object in $\mathcal{C}^{\geq 0}$. Since $\mathcal{C}$ is almost
compactly generated and $j^{*}\otimes\operatorname{id}_{\mathcal{C}}$
preserves almost compact objects, it will follow that $c$ is almost compact in
$\mathcal{C}_{U}$. To prove the claim, write

$(j_{*}\otimes\operatorname{id}_{\mathcal{C}})(c)\cong\underset{\alpha}{\operatorname{colim}}\
c_{\alpha}$

as a filtered colimit of compact objects $c_{\alpha}$ in $\mathcal{C}^{\geq
0}$. Then we have

$c\cong\underset{\alpha}{\operatorname{colim}}\
(j^{*}\otimes\operatorname{id}_{\mathcal{C}})(c_{\alpha})$

in $(\mathcal{C}_{U})^{\geq 0}$. Compactness of $c$ in
$(\mathcal{C}_{U})^{\geq 0}$ now implies that $c$ is a direct summand of
$(j^{*}\otimes\operatorname{id}_{\mathcal{C}})(c_{\alpha})$ for some $\alpha$.

Renormalization of the above adjunction yields a $\mathsf{D}(S)$-linear
adjunction

$(j^{*}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}}:\mathcal{C}^{\operatorname{ren}}\rightleftarrows(\mathcal{C}_{U})^{\operatorname{ren}}:(j_{*}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}}.$

The right adjoint is continuous and fully faithful, so in particular this
adjunction is monadic. Note that
$(j_{*}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}}$ factors
through

$j_{*}\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}}:(\mathcal{C}^{\operatorname{ren}})_{U}\longrightarrow\mathcal{C}^{\operatorname{ren}},$

since this is the case after restricting to $(\mathcal{C}_{U})^{+}$. The
resulting functor

$(\mathcal{C}_{U})^{\operatorname{ren}}\longrightarrow(\mathcal{C}^{\operatorname{ren}})_{U}$

corresponds to a morphism of (left t-exact) monads on
$\mathcal{C}^{\operatorname{ren}}$, which is an isomorphism because it is such
after restricting to $\mathcal{C}^{+}$.

The assertion that $\mathcal{C}_{U}$ is almost ULA generated over $U$ follows
readily from the above.

∎

###  2.10. Change of base: closed embedding

Suppose that we are given a smooth closed subvariety $i:T\to S$. Writing

$\mathcal{C}_{T}:=\mathsf{D}(T)\underset{\mathsf{D}(S)}{\otimes}\mathcal{C}$

for any $\mathsf{D}(S)$-module category $\mathcal{C}$, note that we have an
adjunction

$i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}_{T}\rightleftarrows\mathcal{C}:i^{!}\otimes\operatorname{id}_{\mathcal{C}},$

with the left adjoint being fully faithful.

######  Lemma 2.10.1.

The functor
$i^{!}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}_{T}$
sends objects ULA over $S$ to objects ULA over $T$. If $\mathcal{C}$ is ULA
generated over $S$, the $\mathcal{C}_{T}$ is ULA generated over $T$ by objects
of the form $(i^{!}\otimes\operatorname{id}_{\mathcal{C}})(c)$, where $c$ is
ULA in $\mathcal{C}$.

###### Proof.

Consider the commutative square of symmetric monoidal categories

${\mathsf{D}(S)}$${\mathsf{D}(T)}$${\mathsf{IndCoh}(S)}$${\mathsf{IndCoh}(T).}$$\scriptstyle{i^{!}}$$\scriptstyle{\operatorname{oblv}_{S}}$$\scriptstyle{\operatorname{oblv}_{T}}$$\scriptstyle{i^{!}}$

This can also be viewed as a commutative square of $\mathsf{D}(S)$-modules,
which we can tensor with $\mathcal{C}$ to obtain

${\mathcal{C}}$${\mathcal{C}_{T}}$${\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}}$${\mathsf{IndCoh}(T)\otimes_{\mathsf{D}(T)}\mathcal{C}_{T}.}$$\scriptstyle{i^{!}\otimes\operatorname{id}_{\mathcal{C}}}$$\scriptstyle{\operatorname{oblv}_{S}}$$\scriptstyle{\operatorname{oblv}_{T}}$$\scriptstyle{i^{!}\otimes\operatorname{id}_{\mathcal{C}}}$

Since $S$ and $T$ are smooth, the functor

$i_{*}:\mathsf{IndCoh}(T)\longrightarrow\mathsf{IndCoh}(S)$

admits a left adjoint $i^{\mathsf{IndCoh},*}$, which agrees with $i^{!}$ up to
tensoring with a cohomologically shifted line bundle. Thus $i^{!}$ admits a
$\mathsf{D}(S)$-linear right adjoint, which implies that the lower horizontal
functor in the above square admits a continuous right adjoint and hence
preserves compact objects. It follows immediately that the upper horizontal
functor preserves ULA objects.

The second claim follows from the observation that the image of

$i^{!}\otimes\operatorname{id}_{\mathcal{C}}:\mathsf{IndCoh}(S)\otimes_{\mathsf{D}(S)}\mathcal{C}\longrightarrow\mathsf{IndCoh}(T)\otimes_{\mathsf{D}(T)}\mathcal{C}_{T}$

generates the target under colimits, which is immediate from the corresponding
property of

$i^{!}:\mathsf{IndCoh}(S)\longrightarrow\mathsf{IndCoh}(T)$

(this being equivalent to the conservativity of its right adjoint, which is
isomorphic to $i_{*}$ tensored with a cohomologically shifted line bundle).

∎

######  Lemma 2.10.2.

Suppose that $\mathcal{C}$ is equipped with a t-structure compatible with the
$\mathsf{D}(S)$-action. Then the image of the fully faithful functor

$i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}_{T}\longrightarrow\mathcal{C}$

is stable under truncations, so $\mathcal{C}_{T}$ admits a unique t-structure
such that $i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}$ is
t-exact. Moreover, the subcategory
$(\mathcal{C}_{T})^{\heartsuit}\subset\mathcal{C}^{\heartsuit}$ is stable
under taking subobjects, and the t-structure on $\mathcal{C}_{T}$ is
compatible with the $\mathsf{D}(T)$-action.

###### Proof.

The stability of $\mathcal{C}_{T}$ under truncations and of
$(\mathcal{C}_{T})^{\heartsuit}$ under taking subobjects both follow formally
from Lemma 2.9.1 (specifically, the fact that
$j^{*}\otimes\operatorname{id}_{\mathcal{C}}$ is t-exact). To see that the
$\mathsf{D}(T)$-action on $\mathcal{C}_{T}$ is compatible with the
t-structure, consider the commutative square

${\mathcal{C}_{T}}$${\mathcal{C}}$${\mathsf{D}(T)\otimes\mathcal{C}_{T}}$${\mathsf{D}(S)\otimes\mathcal{C}.}$$\scriptstyle{i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}}$$\scriptstyle{\operatorname{coact}_{\mathcal{C}}}$$\scriptstyle{\operatorname{coact}_{\mathcal{C}}}$$\scriptstyle{i_{\operatorname{dR},*}\otimes
i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}}$

The upper horizontal and right vertical functors are t-exact, so it suffices
to show that that the lower horizontal functor is conservative and t-exact. In
fact, it is fully faithful, being the tensor product of two fully faithful
functors which admit continuous right adjoints. To see the t-exactness, note
that this functor can be decomposed as

$\mathsf{D}(T)\otimes\mathcal{C}_{T}\xrightarrow{i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}_{T}}}\mathsf{D}(S)\otimes\mathcal{C}_{T}\xrightarrow{\operatorname{id}_{\mathsf{D}(S)}\otimes
i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}}\mathsf{D}(S)\otimes\mathcal{C}.$

Since the t-structures on $\mathsf{D}(S)$ and $\mathsf{D}(T)$ are compactly
generated, the t-exactness of these two functors follows from that of
$i_{\operatorname{dR},*}$ and
$i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}}:\mathcal{C}_{T}\to\mathcal{C}$
by Lemma 2.2.1.

∎

If $\mathcal{C}_{T}$ and $\mathcal{C}$ are almost ULA generated, then we
obtain an adjunction

$(i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}}:(\mathcal{C}_{T})^{\operatorname{ren}}\rightleftarrows\mathcal{C}^{\operatorname{ren}}:(i^{!}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}},$

which takes place in $\mathsf{D}(S)$-modules by Proposition 2.7.1.

######  Proposition 2.10.3.

Suppose that $\mathcal{C}$ and $\mathcal{C}_{T}$ are almost ULA generated
(with respect to $S$ and $T$ respectively). Then the functor
$(i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}}$
factors through an equivalence

$(\mathcal{C}_{T})^{\operatorname{ren}}\tilde{\longrightarrow}(\mathcal{C}^{\operatorname{ren}})_{T}.$

###### Proof.

Note that $(\mathcal{C}^{\operatorname{ren}})_{T}$ is the kernel of the monad
$(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}}$,
which fits into the commutative square

${\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}}$${\mathcal{C}.}$$\scriptstyle{(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}}}$$\scriptstyle{(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}}}$

Since $\mathcal{C}^{\operatorname{ren},+}\tilde{\to}\mathcal{C}^{+}$, and
$(\mathcal{C}_{T})^{\operatorname{coh}}\subset\mathcal{C}_{T}$ is annihilated
by $(j_{*}j^{*})\otimes\operatorname{id}_{\mathcal{C}}$, it follows that
$(i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}})^{\operatorname{ren}}$
factors through

$i_{\operatorname{dR},*}\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}}:(\mathcal{C}^{\operatorname{ren}})_{T}\longrightarrow\mathcal{C}^{\operatorname{ren}}.$

To see that the resulting functor

$(\mathcal{C}_{T})^{\operatorname{ren}}\longrightarrow(\mathcal{C}^{\operatorname{ren}})_{T}$

is essentially surjective, observe that Proposition 2.8.2 and Lemma 2.10.1
together imply that $(\mathcal{C}^{\operatorname{ren}})_{T}$ is ULA generated
over $T$ by objects of the form
$(i^{!}\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}})(c)$, where
$c$ is an object in $\mathcal{C}^{\operatorname{ren}}$ which is ULA over $S$.
Now we appeal to the commutative square

${\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}^{\operatorname{ren}}}$${\mathcal{C}}$${\mathcal{C},}$$\scriptstyle{(i_{\operatorname{dR},*}i^{!})\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}}}$$\scriptstyle{(i_{\operatorname{dR},*}i^{!})\otimes\operatorname{id}_{\mathcal{C}}}$

which shows that
$((i_{\operatorname{dR},*}i^{!})\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}})(c)$
agrees with
$((i_{\operatorname{dR},*}i^{!})\otimes\operatorname{id}_{\mathcal{C}})(c)$
under the identification
$\mathcal{C}^{\operatorname{ren},+}\tilde{\to}\mathcal{C}^{+}$ (in the latter
formula, we view $c$ as an object of $\mathcal{C}^{+}$). But the object
$((i_{\operatorname{dR},*}i^{!})\otimes\operatorname{id}_{\mathcal{C}})(c)$
belongs to $(\mathcal{C}_{T})^{\operatorname{coh}}\subset\mathcal{C}$, which
shows that
$(i^{!}\otimes\operatorname{id}_{\mathcal{C}^{\operatorname{ren}}})(c)$
belongs to the essential image of $(\mathcal{C}_{T})^{\operatorname{ren}}$ as
needed.

∎

##  3\. Preliminaries on factorization

This section covers foundational technical material on factorization
categories, algebras, and modules. The key result is Proposition 3.9.2, which
gives sufficient conditions under which it is possible to renormalize a
factorization category.

### 3.1.

We refer the reader to §6 of [Ras15a] for careful definitions of the objects
introduced in this section. An alternative but equivalent set of definitions
can be found in §10 of [CF21b] (see Remark 10.5.11 there, which compares the
two).

We will denote by $\mathsf{FactCat}$ the category of unital factorization
categories on $X$, with morphisms given by strictly unital factorizable
functors. We can also consider the category
$\mathsf{FactCat}^{\operatorname{lax-fact}}$ consisting of lax factorization
categories. Both of these categories admit variants in which functors are
allowed to be lax unital, and together these assemble into a cartesian square

${\mathsf{FactCat}}$${\mathsf{FactCat}^{\operatorname{lax-
fact}}}$${\mathsf{FactCat}_{\operatorname{lax-
untl}}}$${\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-
untl}}.}$

Here the horizontal functors are fully faithful, while the vertical functors
are only $1$-fully faithful but act as the identity on objects.

By definition, we have a forgetful functor

$\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-
untl}}\longrightarrow\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}}),$

where $\operatorname{Ran}^{\operatorname{un}}$ denotes the unital Ran space of
$X$, i.e. the lax prestack parameterizing finite subsets of $X$ and inclusions
between them. The category $\mathsf{FactCat}^{\operatorname{lax-
fact}}_{\operatorname{lax-untl}}$ admits a natural symmetric monoidal
structure which makes the forgetful functor symmetric monoidal. This restricts
to a symmetric monoidal structure on each of the $1$-full subcategories of
$\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$
considered above.

Given an object $\mathcal{C}$ in $\mathsf{FactCat}^{\operatorname{lax-
fact}}_{\operatorname{lax-untl}}$, we will write

$\mathcal{C}_{Y}:=\Gamma(Y,\mathcal{C})$

for any prestack $Y$ mapping to
$\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}}$. For example, for
any finite set $I$ we have the $\mathsf{D}(X^{I})$-module category
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ and the
$\mathsf{IndCoh}(X^{I})$-module category

$\mathcal{C}_{X^{I}}=\mathsf{IndCoh}(X^{I})\underset{\mathsf{D}(X^{I})}{\otimes}\mathcal{C}_{X^{I}_{\operatorname{dR}}}.$

###  3.2. A combinatorial presentation for factorization categories

We denote by $\mathsf{fSet}$ the category of finite sets, and write
$\mathsf{fSet}_{\operatorname{surj}}\subset\mathsf{fSet}$ for the subcategory
consisting of nonempty finite sets and surjections between them.

Recall from [Ras15a] that for a lax prestack $Y$, there is a certain $1$-full
subcategory

$\mathsf{ShvCat}(Y)^{\operatorname{naive}}\subset\mathsf{ShvCat}(Y)$

with the same objects but fewer morphisms. In the case of unital Ran space,
the subcategory

$\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}})^{\operatorname{naive}}\subset\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}})$

contains only those morphisms which are “strictly unital,” i.e.

$\mathsf{FactCat}^{\operatorname{lax-
fact}}=\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-
untl}}\underset{\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}})}{\times}\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}})^{\operatorname{naive}}.$

From the colimit presentation

$\operatorname{Ran}=\underset{I\in\mathsf{fSet}^{\operatorname{op}}_{\operatorname{surj}}}{\operatorname{colim}}X^{I}$

and the $1$-affineness of $X^{I}_{\operatorname{dR}}$, we obtain a limit
presentation

$\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}})=\lim_{I\in\mathsf{fSet}_{\operatorname{surj}}}\mathsf{D}(X^{I})\text{\textendash}\mathsf{mod}.$

The following result supplies an analogous presentation of
$\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}})^{\operatorname{naive}}$
as a lax limit.

######  Proposition 3.2.1.

There is a canonical fully faithful embedding of symmetric monoidal categories

$\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}})^{\operatorname{naive}}\longrightarrow\underset{I\in\mathsf{fSet}}{\operatorname{lax}\,\lim}\
\mathsf{D}(X^{I})\text{\textendash}\mathsf{mod}.$

The image consists of those objects
$\\{\mathcal{C}_{X^{I}_{\operatorname{dR}}}\\}$ such that, for any surjection
$I\to J$, the functor

$\mathsf{D}(X^{J})\underset{\mathsf{D}(X^{I})}{\otimes}\mathcal{C}_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathcal{C}_{X^{J}_{\operatorname{dR}}}$

is an equivalence.

###### Proof.

This is Corollary 4.6.1 in [Ras15a].

∎

Following _loc. cit._ , we write
$\operatorname{Part}\subset\operatorname{Part}_{\operatorname{un}}$ for the
twisted arrows categories of
$\mathsf{fSet}_{\operatorname{surj}}\subset\mathsf{fSet}$. For any $p:I\to J$
in $\operatorname{Part}_{\operatorname{un}}$, there is a corresponding open
subscheme $U(p)\subset X^{I}$, and given $\epsilon:p_{1}\to p_{2}$ in
$\operatorname{Part}_{\operatorname{un}}$, there is a unique morphism
$U(\epsilon):U(p_{2})\to U(p_{1})$ which makes the following square commute:

${U(p_{2})}$${U(p_{1})}$${X^{I_{2}}}$${X^{I_{1}}.}$$\scriptstyle{U(\epsilon)}$

Thus we obtain a functor $U$ from
$\operatorname{Part}_{\operatorname{un}}^{\operatorname{op}}$ into schemes.

If we equip $\operatorname{Part}_{\operatorname{un}}$ with the symmetric
monoidal structure given by $(p_{1},p_{2})\mapsto p_{1}\sqcup p_{2}$, then $U$
is canonically oplax symmetric monoidal, since

$U(p_{1}\sqcup p_{2})\subset U(p_{1})\times U(p_{2})$

as subschemes of $X^{I_{1}\sqcup I_{2}}=X^{I_{1}}\times X^{I_{2}}$. This
endows the resulting functor

$\displaystyle\operatorname{Part}_{\operatorname{un}}$
$\displaystyle\longrightarrow\mathsf{Cat}$ $\displaystyle p$
$\displaystyle\mapsto\mathsf{D}(U(p))\text{\textendash}\mathsf{mod}$

with a lax symmetric monoidal structure, and hence its straightening (the
corresponding cocartesian fibration)

$\underset{p\in\operatorname{Part}_{\operatorname{un}}}{\operatorname{lax}\,\operatorname{colim}}\
\mathsf{D}(U(p))\text{\textendash}\mathsf{mod}\longrightarrow\operatorname{Part}_{\operatorname{un}}$
(3.2.1)

is naturally symmetric monoidal (in particular, the lax colimit admits a
symmetric monoidal structure).

######  Proposition 3.2.2.

There is canonical equivalence from $\mathsf{FactCat}^{\operatorname{lax-
fact}}$ to the full subcategory of lax symmetric monoidal sections of (3.2.1)
satisfying the following conditions:

  1. (i)

morphisms in
$\operatorname{Part}\subset\operatorname{Part}_{\operatorname{un}}$ map to
cocartesian morphisms;

  2. (ii)

morphisms between objects of the form $\varnothing\to I$ map to cocartesian
morphisms.

###### Proof.

This is a slight variant of Proposition-Construction 8.4.1 (cf. also §8.12) in
[Ras15a].

∎

###  3.3. Dualizable factorization categories

We will need the following technical result.

######  Proposition 3.3.1.

Suppose that $\mathcal{C}$ in $\mathsf{FactCat}$ has the property that
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ is dualizable as a
$\mathsf{D}(X^{I})$-module for any finite set $I$. Assume also that for any
injection $J\to I$, the unital structure functor

$\mathsf{D}(X^{I})\underset{\mathsf{D}(X^{J})}{\otimes}\mathcal{C}_{X^{J}_{\operatorname{dR}}}\longrightarrow\mathcal{C}_{X^{I}_{\operatorname{dR}}}$

admits a $\mathsf{D}(X^{I})$-linear right adjoint. Then $\mathcal{C}$ is
dualizable as an object of $\mathsf{FactCat}_{\operatorname{lax-untl}}$.

###### Proof.

First, observe that the hypothesis in the proposition implies that

$\mathsf{D}(X^{I})\underset{\mathsf{D}(X^{J})}{\otimes}\mathcal{C}_{X^{J}_{\operatorname{dR}}}\longrightarrow\mathcal{C}_{X^{I}_{\operatorname{dR}}}$

admits a $\mathsf{D}(X^{I})$-linear right adjoint for any (not necessarily
injective) map $I\to J$ in $\mathsf{fSet}$. Namely, any such map can written
as the composition of a surjection and an injection, and the functor in
question is an equivalence if $I\to J$ is surjective. It follows that the
sheaf of categories underlying $\mathcal{C}$ is dualizable as an object of
$\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}})$,
with dual object given by dualizing termwise and then passing to left adjoints
of the structure functors.

A lax factorization structure on $\mathcal{C}$ induces an oplax factorization
structure on $\mathcal{C}^{\vee}$, and if the factorization structure on
$\mathcal{C}$ is strict then so is the one on $\mathcal{C}^{\vee}$. It is not
difficult to see that the evaluation

$\mathcal{C}^{\vee}\otimes\mathcal{C}\longrightarrow\operatorname{Vect}$

and coevaluation

$\operatorname{Vect}\longrightarrow\mathcal{C}^{\vee}\otimes\mathcal{C}$

morphisms lift from
$\mathsf{ShvCat}(\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}})$
to $\mathsf{FactCat}_{\operatorname{lax-untl}}$.

∎

######  Remark 3.3.2.

We emphasize that it is necessary to take the dual of $\mathcal{C}$ in
$\mathsf{FactCat}_{\operatorname{lax-untl}}$ rather than $\mathsf{FactCat}$.
The point is that the coevaluation functor

$\operatorname{Vect}\longrightarrow\mathcal{C}^{\vee}\otimes\mathcal{C}$

is typically only lax unital.

###  3.4. Commutative factorization categories

As shown in [Ras15a] §7.5, one can functorially attach an object of
$\mathsf{FactCat}$ to any symmetric monoidal DG category. Given a symmetric
monoidal DG category, we denote the associated factorization category by the
same symbol to avoid cluttering the notation. We will take care to avoid
confusion by specifying which object we are referring to in each case.

Let $\mathcal{C}$ be a symmetric monoidal category. For any finite set $I$, we
have a symmetric monoidal functor

$\operatorname{Loc}_{X^{I}}:\mathcal{C}^{\otimes
I}\longrightarrow\mathcal{C}_{X^{I}_{\operatorname{dR}}},$

defined in Remark 6.8.4 of [Ras21a].

######  Proposition 3.4.1.

Suppose that $\mathcal{C}$ is a rigid symmetric monoidal category whose
underlying DG category is compactly generated.

  1. (i)

The functor $\operatorname{Loc}_{X^{I}}$ sends compact objects to objects ULA
over $X^{I}$. If $\\{c_{\alpha}\\}$ is a set of compact generators for
$\mathcal{C}^{\otimes I}$, then $\\{\operatorname{Loc}_{X^{I}}(c_{\alpha})\\}$
is a set of ULA generators for $\mathcal{C}_{X^{I}_{\operatorname{dR}}}$.

  2. (ii)

The factorization category attached to $\mathcal{C}$ is dualizable and self-
dual as an object of $\mathsf{FactCat}_{\operatorname{lax-untl}}$.

###### Proof.

For (i), see Proposition 6.16.1 and the proof of Theorem 6.7.1 in _loc. cit._

By Corollary 6.18.2 of [Ras21a], the $\mathsf{D}(X^{I})$-module category
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ is dualizable and self-dual for any
finite set $I$. According to Proposition 3.3.1, to prove (ii) it remains to
show that for any injection $J\to I$, the structure map

$\mathsf{D}(X^{I})\underset{\mathsf{D}(X^{J})}{\otimes}\mathcal{C}_{X^{J}_{\operatorname{dR}}}\longrightarrow\mathcal{C}_{X^{I}_{\operatorname{dR}}}$

admits a $\mathsf{D}(X^{I})$-linear right adjoint, or equivalently preserves
ULA objects. This follows from the commutative square

${\mathcal{C}^{\otimes J}}$${\mathcal{C}^{\otimes
I}}$${\mathsf{D}(X^{I})\underset{\mathsf{D}(X^{J})}{\otimes}\mathcal{C}_{X^{J}_{\operatorname{dR}}}}$${\mathcal{C}_{X^{I}_{\operatorname{dR}}},}$$\scriptstyle{\operatorname{id}\otimes\operatorname{Loc}_{X^{J}}}$$\scriptstyle{\operatorname{Loc}_{X^{I}}}$

where the upper horizontal arrow acts as the identity on $\mathcal{C}^{\otimes
J}$ and inserts the unit in $\mathcal{C}^{\otimes(I\setminus J)}$.

∎

The following lemma will also be useful.

######  Lemma 3.4.2.

Let $F:\mathcal{C}\to\mathcal{D}$ be a symmetric monoidal functor. Assume that
$\mathcal{C}$ is rigid and compactly generated, and that the unit object in
$\mathcal{D}$ is compact, so that in particular the right adjoint
$F^{\operatorname{R}}$ of $F$ is continuous. Then the morphism in
$\mathsf{FactCat}$ arising from $F$ admits a right adjoint in
$\mathsf{FactCat}_{\operatorname{lax-untl}}$, which in particular agrees with
$F^{\operatorname{R}}$ fiberwise.

###### Proof.

It suffices to prove that for any finite set $I$, the
$\mathsf{D}(X^{I})$-linear functor

$F:\mathcal{C}_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathcal{D}_{X^{I}_{\operatorname{dR}}}$

preserves ULA objects. Using the commutative square

${\mathcal{C}^{\otimes I}}$${\mathcal{D}^{\otimes
I}}$${\mathcal{C}_{X^{I}_{\operatorname{dR}}}}$${\mathcal{D}_{X^{I}_{\operatorname{dR}}}}$$\scriptstyle{F^{\otimes
I}}$$\scriptstyle{\operatorname{Loc}_{X^{I}}}$$\scriptstyle{\operatorname{Loc}_{X^{I}}}$$\scriptstyle{F}$

and Proposition 3.4.1, it is enough to prove that the composition

$\mathcal{C}^{\otimes I}\xrightarrow{F^{\otimes I}}\mathcal{D}^{\otimes
I}\xrightarrow{\operatorname{Loc}_{X^{I}}}\mathcal{D}_{X^{I}_{\operatorname{dR}}}$

sends compact objects to objects ULA over $X^{I}$. This functor is symmetric
monoidal, and the proof of Lemma 6.16.2 in [Ras21a] shows that the unit object
in $\mathcal{D}_{X^{I}_{\operatorname{dR}}}$ is ULA, so the claim follows from
the rigidity of $\mathcal{C}^{\otimes I}$.

∎

###  3.5. Functoriality of factorization modules

Consider the category $(\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}}$, whose objects are pairs
$(\mathcal{C},A)$ consisting of a factorization category $\mathcal{C}$ and an
(always unital unless otherwise specified) factorization algebra $A$ in
$\mathcal{C}$. A morphism $(\mathcal{C},A)\to(\mathcal{D},B)$ is given by a
lax unital factorization functor $F:\mathcal{C}\to\mathcal{D}$ and a morphism
$B\to F(A)$ of factorization algebras in $\mathcal{D}$.

For a given pair $(\mathcal{C},A)$ as above we can consider the lax
factorization category of (unital) factorization modules
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})$. This
construction extends to a functor

$(\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}}\longrightarrow\mathsf{FactCat}^{\operatorname{lax-
fact}}_{\operatorname{lax-untl}}.$ (3.5.1)

In particular, given a morphism $(\mathcal{C},A)\to(\mathcal{D},B)$ consisting
of $F:\mathcal{C}\to\mathcal{D}$ and $B\to F(A)$, we have the resulting
functor

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})\longrightarrow
F(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{D})\longrightarrow
B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{D}).$

Here the first functor is induced by $F$, and the second is restriction of the
factorization module structure along $B\to F(A)$ (cf. [CF21b] §10.6).

We observe that $(\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}}$ admits a natural symmetric
monoidal structure, given on objects by

$(\mathcal{C},A)\otimes(\mathcal{D},B)=(\mathcal{C}\otimes\mathcal{D},A\boxtimes
B).$

Moreover, the functor (3.5.1) is naturally lax symmetric monoidal: in
particular, for $(\mathcal{C},A)$ and $(\mathcal{D},B)$ in
$(\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}}$ we have a canonical morphism

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})\otimes
B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{D})\longrightarrow(A\boxtimes
B)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C}\otimes\mathcal{D})$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$, which intertwines the
forgetful functors to $\mathcal{C}\otimes\mathcal{D}$.

### 3.6.

We will also need the following lemma. Let $\mathcal{C}$ be a factorization
category and $A\to B$ a morphism of factorization algebras in $\mathcal{C}$.
The resulting morphism

$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}_{\operatorname{lax-untl}}$
sends the vacuum module $B$ to a factorization algebra in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})$ which we
denote by the same symbol $B$.

######  Lemma 3.6.1.

Applying (3.5.1) to the forgetful functor

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})\longrightarrow\mathcal{C}$

and the factorization module $B$ induces an isomorphism

$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C}))\longrightarrow
B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})$

in $\mathsf{FactCat}^{\operatorname{lax-fact}}$.

###### Proof.

The inverse is given by applying (3.5.1) to the restriction functor

$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})$

and the factorization module $B$.

∎

### 3.7.

Observe that the projection $(\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}}\to\mathsf{FactCat}_{\operatorname{lax-
untl}}$ is symmetric monoidal, and in particular induces a functor

$\mathsf{AssocAlg}((\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}})\longrightarrow\mathsf{AssocAlg}(\mathsf{FactCat}_{\operatorname{lax-
untl}}).$

This functor admits a canonical section

$\mathsf{AssocAlg}(\mathsf{FactCat}_{\operatorname{lax-
untl}})\longrightarrow\mathsf{AssocAlg}((\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}})$ (3.7.1)

given on objects by
$\mathcal{A}\mapsto(\mathcal{A},\mathds{1}_{\mathcal{A}})$, where
$\mathds{1}_{\mathcal{A}}$ denotes the monoidal unit of $\mathcal{A}$ viewed
as a factorization algebra.

The lax symmetric monoidal structure on (3.5.1) gives rise to a functor

$\mathsf{AssocAlg}((\mathsf{FactCat}_{\operatorname{lax-
untl}})^{\mathsf{FactAlg}^{\operatorname{op}}})\longrightarrow\mathsf{AssocAlg}(\mathsf{FactCat}^{\operatorname{lax-
fact}}_{\operatorname{lax-untl}}).$

Note that the composition of this functor with (3.7.1) factors through the
$1$-full subcategory

$\mathsf{AssocAlg}(\mathsf{FactCat}^{\operatorname{lax-
fact}})\longrightarrow\mathsf{AssocAlg}(\mathsf{FactCat}^{\operatorname{lax-
fact}}_{\operatorname{lax-untl}}),$

and hence can be viewed as a functor

$\displaystyle\mathsf{AssocAlg}(\mathsf{FactCat}_{\operatorname{lax-untl}})$
$\displaystyle\longrightarrow\mathsf{AssocAlg}(\mathsf{FactCat}^{\operatorname{lax-
fact}})$ (3.7.2) $\displaystyle\mathcal{A}$
$\displaystyle\mapsto\mathds{1}_{\mathcal{A}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{A}).$

Namely, the factorization unit and the monoidal unit in
$\mathds{1}_{\mathcal{A}}\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{A})$
coincide, both being the vacuum module.

###  3.8. Unital vs. non-unital factorization modules

Fix a factorization category $\mathcal{C}$. We denote by
$\mathsf{FactAlg}(\mathcal{C})$ and $\mathsf{FactAlg}^{\operatorname{non-
untl}}(\mathcal{C})$ the categories of unital and non-unital factorization
algebras, respectively.

######  Proposition 3.8.1.

The forgetful functor

$\operatorname{OblvUnit}:\mathsf{FactAlg}(\mathcal{C})\longrightarrow\mathsf{FactAlg}^{\operatorname{non-
untl}}(\mathcal{C})$

admits a left adjoint $\operatorname{AddUnit}$. The comonad

$\operatorname{AddUnit}\circ\operatorname{OblvUnit}:\mathsf{FactAlg}(\mathcal{C})\longrightarrow\mathsf{FactAlg}(\mathcal{C})$

is given by $A\mapsto A\times\operatorname{unit}_{\mathcal{C}}$.

As in the setting of commutative algebras, a unital
$\operatorname{AddUnit}(A)$-module structure is the same datum as a non-unital
$A$-module structure.

######  Proposition 3.8.2.

Given $A$ in $\mathsf{FactAlg}^{\operatorname{non-untl}}(\mathcal{C})$, the
composition

$\operatorname{AddUnit}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\longrightarrow\operatorname{AddUnit}(A)\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}$

is an equivalence for any nonempty finite set $I$.

Modules over cartesian products of factorization algebras also behave as
expected.

######  Proposition 3.8.3.

For any $A$, $B$ in $\mathsf{FactAlg}(\mathcal{C})$, the functor

$\operatorname{res}^{A\times B}_{A}\oplus\operatorname{res}^{A\times
B}_{B}:A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\times
B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\longrightarrow(A\times
B)\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}$

is an equivalence for any finite set $I$.

Combining the above, we obtain the following useful result.

######  Proposition 3.8.4.

For any $A$ in $\mathsf{FactAlg}(\mathcal{C})$, the forgetful functor

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}$

factors canonically as

${A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}}$${A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}\times\mathcal{C}_{X^{I}_{\operatorname{dR}}}}$${A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathcal{C})_{X^{I}_{\operatorname{dR}}}}$$\scriptstyle{(\operatorname{id},0)}$

where the vertical functor is an equivalence. In particular, this forgetful
functor is fully faithful.

###  3.9. Renormalization

Finally, we formulate conditions under which it is possible to renormalize a
factorization category.

######  Proposition 3.9.1.

Let $\mathcal{C}$ be a sheaf of categories on
$\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}}$, and suppose we
are given a t-structure on $\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ for each
finite set $I$, compatible with the $\mathsf{D}(X^{I})$-action. Assume also
that the following conditions are satisfied:

  1. (i)

$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ is almost ULA generated for any $I$
in $\mathsf{fSet}$;

  2. (ii)

for any surjection $I\to J$, the functor
$\mathcal{C}_{X^{J}_{\operatorname{dR}}}\to\mathcal{C}_{X^{I}_{\operatorname{dR}}}$
left adjoint to the structure functor is t-exact;

  3. (iii)

for any injection $I\to J$, the structure functor
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}\to\mathcal{C}_{X^{J}_{\operatorname{dR}}}$
has finite homological amplitude.

Then there exists a sheaf of categories $\mathcal{C}^{\operatorname{ren}}$ on
$\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}}$ equipped with a
morphism $\mathcal{C}^{\operatorname{ren}}\to\mathcal{C}$, such that for any
finite set $I$ the functor
$(\mathcal{C}^{\operatorname{ren}})_{X^{I}_{\operatorname{dR}}}\to\mathcal{C}_{X^{I}_{\operatorname{dR}}}$
realizes $(\mathcal{C}^{\operatorname{ren}})_{X^{I}_{\operatorname{dR}}}$ as
the renormalization of $\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ constructed
in Proposition 2.7.1.

###### Proof.

The almost ULA generation condition implies in particular that each
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ is almost compactly generated, and
conditions (ii-iii) imply that the structure functor
$\mathsf{D}(X^{I})\to\mathsf{D}(X^{J})$ has finite homological amplitude for
any map of finite sets $I\to J$. Applying the description of
$\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}})^{\operatorname{naive}}$
from Proposition 3.2.1 and the construction of Proposition 2.7.1, we obtain an
object of

$\underset{I\in\mathsf{fSet}}{\operatorname{lax}\,\lim}\
\mathsf{D}(X^{I})\text{\textendash}\mathsf{mod},$

which lies in the image of
$\mathsf{ShvCat}(\operatorname{Ran}^{\operatorname{un}}_{\operatorname{dR}})^{\operatorname{naive}}$
by Propositions 2.8.2 and 2.10.3.

∎

######  Proposition 3.9.2.

Let $\mathcal{C}$ be an object of $\mathsf{FactCat}^{\operatorname{lax-
fact}}$. Suppose that we are given a t-structure on
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ for any finite set $I$ satisfying
the conditions of Proposition 3.9.1, and that for any finite sets
$I_{1},I_{2}$, the image of the composite functor

$\mathcal{C}_{X^{I_{1}}_{\operatorname{dR}}}^{+}\times\mathcal{C}_{X^{I_{2}}_{\operatorname{dR}}}^{+}\longrightarrow\mathcal{C}_{X^{I_{1}}_{\operatorname{dR}}}\otimes\mathcal{C}_{X^{I_{2}}_{\operatorname{dR}}}\longrightarrow\mathsf{D}((X^{I_{1}}\times
X^{I_{2}})_{\operatorname{disj}})\underset{\mathsf{D}(X^{I_{1}\sqcup
I_{2}})}{\otimes}\mathcal{C}_{X^{I_{1}\sqcup I_{2}}_{\operatorname{dR}}}$

is contained in

$(\mathsf{D}((X^{I_{1}}\times
X^{I_{2}})_{\operatorname{disj}})\underset{\mathsf{D}(X^{I_{1}\sqcup
I_{2}})}{\otimes}\mathcal{C}_{X^{I_{1}\sqcup
I_{2}}_{\operatorname{dR}}})^{+}.$

Then the sheaf of categories $\mathcal{C}^{\operatorname{ren}}$ on
$\operatorname{Ran}_{\operatorname{dR}}^{\operatorname{un}}$ from Proposition
3.9.1 naturally lifts to an object of $\mathsf{FactCat}^{\operatorname{lax-
fact}}$.

###### Proof.

Apply Proposition 3.2.2 to view $\mathcal{C}$ as a lax symmetric monoidal
section of (3.2.1). By Proposition 2.9.2, for any object $p:I\to J$ in
$\operatorname{Part}_{\operatorname{un}}$, the category

$\mathcal{C}_{U(p)}=\mathsf{D}(U(p))\underset{\mathsf{D}(X^{I})}{\otimes}\mathcal{C}_{X^{I}_{\operatorname{dR}}}$

is almost ULA generated over $U(p)$ with respect to the unique t-structure
that makes $\mathcal{C}_{X^{I}_{\operatorname{dR}}}\to\mathcal{C}_{U(p)}$
t-exact. Given objects $p_{1}:I_{1}\to J_{1}$, $p_{2}:I_{2}\to J_{2}$ and a
morphism $\epsilon:p_{1}\to p_{2}$ in
$\operatorname{Part}_{\operatorname{un}}$, we claim that the corresponding
structure functor $\mathcal{C}_{U(p_{1})}\to\mathcal{C}_{U(p_{2})}$ has finite
homological amplitude. This follows from hypotheses (ii-iii) of Proposition
3.9.1 and the commutative square

${\mathcal{C}_{X^{I_{1}}_{\operatorname{dR}}}}$${\mathcal{C}_{X^{I_{2}}_{\operatorname{dR}}}}$${\mathcal{C}_{U(p_{1})}}$${\mathcal{C}_{U(p_{2})},}$

since the vertical functors are t-exact with fully faithful and left t-exact
right adjoints. Similarly, the structure functor
$\mathcal{C}_{U(p_{1})}\otimes\mathcal{C}_{U(p_{2})}\to\mathcal{C}_{U(p_{1}\sqcup
p_{2})}$ coming from the lax symmetric monoidal structure sends
$\mathcal{C}_{U(p_{1})}^{+}\times\mathcal{C}_{U(p_{2})}^{+}$ into
$\mathcal{C}_{U(p_{1}\sqcup p_{2})}^{+}$, which we can see from the
commutative square

${\mathcal{C}_{X^{I_{1}}_{\operatorname{dR}}}\otimes\mathcal{C}_{X^{I_{1}}_{\operatorname{dR}}}}$${\mathsf{D}((X^{I_{1}}\times
X^{I_{2}})_{\operatorname{disj}})\underset{\mathsf{D}(X^{I_{1}\sqcup
I_{2}})}{\otimes}\mathcal{C}_{X^{I_{1}\sqcup
I_{2}}_{\operatorname{dR}}}}$${\mathcal{C}_{U(p_{1})}\otimes\mathcal{C}_{U(p_{2})}}$${\mathcal{C}_{U(p_{1}\sqcup
p_{2})}.}$

Combining the above observations, it follows that applying (2.6.1) termwise to
the categories $\mathcal{C}_{U(p)}$ yields another lax symmetric monoidal
section of (3.2.1). This section satisfies condition (i) of Proposition 3.2.2
by Proposition 2.10.3, and condition (ii) is trivial to verify.

∎

Let $(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{t-str}}$ be the pseudo-tensor category whose
objects consist of $\mathcal{C}$ in $\mathsf{FactCat}^{\operatorname{lax-
fact}}$ together with a t-structure on
$\mathcal{C}_{X^{I}_{\operatorname{dR}}}$ for every finite set $I$, and whose
$n$-ary morphisms are the same as those in the symmetric monoidal category
$\mathsf{FactCat}^{\operatorname{lax-fact}}$, i.e. we do not impose any
t-exactness conditions on functors.

Denote by

$(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{aULA}}\subset(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{t-str}}$

the pseudo-tensor subcategory whose objects $\mathcal{C}$ satisfy the
conditions of Propositions 3.9.1 and 3.9.2, and whose $n$-ary morphisms
$(\mathcal{C}_{1},\cdots\\!,\mathcal{C}_{n})\to\mathcal{D}$ satisfy the
condition that the image of

$(\mathcal{C}_{1})_{X^{I}_{\operatorname{dR}}}^{+}\times\cdots\times(\mathcal{C}_{n})_{X^{I}_{\operatorname{dR}}}^{+}\longrightarrow(\mathcal{C}_{1})_{X^{I}_{\operatorname{dR}}}\underset{\mathsf{D}(X^{I})}{\otimes}\cdots\underset{\mathsf{D}(X^{I})}{\otimes}(\mathcal{C}_{n})_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathcal{D}_{X^{I}_{\operatorname{dR}}}$

is contained in $\mathcal{D}_{X^{I}_{\operatorname{dR}}}^{+}$ for any finite
set $I$.

######  Proposition 3.9.3.

The assignment $\mathcal{C}\mapsto\mathcal{C}^{\operatorname{ren}}$ of
Proposition 3.9.2 naturally lifts to a morphism of pseudo-tensor categories

$(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{aULA}}\longrightarrow(\mathsf{FactCat}^{\operatorname{lax-
fact}})^{\otimes}_{\operatorname{aULA}}.$

###### Proof.

This is immediate, since the construction of Proposition 3.9.2 is induced by
the morphism of pseudo-tensor categories (2.6.1).

∎

##  4\. A local acyclity theorem

The main goal of this section is to prove Theorem 4.6.1, which says in
particular that the vacuum factorization module for a commutative algebra
almost of finite type is almost ULA. From this we deduce Corollary 4.6.1.1,
which says that the category of factorization modules over a commutative
algebra of finite type satisfies the conditions of Proposition 3.9.2 and hence
can be renormalized, provided that the same is true for the category of
commutative modules over that algebra.

###  4.1. Representations as a factorization category

Below we collect some facts about the factorization category
$\mathsf{Rep}(H)$, where $H$ is an algebraic group.

The forgetful functor
$\operatorname{oblv}_{H}:\mathsf{Rep}(H)\to\operatorname{Vect}$ is symmetric
monoidal, and hence gives rise to a morphism in $\mathsf{FactCat}$.

######  Proposition 4.1.1.

For any finite set $I$, the functor

$\operatorname{oblv}_{H}:\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{D}(X^{I})$

is conservative, and there is a unique t-structure on
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ which makes this functor
t-exact. The morphism $\operatorname{oblv}_{H}$ admits a right adjoint
$\operatorname{coind}_{H}$ in $\mathsf{FactCat}_{\operatorname{lax-untl}}$,
and for any finite set $I$, the functor

$\operatorname{coind}_{H}:\mathsf{D}(X^{I})\longrightarrow\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$

is t-exact.

###### Proof.

See Propositions 6.22.1 and 6.24.1 of [Ras21a].

∎

######  Proposition 4.1.2.

The factorization category $\mathsf{Rep}(H)$, with the t-structure on
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ defined as above for each finite
set $I$, satisfies the conditions of Proposition 3.9.2, and

$\mathsf{Rep}(H)^{\operatorname{ren}}\longrightarrow\mathsf{Rep}(H)$

is an isomorphism.

###### Proof.

The t-structure on $\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ inherits
compatibility with filtered colimits and right completeness from
$\mathsf{D}(X^{I})$ via the conservative and t-exact functor
$\operatorname{oblv}_{H}$. To see that this t-structure is almost compactly
generated, it remains to prove that if an object $\mathcal{M}$ is compact in
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}^{\geq 0}$, then it is almost
compact in $\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$. By the previous
proposition, we have the $\mathsf{D}(X^{I})$-linear adjunction

$\operatorname{oblv}_{H}:\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}\rightleftarrows\mathsf{D}(X^{I}):\operatorname{coind}_{H}$

with both functors t-exact. Thus $\operatorname{oblv}_{H}(\mathcal{M})$ is
compact in $\mathsf{D}(X^{I})^{\geq 0}$, hence almost compact because
$\mathsf{D}(X^{I})$ is almost compactly generated. Now Lemma 2.4.1 implies
that $\mathcal{M}$ is almost compact.

If $\\{V_{\alpha}\\}$ is a set of compact generators for
$\mathsf{Rep}(H)^{\otimes I}$, then
$\\{\operatorname{Loc}_{X^{I}}(V_{\alpha})\\}$ is a set of ULA generators for
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$. Note that
$\operatorname{Loc}_{X^{I}}$ is t-exact up to shift, and in particular each of
these objects is eventually coconnective. It follows that
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ is almost ULA generated and that

$(\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}})^{\operatorname{ren}}\tilde{\longrightarrow}\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}.$

It is not difficult to see that the other conditions in Propositions 3.9.1 and
3.9.2 are are inherited from $\mathsf{D}(X^{I})$ via
$\operatorname{oblv}_{H}$.

∎

###  4.2. The t-structure on factorization modules

Let $A$ be a commutative algebra in $\mathsf{Rep}(H)^{\leq 0}$.

######  Proposition 4.2.1.

For any finite set $I$, there is a unique t-structure on
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
which makes the forgetful functor

$\operatorname{oblv}^{\operatorname{fact}}:A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$

t-exact.

We will need the following lemma.

######  Lemma 4.2.2.

If $Y$ and $Z$ are classical local complete intersection schemes and $f:Y\to
Z$ is finite flat, then $f^{!}:\mathsf{D}(Z)\to\mathsf{D}(Y)$ is t-exact.

###### Proof.

First, we claim that the forgetful functor

$\operatorname{oblv}_{Y}:\mathsf{D}(Y)\longrightarrow\mathsf{IndCoh}(Y)$

is t-exact. Recall that for any scheme $Y$ the functor
$\operatorname{oblv}_{Y}$ is left t-exact, so it suffices to prove right
t-exactness. Since the claim is Zariski local on $Y$, we can assume that there
exists a regular closed embedding $i:Y\to W$ where $W$ is smooth. The
composite functor

$\mathsf{D}(Y)\xrightarrow{i_{\operatorname{dR},*}}\mathsf{D}(W)\xrightarrow{\operatorname{oblv}_{W}}\mathsf{IndCoh}(W)$

is t-exact, and admits a nonnegative filtration with associated graded

$\operatorname{Sym}_{\mathcal{O}_{W}}(\operatorname{N}(Y/W))\otimes
i_{*}\operatorname{oblv}_{Y}(-)$

(here $\operatorname{N}(Y/W)$ denotes the normal bundle, and the tensor
product denotes the action of $\mathsf{QCoh}(W)$ on $\mathsf{IndCoh}(W)$).
Since this associated graded contains $i_{*}\operatorname{oblv}_{Y}(-)$ as a
direct summand and $i_{*}$ is t-exact and conservative, it follows that
$\operatorname{oblv}_{Y}$ is right t-exact.

Using the commutative square

${\mathsf{D}(Z)}$${\mathsf{D}(Y)}$${\mathsf{IndCoh}(Z)}$${\mathsf{IndCoh}(Y),}$$\scriptstyle{f^{!}}$$\scriptstyle{\operatorname{oblv}_{Z}}$$\scriptstyle{\operatorname{oblv}_{Y}}$$\scriptstyle{f^{!}}$

it is therefore enough to prove that
$f^{!}:\mathsf{IndCoh}(Z)\longrightarrow\mathsf{IndCoh}(Y)$ is t-exact. This
functor is left t-exact because $f$ is finite. On the other hand, since $Y$ is
Gorenstein, so is the morphism $f$, which implies right t-exactness.

∎

Now we can prove the proposition.

###### Proof of Proposition 4.2.1.

First, we claim that if such a t-structure exists on

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}},$

then the proposition will follow. Namely, by Proposition 3.8.4 we have

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\cong
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\times\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}.$

It therefore suffices to observe that the direct factor
$0\times\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ is stable under
truncation functors.

Thus we are reduced to proving the proposition for non-unital factorization
modules. In that case, by [FG12] Corollary 7.3.5 we have the Koszul duality
equivalence

$A\text{\textendash}\mathsf{mod}^{\operatorname{fact},\operatorname{non-
untl}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\tilde{\longrightarrow}(A\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}},$

where $(A\otimes\omega_{X}[-1])$ is the commutative chiral algebra attached to
$A$. This equivalence intertwines the forgetful functors to
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$. If $\mathcal{M}$ is an object
of

$(A\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}},$

we claim that the truncation $\tau^{\leq 0}\mathcal{M}$ taken in
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ admits a chiral
$(A\otimes\omega_{X}[-1])$-module structure compatible with the morphism
$\tau^{\leq 0}\mathcal{M}\to\mathcal{M}$. It will then follow that

$\operatorname{Hom}_{(A\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}}(\mathcal{N},\tau^{\leq
0}\mathcal{M})\tilde{\longrightarrow}\operatorname{Hom}_{(A\otimes\omega_{X}[-1])\text{\textendash}\mathsf{mod}^{\operatorname{ch}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}}(\mathcal{N},\mathcal{M})$

for any $\mathcal{N}$ which is connective in
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$, since

$\operatorname{Hom}_{\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}}(\mathcal{N},\tau^{>0}\mathcal{M})=0$

and $\operatorname{oblv}_{A\otimes\omega_{X}[-1]}$ is conservative.

Let $i:Z_{I}\to X\times X^{I}$ denote the inclusion of the incidence divisor
and $j:(X\times X^{I})_{\operatorname{disj}}\to X\times X^{I}$ its complement,
and write $p:Z_{I}\to X^{I}$ for the projection. Note that by Lemma 4.2.2, the
functor

$i_{\operatorname{dR},*}p^{!}:\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{X_{\operatorname{dR}}}\otimes\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$

is t-exact. The structure map of $\mathcal{M}$ is a morphism

$j_{*}j^{*}((A\otimes\omega_{X}[-1])\boxtimes\mathcal{M})\longrightarrow
i_{\operatorname{dR},*}p^{!}\mathcal{M}$

in
$\mathsf{Rep}(H)_{X_{\operatorname{dR}}}\otimes\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$.
The object

$j_{*}j^{*}((A\otimes\omega_{X}[-1])\boxtimes\tau^{\leq 0}\mathcal{M})$

is connective, and hence the composition

$j_{*}j^{*}((A\otimes\omega_{X}[-1])\boxtimes\tau^{\leq
0}\mathcal{M})\longrightarrow
j_{*}j^{*}((A\otimes\omega_{X}[-1])\boxtimes\mathcal{M})\longrightarrow
i_{\operatorname{dR},*}p^{!}\mathcal{M}$

factors uniquely through

$\tau^{\leq
0}i_{\operatorname{dR},*}p^{!}\mathcal{M}=i_{\operatorname{dR},*}p^{!}\tau^{\leq
0}\mathcal{M}.$

The same reasoning applies to $n$-ary chiral operations for $n>1$, which
yields the desired chiral module structure on $\tau^{\leq 0}\mathcal{M}$.

∎

###  4.3. Commutative factorization modules

We continue to assume that $A$ is connective. Recall that $A$ determines a
commutative factorization algebra in the commutative factorization category
$\mathsf{Rep}(H)$, so that in particular we obtain an object $A_{X^{I}}$ in
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ for any finite set $I$.

######  Lemma 4.3.1.

For any finite set $I$, we have $H^{i}(A_{X^{I}})=0$ for all $i>-\\#I$. If $A$
is eventually coconnective, then so is $A_{X^{I}}$.

###### Proof.

It suffices to prove both claims after applying the t-exact and conservative
functor

$\operatorname{oblv}_{H}:\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{D}(X^{I}),$

so we can assume that $H$ is trivial.

The first assertion can be proved by the same argument as Lemma 6.24.3 of
[Ras21a].

The second assertion is clear from the Cousin filtration, whose subquotients
have the form $A^{\otimes J}\otimes j_{*}j^{!}\omega_{Z(p)}$ where $p:I\to J$
is surjective and $j:Z(p)\to X^{I}$ is the inclusion of the corresponding
stratum.

∎

The symmetric monoidal functor

$\operatorname{ind}_{A}:\mathsf{Rep}(H)\longrightarrow
A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))$

gives rise to a morphism of factorization categories. Since these symmetric
monoidal categories are rigid, Lemma 3.4.2 says that $\operatorname{ind}_{A}$
admits a right adjoint

$\operatorname{oblv}_{A}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))\longrightarrow\mathsf{Rep}(H)$

in $\mathsf{FactCat}_{\operatorname{lax-untl}}$.

######  Proposition 4.3.2.

For any finite set $I$, the category
$A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$
admits a unique t-structure such that

$\operatorname{oblv}_{A}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\longrightarrow\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$

is t-exact.

###### Proof.

The functor $\operatorname{oblv}_{A}$ admits the $\mathsf{D}(X^{I})$-linear
left adjoint $\operatorname{ind}_{A}$, and the monad
$\operatorname{oblv}_{A}\circ\operatorname{ind}_{A}$ on
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ is given by tensoring with the
commutative algebra $A_{X^{I}}$. Since the tensor product in
$\mathsf{D}(X^{I})$ is has homological amplitude $\\#I$, the same holds for
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$, so Lemma 4.3.1 implies that
this monad is right t-exact. The proposition follows.

∎

The forgetful functor from commutative to factorization $A$-modules is a
morphism

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))$

in $\mathsf{FactCat}$. It is compatible with the forgetful functors to
$\mathsf{Rep}(H)$, and in particular the functor

$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}:A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}$

is t-exact for any finite set $I$.

By definition, the _vacuum factorization module_ $\operatorname{Vac}_{A}$ in
$A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))$ is
the unit object for the factorization structure. The unitality of
$\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}$ implies that

$\operatorname{Vac}_{A}=\operatorname{oblv}^{\operatorname{com}\to\operatorname{fact}}A.$

### 4.4.

Let $\mathcal{A}$ be a symmetric monoidal DG category, $\mathcal{C}$ an
$\mathcal{A}$-module category, and $\mathcal{C}_{0}\subset\mathcal{C}$ a (not
necessarily stable) full subcategory. For any associative algebra $A$ in
$\mathcal{A}$, we can consider

$A\text{\textendash}\mathsf{mod}(\mathcal{C}_{0}):=A\text{\textendash}\mathsf{mod}(\mathcal{C})\underset{\mathcal{C}}{\times}\mathcal{C}_{0}.$

We will use the following formal lemma.

######  Lemma 4.4.1.

Suppose that $A\to B$ is a morphism of associative algebras in $\mathcal{A}$
with the property that

$\operatorname{Hom}_{\mathcal{C}}(B^{\otimes n}\otimes
M,N)\longrightarrow\operatorname{Hom}_{\mathcal{C}}(A^{\otimes n}\otimes M,N)$

is an isomorphism for all $n\geq 1$ and all $M,N$ in $\mathcal{C}_{0}$. Then
the restriction of scalars functor

$B\text{\textendash}\mathsf{mod}(\mathcal{C}_{0})\longrightarrow
A\text{\textendash}\mathsf{mod}(\mathcal{C}_{0})$

is an equivalence.

######  Lemma 4.4.2.

Fix $m\geq 0$. Let $A\to B$ be a morphism of connective commutative algebras
in $\mathsf{Rep}(H)$ such that $H^{i}(A)\tilde{\to}H^{i}(B)$ for all
$i\geq-m$. Then for any finite set $I$, we have

$H^{i}(A_{X^{I}})\tilde{\longrightarrow}H^{i}(B_{X^{I}})$

for all $i\geq-\\#I-m$.

###### Proof.

As in the proof of Lemma 4.3.1, we can assume that $H$ is trivial.

Let us treat the case $\\#I=2$ for simplicity, the general case being similar.
Fix $i\geq-2-m$. Our hypothesis implies that

$H^{i+1}(A\otimes\Delta_{\operatorname{dR},*}\omega_{X})=H^{i+2}(A)\otimes\Delta_{\operatorname{dR},*}\omega_{X}[-1]\longrightarrow
H^{i+2}(B)\otimes\Delta_{\operatorname{dR},*}\omega_{X}[-1]=H^{i+1}(B\otimes\Delta_{\operatorname{dR},*}\omega_{X})$

is an isomorphism. Since $A$ and $B$ are connective, we have

$H^{i+2}(A\otimes A)\tilde{\longrightarrow}H^{i+2}(B\otimes B),$

from which we deduce that

$H^{i}(A\otimes A\otimes
j_{*}j^{*}\omega_{X^{2}})\tilde{\longrightarrow}H^{i}(B\otimes B\otimes
j_{*}j^{*}\omega_{X^{2}}).$

On the other hand, the surjectivity of $H^{i+2}(A\otimes A)\to H^{i+2}(A)$
implies that the connecting homomorphism

$H^{i}(A\otimes A\otimes j_{*}j^{*}\omega_{X^{2}})\longrightarrow
H^{i+1}(A\otimes\Delta_{\operatorname{dR},*}\omega_{X})$

for the Cousin triangle is surjective, and likewise for $B$, which implies the
desired isomorphism.

∎

######  Lemma 4.4.3.

In the situation of the previous lemma, the restriction of scalars functors

$B\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\leq
0,\geq-m}\longrightarrow
A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\leq
0,\geq-m}$

and

$B\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\leq
0,\geq-m}\longrightarrow
A\text{\textendash}\mathsf{mod}^{\operatorname{fact}}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}^{\leq
0,\geq-m}$

are equivalences.

###### Proof.

For the commutative case, we saw in the proof of Proposition 4.3.2 that

$A_{X^{I}}\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}})\tilde{\longrightarrow}A\text{\textendash}\mathsf{mod}(\mathsf{Rep}(H))_{X^{I}_{\operatorname{dR}}}.$

For any $\mathcal{M}$ in $\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}^{\leq
0,\geq-m}$ and any $n\geq 1$, Lemmas 4.3.1 and 4.4.2 imply that

$H^{i}(A_{X^{I}}^{\otimes
n}\overset{!}{\otimes}\mathcal{M})\tilde{\longrightarrow}H^{i}(B_{X^{I}}^{\otimes
n}\overset{!}{\otimes}\mathcal{M})$

for all $i\geq-m$, since the tensor product in
$\mathsf{Rep}(H)_{X^{I}_{\operatorname{dR}}}$ has cohomological amplitude
$\\#I$. Now apply Lemma 4.4.1.