# Generalized local jocabians and commutative group stacks

Bertrand Toën

(Date: August 2023)

###### Abstract.

In [CS01, Page 109] Grothendieck sketches the construction of a complex
$J_{*}(X)$ made of commutative pro-algebraic groups, associated to a smooth
variety $X$, and for which each $J_{i}(X)$ is a product of local factors
called the _local generalized jacobians_. The complex $J_{*}(X)$ has the
property to compute Zariski cohomology of $X$ with coefficients in algebraic
groups and play the role of a homology of $X$. The purpose of this note is to
recast this construction in the setting of higher algebraic group stacks for
the fppf topology instead. For this, we introduce a notion of algebraic
homology associated to a scheme which is a universal object computing fppf
cohomology with coefficients in group schemes. We endow this algebraic
homology with a filtration by dimension of supports, and prove that, when $X$
is smooth, $J_{*}(X)$ appears as the $E_{1}$-page of the corresponding
spectral sequence. In a final part we partially extends our constructions and
results over arbitrary bases.

###### Contents

  1. 1 Commutative algebraic group stacks
    1. 1.1 Reminders on abelian group schemes
    2. 1.2 Reminders on purity for flat cohomology
    3. 1.3 Commutative algebraic group stacks
  2. 2 Algebraic homology
  3. 3 Grothendieck’s generalized local jacobians
    1. 3.1 The filtration by (co)dimension
    2. 3.2 Grothendieck’s construction
  4. 4 Partial generalization to more general schemes
    1. 4.1 Affine homology
    2. 4.2 Affine homology as a dg-module

## Introduction

The main purpose of this note is to introduce a notion of _algebraic homology_
, associated to schemes over a perfect field $k$, which is a universal object
computing flat cohomology with coefficients in various commutative group
schemes. The idea of algebraic homology is certainly not new, and was already
sketched in a slight different setting in a letter of Grothendieck to Serre
(see letter from 8.9.1960 in [CS01, page 109]), as well as in [Gro] (cote
134-2 page 60) and in [Mil76] for instance. However, in the present work we
realize algebraic homology as a universal (pro)object in the $\infty$-category
of commutative groups in algebraic higher stacks, whereas in the above
references it is constructed as a complex of commutative pro-algebraic groups.
The main result of this work is a comparison statement for smooth schemes, and
states that our notion of algebraic homology recovers Grothendieck’s orginal
construction. This comparison is not a formal statement as it relates two
objects living in two non-equivalent categories, and requires to use an extra
piece of data in the form of a filtration by dimension of supports.
Grothendieck’s original construction is shown to appear as the $0$-th
truncation of algebraic homology, for the Beilinson’s t-structure on filtered
objects (see for instance [Pav23]). This is a fancy manner to states that
Grothendieck’s complex of pro-algebraic groups appears as the $E_{1}$-page of
the spectral sequence associated to the filtration by dimension of supports on
algebraic homology. We also show that both objects can be used in order to
compute fppf cohomology with coefficients in commutative group schemes (with
some affineness and unipotency conditions for higher cohomology groups). This
last statement is true by definition for algebraic homology but requires a
proof in the case of Grothendieck’s construction which is done by gluing
together the so-called _local generalized jacobians_ appearing in our title.

We work over a perfect field $k$, and in the $\infty$-topos of hypercomplete
fppf stacks $\mathbf{St}_{k}$. In a first part of this work we introduce the
notions of _commutative algebraic group stacks over $k$_ (see definition 1.3),
which are the commutative group objects inside the $\infty$-category of Artin
stacks locally of finite type over $k$ and whose diagonal are quasi-compact
(in a strong sense). These are higher stacks analogues of _locally algebraic
groups over $k$_, that is separated commutative group schemes locally of
finite type. The commutative algebraic group stacks can also be characterized
as connective complexes of sheaves of abelian groups $E$ of the big fppf site,
whose cohomology sheaves $H^{i}(E)$ are all representable by locally algebraic
groups, and are moreover quasi-projective if $i<0$. The algebraic homology of
a scheme $X$ over $k$ (or any stack over $k$) is then defined as follows (see
definition 3.1).

######  Definition 0.1.

For a scheme $X$ over $k$, its algebraic homology $\mathbb{H}^{alg}(X)$ is the
pro-object in commutative algebraic group stacks which pro-represents the
$\infty$-functor $E\mapsto\mathbf{Map}(X,E)$.

The existence of the pro-object $\mathbb{H}^{alg}(X)$ in the definition above
follows formally from the left exactness of the $\infty$-functor
$E\mapsto\mathbf{Map}(X,E)$. However, its existence already carries
interesting cohomological information. For instance, the universal morphism of
stacks $X\to\mathbb{H}^{alg}(X)$ can be seen to be a generalization of the
Albanese morphism of $X$ which includes higher cohomological data in the
picture. In a way, $\mathbb{H}^{alg}(X)$ could also be considered as a derived
version of the classical Albanese variety of $X$ (see end of §2).

In order to express $\mathbb{H}^{alg}(X)$ in more explicit terms we introduce
the standard filtration by dimension of supports

$0\to F_{dimX}\mathbb{H}^{alg}(X)\dots\to\dots F_{i}\mathbb{H}^{alg}(X)\to
F_{i-1}\mathbb{H}^{alg}(X)\dots\to
F_{0}\mathbb{H}^{alg}(X)=\mathbb{H}^{alg}(X)$

where $F_{i}\mathbb{H}^{alg}(X)$ is the part of the algebraic homology which
is supported in codimension $i$ in $X$. The $d$-th graded piece of this
filtration is given by the product of all local algebraic homologies over all
the points of codimension $d$ in $X$ (see proposition 3.2)

$Gr_{d}\mathbb{H}^{alg}(X)\simeq\prod_{x\in
X^{(dimX-d)}}\mathbb{H}^{alg}_{x}(X).$

When $X$ is furthermore smooth over $k$, purity implies that each graded piece
$Gr_{d}\mathbb{H}^{alg}(X)$ is in fact $(n-d)$-connective (see corollary 3.4).
In another word, the filtered object $F_{*}\mathbb{H}^{alg}(X)$ is
$0$-connective with respect to Beilinson’s t-structure on filtered objects. As
being connective, we get a canonical projection of filtered objects

$F_{*}\mathbb{H}^{alg}(X)\longrightarrow H^{0}_{B}(F_{*}\mathbb{H}^{alg}(X))$

where $H^{0}_{B}$ refers to the $0$-th cohomology object for the Beilinson
t-structure. The object $H^{0}_{B}(F_{*}\mathbb{H}^{alg}(X))$ is a genuine
complex made of commutative pro-algebraic groups, and as such can be compared
to Grothendieck’s original construction $J_{*}(X)$. In the complex $J_{*}(X)$,
the degree $d$ part is the product of local generalized jacobian $J_{x}(X)$
over all points $x$ of codimension $d$ in $X$ which are defined to pro-
represent local cohomology in degree $d$ (see definition 3.5).

Our main comparison result is the following theorem.

######  Theorem 0.2.

We have a natural isomorphism of complexes of pro-algebraic groups over $k$

$H^{0}_{B}(F_{*}\mathbb{H}^{alg}(X))\simeq J_{*}(X).$

The theorem below, shows that $J_{*}(X)$ appears as the $0$-th truncation of
$F_{*}\mathbb{H}^{alg}(X)$. However, the natural projection
$F_{*}\mathbb{H}^{alg}(X)\to J_{*}(X)$ is not an equivalence in general, and
it is thus surprising that both objects still compute the same cohomologies
with unipotent coefficients.

######  Theorem 0.3.

Let $i\geq 0$ and $G$ a commutative algebraic group $G$, affine if $i>0$ and
unipotent if $i>1$. Then, the canonical projection $\mathbb{H}^{alg}(X)\to
J_{*}(X)$ induces isomorphisms

$Ext^{i}_{naive}(J_{*}(X),G)\simeq Ext^{i}_{fppf}(\mathbb{H}^{alg}(X),G)\simeq
H^{i}_{fppf}(X,G).$

In the theorem above the $Ext^{i}_{naive}$ refers to the Yoneda ext-groups
computed inside the derived category of complexes of commutative pro-algebraic
groups. These ext-groups differ from the ext-groups computed as fppf sheaves,
and thus are not given by ext-groups computed inside our category of
commutative group stacks. We note that the theorem above is already stated in
[CS01, page 109] where the fppf topology is replaced with the Zariski
topology.

All the results mentioned above are true over a perfect field $k$. We do not
know a way to extend our results and constructions over more general base
schemes, at least in an interesting manner. The reader will notice that a key
ingredient for our results is purity for local cohomology of regular schemes
with coefficients in group schemes. Very general purity theorems have been
proven recently in [CS24], but we would rather need purity results with
coefficients in commutative algebraic group stacks, and we do not see a simple
manner to obtained those from the case of flat group schemes. We hope to be
able to come back to this question in a future work.

However, when restricted to unipotent coefficients, we propose a
generalization of algebraic homology over any base ring, based on our notion
of affine stacks (see [Toe06]) and their abelian counter-parts which we call
_affine homology types_. We prove that any (small) stack $X$, in particular
any scheme, maps to a universal affine homology types called its affine
homology and denoted by $\mathbb{H}^{u}(X)$ (see proposition 4.6 and
definition 4.8). Over a field, $\mathbb{H}^{u}(X)$ is the unipotent completion
of algebraic homology, and its advantage is that it makes sense over any base.
Affine homology comes equipped with a filtration by dimension of supports
whose associated graded are given by local affine homologies. Moreover, we
prove that $\mathbb{H}^{u}(X)$ is completely characterized by the complex
$\mathbb{R}\Gamma(X,\mathbb{G}_{a})$ considered as a dg-algebra over
$\mathbb{R}End(\mathbb{G}_{a})$, which we consider as a form of Dieudonné
theory for affine homology types.

To finish this introduction we would like to mention several related works. As
we already mentioned the notion of algebraic homology already appears in
[CS01, page 109]) and [Mil76]. The results of the present work does not
pretend to be very original and the main point of this note was rather to
recast these works in the more modern language of higher stacks. The only
exception is possibly our last section on affine homology, which we think can
be of independent interest. After posting a first version of this work O.
Gwilliam pointed to us the work [Mir17] in which a general setting for
algebraic homology based on higher commutative group stacks is presented.

In another direction, the results of this work have some limitations. First of
all the fact that we work over a base field is restrictive. Over more general
bases, the $1$-truncation of algebraic homology essentially appears in the
work of S. Brochard in [Bro21]. Extending the results of [Bro21] seems however
difficult because of the presence of complicated higher ext groups between
commutative group schemes which prevent to a perfect duality by mapping to
$\mathbb{G}_{m}$. Also, the approach in [Bro21] is based on duality, which is
an aspect we havent touched at all in this note. Grothendieck already mentions
duality in [Gro, 134-2], and in degree $0$ this duality of local generalized
jacobians have been studied in [CC90] (it also appears in [Mir17]).

Our comparison requires $X$ to be smooth, and in his letter Grothendieck
already suggests an extension of the definition of $J_{*}(X)$ to the singular
setting as well, but we could not reconstruct from it anything meaningful at
the moment.

Finally, there are also relations with algebraic homotopy types, as studies
for instance in [Toe06, KPT09, BV15, MR23]. Our notion of affine homology is
the ablelian analogue of these, and the two should be related by means of the
Hurewitz morphism. As a final comment, affine stacks are duals to cosimplicial
commutative algebras, also called $LSym$-algebras. These are Koszul dual to
partition Lie algebras, thanks to [BM23], and therefore our notion of affine
homology should be closely related to that of abelian partition Lie algebras.

Acknowledgments: In a first version of this work algebraic homology was only
studied with some extra artificial unipotency conditions. I warmly thank B.
Kahn for his comments during a lecture on the content of this work which has
convinced me that these extra conditions were not necessary at all.

##  1\. Commutative algebraic group stacks

We work over a perfect base field $k$. We denote by $\mathbf{Aff}_{k}$ the
category of affine $k$-schemes, which will be endowed with the fppf topology.
The $\infty$-category of (hypercomplete) stacks on $\mathbf{Aff}_{k}$ is
denoted by $\mathbf{St}_{k}$.

###  1.1. Reminders on abelian group schemes

We denote by $\mathbf{AbSch}_{k}$, the category of commutative group schemes
over $k$ which are separated and of finite type over $k$ (or equivalently
quasi-projective over $k$, see for instance [sta] lemma 39.8.2, tag 0BF6). The
Yoneda embedding provides a fully faithful functor

$\mathbf{AbSch}_{k}\hookrightarrow\mathbf{Ab}_{k},$

from $\mathbf{AbSch}_{k}$ to the category $\mathbf{Ab}_{k}$ of sheaves of
abelian groups on $\mathbf{Aff}_{k}$ endowed with the $fppf$-topology. It is
well known that this full embedding commutes with the formations of kernels
and cokernels, and that its essential image is also stable by extensions. We
do not know a precise reference for this, so we include it as a lemma.

######  Lemma 1.1.

The full sub-category of $\mathbf{Ab}_{k}$ formed by sheaves which are
representable by commutative separated group schemes of finite type over $k$,
is stable by taking kernels, cokernels and extensions.

Proof. The stability by kernels is obvious as the Yoneda embedding is left
exact. For the stability by cokernels, the quotient fppf-sheaf $G/H$, for a
monomorphism of commutative algebraic groups $H\hookrightarrow G$, is always
representable by an Artin stack whose diagonal is a monomorphism, that is by
an algebraic space. This algebraic space is moreover quasi-separated (because
$H$ is quasi-compact) and of finite type. We can thus use Artin’s theorem
stating that a group object in the category of quasi-separated algebraic
spaces over a field is representable by a (separated) group scheme (see
[Art69, Lem. 4.2]). Finally, if a sheaf of abelian groups is an extension of
algebraic groups, then it is, by descent, representable by a quasi-separated
algebraic space, and we can conclude using again Artin’s result. $\Box$

The Barsotti-Chevalley decomposition theorem (see [Mil13]) states that any
object $G\in\mathbf{AbSch}_{k}$ fits in an exact sequence

$0\to G_{0}\to G\to A\to 0,$

where $G_{0}$ is affine and $A$ is an abelian variety. Moreover, by [DG70,
II-5 Cor. 2.3] $G_{0}$ itself sits in an exact sequence $0\to K\to G_{0}\to
H\to 0,$ where $K$ is smooth and affine, and $H$ is finite infinitesimal (i.e.
$H_{red}=0$). We also recall from [DG70, IV-2 Porp. 2.5] that
$G\in\mathbf{AbSch}_{k}$ is unipotent if it admits a finite filtration by sub-
group schemes

$\dots\subset G_{i+1}\subset G_{i}\dots\subset G_{1}\subset G_{0}=G$

such that each $G_{i}/G_{i+1}$ can be realized as a sub-group scheme the
additive group $\mathbb{G}_{a}$. We can moreover arrange things so that each
$G_{i}/G_{i+1}$ is a twisted form of $\mathbb{Z}/p$ or of $\alpha_{p}$.
Unipotent group schemes are automatically affine, and form a full sub-category
of $\mathbf{AbSch}_{k}$ stable by taking kernels, cokernels and extensions.

Finally, an affine commutative group scheme $G$ of finite type over $k$ is
always isomorphic to a product $G^{u}\times G^{m}$ of a unipotent group scheme
$G^{u}$ and a group scheme of multiplicative type $G^{m}$. When $k$ is
algebraically closed, such a group scheme $G^{m}$ is always a product of
multiplicative groups $\mathbb{G}_{m}$ and roots of unity $\mu_{n}$.

###  1.2. Reminders on purity for flat cohomology

We remind here the purity theorem we will use in a crucial manner in order to
compare algebraic homology with Grothendieck’s original definition. The most
general results on purity can be found in [CS24], but for smooth schemes $X$
over a perfect $k$ these can be proven with simple technology, which we recall
below.

Recall that for a point $x$ in a scheme $X$ we define complex of local
cohomology of $X$ at $x$, noted $\mathbb{R}\Gamma_{x}(X,A)$, with coefficient
in an abelian fppf-sheaf $A\in Ab_{k}$, as the homotopy fiber of the
restriction map

$\mathbb{R}\Gamma_{fppf}(X_{x},A)\to\mathbb{R}\Gamma_{fppf}(X_{x}-\\{x\\},A)$

where $X_{x}=\mathbf{Spec}\,\mathcal{O}_{X,x}$ is the local scheme at $x$. It
cohomology groups are denoted as usual by
$H^{i}_{x}(X,A):=H^{i}(\mathbb{R}\Gamma_{x}(X,A))$.

######  Propositon 1.2.

Let $X$ be a smooth and connected scheme of dimension $n$ over $k$. For any
point $x\in X$ of dimension $d$, and any commutative algebraic group scheme
$G$ of finite presentation over $k$, we have

$H^{i}_{x}(X,G)=0\qquad for\,i<d.$

Proof. By Galois descent we can assume that $k$ is algebraically closed. Let
us fix $x\in X$ of dimension $d$ as in the statement. We start by the easy
observation that the class of groups $G$ satisfying the conclusion of the
proposition is stable by extensions, and by taking kernels of epimorphic
morphisms. We can thus apply several standard devissages on $G$ and reduce the
statement to the following elementary cases:

  1. (1)

$G$ is unipotent

  2. (2)

$G$ is an abelian variety

  3. (3)

$G$ is of multiplicative type.

For the case $(1)$ we can use again devissage and thus reduce to the case
where $G=\mathbb{G}_{a}$ is the additive group. Similarly, for the case $(3)$
we reduce to the case of the multiplicative group $G=\mathbb{G}_{m}$.
Moreover, the case $G=\mathbb{G}_{a}$ follows from the usual local coherent
duality: the complex $\mathbb{R}\Gamma_{x}(X,\mathbb{G}_{a})$ is
cohomologically concentrated in the single degree $d$.

Case $G=\mathbb{G}_{m}$. For $d=n$ there is nothing to prove, and when $d=n-1$
the statement simply follows from the fact that $X_{x}-\\{x\\}$ is dense in
$X_{x}$. We thus assume that $d<n-1$. The case $i=1$ is true by Hartogs, as
the restriction map $Pic(X_{x})\to Pic(X_{x}-\\{x\\})$ is an equivalence of
groupoids. For $i=2$ the proposition follows from the fact that
$H^{2}_{et}(X_{x},\mathbb{G}_{m})\to H^{2}_{et}(X_{x}-\\{x\\},\mathbb{G}_{m})$
is always injective (see [Gro68, Cor. 1.10]). When $i>2$, the fact that
$\mathbb{G}_{m}$ has torsion cohomology in degree $2$ or higher (see [Gro68,
Prop. 1.4]) implies that $H^{i}_{x}(X,\mathbb{G}_{m})$ is always torsion. We
are thus led to prove the statement for the group $\mu_{n}$ for some $n$, and
by devissage we can even assume that $n$ is prime. If $n$ is invertible in
$k$, then the purity theorem in étale cohomology (see for instance [Mil80, §VI
Thm. 5.1]) implies the result. When $n=p$ is the characteristic of the base
field, we use the following standard "resolution" of $\mu_{p}$: on the small
fppf site of $X$, there is an equivalence

$\mu_{p}[1]\simeq(\Omega_{X/k}^{1,cl}\to\Omega_{X/k}^{1}),$

where $\Omega_{X/k}^{1,cl}\subset\Omega_{X/k}^{1}$ is the subsheaf of closed
$1$-forms and sits in cohomological degrees $0$ (see [Mil80, §III Prop.
4.14]). The differential $\Omega_{X/k}^{1,cl}\to\Omega_{X/k}^{1}$ is here
$i-C$, where $i$ is the standard inclusion and $C$ is the composition of the
projection $\Omega_{X/k}^{1,cl}\to H^{1}_{DR}(X/k)$ and the Cartier
isomorphism $C:H^{1}_{DR}(X/k)\simeq\Omega^{1}_{X/k}$. Both abelian sheaves
$\Omega_{X/k}^{1,cl}$ and $\Omega_{X/k}^{1}$ are quasi-coherent (the first one
on the Frobenius twists $X^{(1)}$ and the second one on $X$), and thus satisfy
purity. We deduce from this that $H^{i}_{x}(X,\mu_{p}[1])\simeq
H^{i+1}_{x}(X,\mu_{p})\simeq 0$ as soon as $i<d$ and in particular that
$H^{i}_{x}(X,\mu_{p})\simeq 0$ when $i<d$.

Case $G$ is an abelian variety. We first use that all the groups
$H^{i}_{et}(X,G)\simeq H^{i}_{fppf}(X,G)$ are torsion for $i>0$. Indeed, let
$i:\eta\to X$ be the generic point of $X$, and $E=\mathbb{R}i_{*}(i^{*}(G))$.
Then $H^{i}(X,E)\simeq H^{i}(\eta,G)$ is Galois cohomology of the field of
fractions of $X$ with coefficients in $G$, and thus is torsion for all $i>0$.
Moreover, the Weil extension theorem implies that the sheaf
$\underline{H}^{0}(E)$ is canonically isomorphic to $G$ (via the adjunction
map $G\to E\simeq\mathbb{R}i_{*}i^{*}(G)$). We consider the triangle $G[0]\to
E\to E_{>0}$, and we notice that all the cohomology sheaves of $E_{>0}$ are
torsion sheaves because are also described in terms of Galois cohomology. The
fibration

$\mathbb{R}\Gamma(X,G)\to\mathbb{R}\Gamma(X,E)\to\mathbb{R}\Gamma(X,E_{>0})$

then implies that all the cohomology groups $H^{i}(X,G)$ must also be torsion.

When $i=0$, $H^{0}_{x}(X,G)=0$ by the density of $X_{x}-\\{x\\}$ in $X_{x}$.
Moreover, when $d<n$, the restriction $G(X_{x})\to G(X_{x}-\\{x\\})$ is
bijective by Weil’s extension theorem, and thus $H^{1}_{x}(X,G)$ identifies
with the kernel of $H^{1}(X_{x},G)\to H^{1}(X_{x}-\\{x\\},G)$ and are thus
torsion groups. Finally, by what we have seen $H^{i}_{x}(X,G)$ are torsions
for $i>1$ as well. Therefore, to prove the proposition we can replace $G$ by
the finite group scheme $G[n]$ which is the kernel of the multiplication by
$n$ on $G$. This finite group scheme can be split into a unipotent part and a
reductive part and thus have already been treated in the cases before. $\Box$

###  1.3. Commutative algebraic group stacks

We denote by $\mathbf{AbSt}_{k}$ the $\infty$-category obtained from
$\mathbf{St}_{k}$ by $\mathbb{Z}$-linearization. It can be described
explicitly as the $\infty$-category of $\infty$-functors

$E:\mathbf{Aff}_{k}^{op}\to\mathbf{dg}^{c}_{\mathbb{Z}},$

from affine $k$-schemes to connective complexes of abelian groups, satisfying
the usual $fppf$ hyper-descent condition. Any object $E\in\mathbf{Ab}_{k}$
possesses an underlying derived stack $|E|\in\mathbf{St}_{k}$ obtained by the
Dold-Kan construction, explicitly defined by

$|E|(S):=\mathbf{Map}_{\mathbf{dg}_{\mathbb{Z}}}(\mathbb{Z},E(S))$

for any $S\in\mathbf{Aff}_{k}$. This defines a forgetful $\infty$-functor
$\mathbf{Ab}_{k}\to\mathbf{dSt}_{k}$, which admits a left adjoint sending $F$
to $\mathbb{Z}\otimes F$. Recall that for $E\in\mathbf{AbSt}_{k}$ we denote by
$\pi_{i}(E)$ the cohomology sheaves $H^{-i}(E)$, which are also the homotopy
sheaves $\pi_{i}(|E|)$ of the underlying stack $|E|$ pointed at $0$.

######  Definition 1.3.

A _commutative algebraic group stack_ is an object $E\in\mathbf{AbSt}_{k}$
such that underlying stack $|E|\in\mathbf{St}_{k}$ is an Artin $n$-stack (for
some $n$), locally of finite presentation and strongly quasi-separated over
$k$. They form, by definition, a full sub-$\infty$-category

$\mathbf{AbSt}^{alg}_{k}\subset\mathbf{AbSt}_{k}.$

Remind that _strongly quasi-separated_ means that the diagonal map $E\to
E\times E$ is strongly quasi-compact in the sense of [TV08]. This means that
all the higher diagonal maps $X\to X^{S^{n}}$ are quasi-compact for $n\geq 0$.
Equivalently, the inertia stack $IE\to E$ is required to be strongly quasi-
compact over $E$. As explained in [Toe05, Cor. 2.9] an Artin stack $E$ locally
of finite presentation which is strongly quasi-separated is such that for all
field value point $s\in E(K)$, all its homotopy sheaves $\pi_{i}(E,s)$ are
quasi-compact and quasi-separated algebraic spaces over $K$, and thus are
representable by group schemes thanks to [Art69, Lem. 4.2]. As we are working
over a field, the converse also holds.

######  Propositon 1.4.

An object $E\in\mathbf{AbSt}_{k}$ is a commutative algebraic group stack if
and only if each homotopy sheaf $\pi_{i}(E)\in Ab_{k}$ is representable by a
group scheme locally of finite type over $k$, which is moreover quasi-
projective for $i>0$ and vanish for all but a finite number of indices $i$.

Proof. Assume first that $E$ is a commutative algebraic group stack. In the
definition 1.3. We have to prove that all the homotopy sheaves $\pi_{i}(E)$
are representable. We consider the inertia stack $IE$ of $E$. We already saw
that $\pi_{i}(E)$ are group schemes for $i>0$ by means of [Toe05, Cor. 2.9],
it remains to treat the case $i=0$. As $E$ is an abelian group object we have
splitting $IE\simeq E\times\Omega_{0}E$, where $\Omega_{0}E$ is the pointed
loop stack at $0$. As $k$ is a field, we thus have that $IE\to E$ is a fppf
cover which is strongly quasi-compact. We can thus use [Toe05, Prop. 2.4],
which remains valid only assuming the diagonal to be strongly quasi-compact,
and deduce that $E$ is a gerbe. In particular, $\pi_{0}(E)$ is representable
by an algebraic space locally of finite type over $k$. This is a group object
in algebraic space, and is moreover quasi-separated by assumptions on $E$, and
by [Art69, Lem. 4.2] is representable by a group scheme locally of finite
type.

Conversely, assume that $E$ satisfies the conditions of the proposition. We
will prove by induction on $i$ that the $i$-th homotopical truncation
$\tau_{\leq i}E$ is an Artin $i$-stack locally of finite presentation and
strongly quasi-separated over $k$. For $i=0$ this is obvious, as $\pi_{0}(E)$
is a scheme locally of finite type by assumption. Consider $E_{\leq i}\to
E_{\leq i-1}$, which fits into a cartesian square

$\textstyle{E_{\leq
i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{\leq
i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\pi_{i}(E),i+1).}$

By assumption, $\pi_{i}(E)$ is a group scheme of finite type over $k$ for
$i>0$, and thus $K(\pi_{i}(E),i+1)$ is an Artin $(i+1)$-stack locally of
finite presentation and strongly quasi-separated $k$. By induction, this
implies that $E_{\leq i}$ is an Artin $i$-stack locally of finite presentation
and strongly quasi-separated. $\Box$

By using the long exact sequence in homotopy, we see that the proposition 1.4
easily implies the following corollary.

######  Corollary 1.5.

The full sub-$\infty$-category
$\mathbf{AbSt}^{alg}_{k}\subset\mathbf{AbSt}_{k}$ is a is stable by finite
limits, retracts and the homotopical truncation $\infty$-functors $\tau_{\leq
i}$.

######  Remark 1.6.

_We warn the reader that if $E$ is a commutative algebraic group stack then
$E[1]=BE$ is in general not a commutative algebraic group stack anymore. It is
so only if $\pi_{0}(E)$ is moreover assumed to be of finite type._

We will also use occasionally the following bigger category.

######  Definition 1.7.

A _commutative locally algebraic group stack_ is an object
$E\in\mathbf{AbSt}_{k}$ for all $i$ the sheaf $\pi_{i}(E)$ is representable by
an algebraic space locally of finite type. The $\infty$-category of
commutative locally algebraic group stacks is denoted by
$\mathbf{AbSt}^{lalg}_{k}\subset\mathbf{AbSt}_{k}$.

As opposed to algebraic group stacks, the locally algebraic stacks are stable
by suspension: if $E\in\mathbf{AbSt}_{k}^{lalg}$, then $BE$, computed in
$\mathbf{AbSt}_{k}$, remains in $\mathbf{AbSt}_{k}^{lalg}$. More generally,
$\mathbf{AbSt}_{k}^{lalg}$ is stable by finite limits and colimits, as this
can be easily seen using the long exact sequence in homotopy.

##  2\. Algebraic homology

In this section we define algebraic homology and study its basic properties.

We let $X$ be any stack over $k$. We consider the $\infty$-functor

$\mathbf{AbSt}_{k}^{alg}\longrightarrow\mathsf{Top}$

defined by $E\mapsto\mathbf{Map}(X,E)$. This $\infty$-functor will be denoted
by $E\mapsto\mathbb{H}(X,E)$. By definition it obviously commutes with finite
limits, and thus is pro-representable by an object

$\mathbb{H}^{alg}(X)\in Pro(\mathbf{AbSt}_{k}^{alg}).$

######  Definition 2.1.

With the notation above, the object $\mathbb{H}^{alg}(X)$ is called the
_algebraic homology of $X$ (relative to $k$)_.

Note that the definition 2.1 makes sense for any stack $X$ over $k$, possibly
not even representable by an Artin stack. However, we will mainly be
interested in the case where $X$ is a scheme of finite type over $k$. Note
also that algebraic homology is a relative notion and depends on the base
field $k$. It should therefore be noted by $\mathbb{H}^{alg}(X/k)$ when
mentioning the base field is important.

By definition we have in particular that for any commutative group scheme $G$,
separated and of finite type over $k$

$H^{i}_{fppf}(X,G)\simeq\pi_{0}(\mathbf{Map}_{\mathbf{St}_{k}}(X,K(G,i))\simeq\pi_{0}(\mathbf{Map}_{Pro(\mathbf{AbSt}_{k})}(\mathbb{H}^{alg}(X),K(G,i))).$

######  Remark 2.2.

_The notion above of algebraic homology is very close to that of flat homology
of[Mil76], but is different. The major difference is that these two objects
live in two different, non-equivalent, $\infty$-categories. Flat homology
lives in the $\infty$-category of complexes of commutative pro-algebraic
groups. This has a canonical $\infty$-functor to $\mathbf{AbSt}_{k}$, by
considering the stack pro-represented by a complex of pro-alegebraic groups,
but this $\infty$-functor is known not to be fully faithful. This is due to
the existence of two different, non-isomorphic, definitions of ext-groups of
commutative algebraic groups: Yoneda ext-groups inside algebraic groups and
ext-groups of the corresponding fppf-sheaves. It is known that these two
notions are different (see for instance [Bre69])._

By definition the algebraic homology possesses the following basic properties.

Functoriality. The construction $X\mapsto\mathbb{H}^{alg}(X)$ is functorial in
$X$, and defines an $\infty$-functor from Artin $k$-stack to pro-object in
derived commutative algebraic groups. Moreover, the identity of
$\mathbb{H}^{alg}(X)$ defines a canonical class
$\alpha_{X}\in\mathbb{H}(X,\mathbb{H}^{alg}(X))$, and thus a canonical
morphism of pro-stacks

$\alpha_{X}:X\to\mathbb{H}^{alg}(X),$

which is functorial in $X$ as well.

Homology of the point. We write $\mathbb{H}^{alg}(k)$ for
$\mathbb{H}^{alg}(\mathbf{Spec}\,k)$. It is easy to see that
$\mathbb{H}^{alg}(k)\simeq\mathbb{Z}$ is the constant fppf sheaf associated to
$\mathbb{Z}$ and pro-constant as a pro-object.

Homology groups. The pro-object $\mathbb{H}^{alg}(X)$ possesses pro-homology
groups $\mathbb{H}_{i}^{alg}(X):=\pi_{i}(\mathbb{H}^{alg}(X))$, which are
commutative pro-algebraic groups over $k$. By definition our notion of
commutative algebraic group stacks $\mathbb{H}_{0}^{alg}(X)$ is a pro
(separated) group scheme locally of finite presentation, and all the
$\mathbb{H}_{i}^{alg}(X)$ are group schemes separated and of finite type for
$i>0$.

In general, the projection $X\to\mathbf{Spec}\,k$ provides a canonical
morphism
$\mathbb{H}^{alg}(X)\to\mathbb{H}^{alg}(\mathbf{Spec}\,k)\simeq\mathbb{Z}$. We
thus have a canonical fibration sequence of commutative algebraic group stacks

$\mathbb{H}^{alg}(X)^{(0)}\to\mathbb{H}^{alg}(X)\to\mathbb{H}^{alg}(k)\simeq\mathbb{Z},$

where $\mathbb{H}^{alg}(X)^{(0)}$ is by definition the _reduced algebraic
homology of $X$_. In general the fibration sequence above does not split (see
below), and a choice of a rational point $x\in X(k)$ provides a canonical
splitting $\mathbb{H}^{alg}(X)\simeq\mathbb{Z}\oplus\mathbb{H}^{alg}(X)^{(0)}$
and corresponding splitting on homology groups.

Note also that by the universal property, if $X$ is geometrically connected
then the pro-étale group of connected components of $\mathbb{H}_{0}^{alg}(X)$
must be isomorphic to the constant sheaf $\mathbb{Z}$ by the above morphism
$\mathbb{H}_{0}^{alg}(X)\to\mathbb{Z}$.

Relation to the Albanese variety. The pro-algebraic group
$\mathbb{H}_{0}^{alg}(X)$ is directly related to the Albanese variety of $X$.
Indeed, for any commutative algebraic group $G$, we have a canonical bijection

$H^{0}(X,G)=Hom(X,G)\simeq Hom(\mathbb{H}_{0}^{alg}(X),G).$

Fixing a point $x\in X(k)$, we get a decomposition
$\mathbb{H}_{0}^{alg}(X)\simeq\mathbb{H}_{0}^{alg}(X)^{(0)}\oplus\mathbb{Z}$.
When $X$ has no $k$-point, the projection $X\to\mathbf{Spec}\,k$ provides a (a
priory non-split) short exact sequence

$0\to\mathbb{H}_{0}^{alg}(X)^{(0)}\to\mathbb{H}_{0}^{alg}(X)\to\mathbb{Z}\to
0.$

We can consider the universal morphism $X\to\mathbb{H}^{alg}(X)$, and check
that the composite morphism $X\to\mathbb{H}_{0}^{alg}(X)\to\mathbb{Z}$ is the
constant function $1$ on $X$ as soon as $X$ is geometrically connected. It
thus factors through a canonical morphism

$alb_{X}:X\to\mathbb{H}_{0}^{alg}(X)^{(1)}$

where $\mathbb{H}_{0}^{alg}(X)^{(1)}$ is the fiber at $1$ of the short exact
sequence above, considered naturally as a torsor over the group
$\mathbb{H}_{0}^{alg}(X)^{(0)}$. This morphism is a generalization of the
usual Albanese morphism to the Albanese variety. When $X$ is smooth and
proper, $\mathbb{H}_{0}^{alg}(X)^{(1)}$ can be identified with the usual
Albanese variety, and $alb_{X}$ with the usual the Albanese map, as these two
objects satisfy the same universal property.

In general, the universal morphism $X\to\mathbb{H}^{alg}(X)$, composed with
the projection $\mathbb{H}^{alg}(X)\to\mathbb{Z}$ is the constant map
$X\to\mathbb{Z}$ equals to $1$. Therefore, we have a canonical morphism
$X\to\mathbb{H}^{alg}(X)^{(1)}$ where $\mathbb{H}^{alg}(X)^{(1)}$ is the fiber
at $1$ of $\mathbb{H}^{alg}(X)\to\mathbb{Z}$. This morphism is a higher
cohomological generalization of the Albanese morphism as being the universal
map towards a torsor over an object in $Pro(\mathbf{AbSt}_{k}^{alg})$.

Relation to the Albanese stack. When $X$ is an Artin stack of finite type over
$X$, the first Postnikov truncation $\mathbb{H}_{\leq 1}^{alg}(X)$ is an
abelian group object in algebraic pro-Artin $1$-stacks. This object is closely
related $D(Pic(X))$, the dual of the Picard stack of $X$ (relative to $k$)
studied in [Bro21]. Indeed,
$Pic(X):=\underline{\mathbf{Map}}(X,B\mathbb{G}_{m})$ is the internal $Hom$
stack from $X$ to $B\mathbb{G}_{m}$, and we thus have a canonical evaluation
morphism $ev:X\times Pic(X)\to B\mathbb{G}_{m}$ which defines a canonical
morphism

$X\to\underline{\mathbf{Map}}(Pic(X),B\mathbb{G}_{m})=D(Pic(X)).$

When $X$ is proper and the dual $D(Pic(X))$ is known to be a representable
group stack and sits in a fibration sequence of commutative group stacks

$0\to D(\pi_{0}(Pic(X)))\to D(Pic(X))\to\mathbb{Z}\to 0.$

The choice of a global point $x\in X(k)$ defines a splitting of this sequence,
and thus the evaluation map $X\to D(Pic(X))$ can be composed to a morphism
$X\to D(H^{0}(Pic(X))$. By the universal property of algebraic homology we get
a canonical morphism of pro-objects in commutative algebraic $1$-stacks

$\mathbb{H}_{\leq 1}^{alg}(X)^{(0)}\to D(H^{0}(Pic(X))).$

In absence of global points we get a morphism $\mathbb{H}_{\leq
1}^{alg}(X)^{(1)}\to D(H^{0}(Pic(X)))$ where $\mathbb{H}_{\leq
1}^{alg}(X)^{(1)}$ is the fiber at $1$ of the canonical fibration sequence

$\mathbb{H}_{\leq 1}^{alg}(X)^{(0)}\to\mathbb{H}_{\leq
1}^{alg}(X)\to\mathbb{H}_{\leq 1}^{alg}(\mathbf{Spec}\,k)$

as described above. This map is an isomorphism when $X$ satisfies some extra
conditions expressed in [Bro21] (e.g. when $X$ is smooth and proper over $k$),
as both objects $\mathbb{H}_{\leq 1}^{alg}(X)^{(1)}$ and $D(H^{0}(Pic(X)))$
then satisfy the same universal property. This provides a direct relation
between algebraic homology and the Albanese morphism constructed in [Bro21].

Change of base fields. The construction $X\mapsto\mathbb{H}^{alg}(X/k)$ is
compatible with change of base field in the following sense. Let
$k^{\prime}/k$ be a finite field extension, and
$X^{\prime}:=X\times_{k}k^{\prime}$ the base change of $X$ over
$\mathbf{Spec}\,k^{\prime}$. By universal property we have a canonical
morphism of pro-objects

$\mathbb{H}^{alg}(X^{\prime}/k^{\prime})\to\mathbb{H}^{alg}(X/k)\times_{k}k^{\prime}.$

We claim that this morphism is in fact an equivalence of pro-object. Indeed,
for any $G\in AbSt_{k}^{alg}$, we have

$Map_{\mathbf{St}_{k^{\prime}}}(\mathbb{H}^{alg}(X/k)\times_{k}k^{\prime},G)\simeq
Map_{\mathbf{St}_{k}}(\mathbb{H}^{alg}(X/k),p_{*}(G)),$

where $p_{*}(G)$ is the direct image (i.e. _restriction à la Weil_) of $G$
along the morphism $p:\mathbf{Spec}\,k^{\prime}\to\mathbf{Spec}\,k$. Because
$p$ is finite and etale it is easy to see that the stack $p_{*}(G)$ is again a
commutative algebraic group stack. Therefore, by the universal property of
algebraic homology we get

$Map_{\mathbf{AbSt}_{k^{\prime}}}(\mathbb{H}^{alg}(X/k)\times_{k}k^{\prime},G)\simeq
Map_{\mathbf{AbSt}_{k}}(\mathbb{H}^{alg}(X/k),p_{*}(G))$ $\simeq
Map_{\mathbf{St}_{k}}(X,p_{*}(G))\simeq
Map_{\mathbf{St}_{k^{\prime}}}(X^{\prime}\times_{k}k^{\prime},G)\simeq
Map_{\mathbf{AbSt}_{k^{\prime}}}(\mathbb{H}^{alg}(X^{\prime}/k^{\prime}),G),$

as required.

##  3\. Grothendieck’s generalized local jacobians

In this section we provide $\mathbb{H}^{alg}(X)$ with a natural filtration,
induced by the standard filtration by codimension on cohomology. The
associated graded to this filtration possesses a description in local terms
involving local cohomology. When working with smooth schemes over $k$, the
local factors will directly related to Grothendieck’s orginial construction of
local generalized jacobians from his letter to Serre (letter from 9.8.1960,
[CS01] page 109).

In this section $X$ is a scheme of finite type over $k$. We denote by
$X^{(d)}$ the subset of $X$ consisting of all points of dimension $d$ in $X$.
Similarly, $X^{(\geq d)}$ will be the subset of points of dimension at least
equal to $d$.

###  3.1. The filtration by (co)dimension

We let $d\geq 0$ be an integer, and consider $Op_{d}(X)$ the poset of opens
$U\subset X$ containing $X^{(\geq d)}$. An open $U$ belongs to $Op_{d}(X)$ if
and only if the closed complement $U-Z$ has dimension at most $d-1$. The only
element in $Op_{0}(X)$ is $X$ itself, whereas $\emptyset\in Op_{dimX+1}(X)$,
and obviously $Op_{d}(X)\subset Op_{d+1}(X)$. We define
$F_{d}\mathbb{H}^{alg}(X)$ to be

$F_{d}\mathbb{H}^{alg}(X):=\lim_{U\in Op_{d}(X)}\mathbb{H}^{alg}(U)$

where the limit is taken over the poset $Op_{d}(X)$ and inside the
$\infty$-category $Pro(\mathbf{AbSt}_{k}^{alg})$ of pro-objects in commutative
algebraic group stacks. Note that
$F_{0}\mathbb{H}^{alg}(X)\simeq\mathbb{H}^{alg}(X)$, and as $\emptyset$ is
initial in $Op_{dimX+1}(X)$ we have $F_{dimX+1}\mathbb{H}^{alg}(X)\simeq 0$.

The object $F_{d}\mathbb{H}^{alg}(X)$ comes equipped with a canonical morphism
$F_{d+1}\mathbb{H}^{alg}(X)\to F_{d}\mathbb{H}^{alg}(X)$, induced from the
inclusion of posets $Op_{d}(X)\to Op_{d+1}(X)$, and we thus obtain this way a
filtration

$0=F_{dimX+1}\mathbb{H}^{alg}(X)=\mathbb{H}^{alg}(X)\to\dots\to
F_{d}\mathbb{H}^{alg}(X)\to F_{d-1}\mathbb{H}^{alg}(X)\to\dots\to
F_{0}\mathbb{H}^{alg}(X)=\mathbb{H}^{alg}(X).$

######  Definition 3.1.

The above filtration is called the _filtration by dimension_ on
$\mathbb{H}^{alg}(X)$.

The $d$-th graded piece $Gr_{d}\mathbb{H}^{alg}(X)$ are defined as the cofiber
of $F_{d+1}\mathbb{H}^{alg}(X)\to F_{d}\mathbb{H}^{alg}(X)$, computed in the
stable $\infty$-category $Pro(\mathbf{AbSt}_{k})$ (see comment 3.3) which is
dual to local cohomology in dimension $d$ on $X$. More precisely, we have the
following universal characterization of $Gr_{d}\mathbb{H}^{alg}(X)$.

######  Propositon 3.2.

With the notation above, and for any commutative algebraic group stack $G$,
there exists a canonical equivalence, functorial in $G$

$\mathbf{Map}(Gr_{d}\mathbb{H}^{alg}(X),G)\simeq\bigoplus_{x\in
X^{(d)}}\mathbb{H}_{x}(X,G),$

where the sum runs over the set $X^{(d)}$ consisting of all points of $X$ of
dimension $d$, and where $\mathbb{H}_{x}(X,G)$ is the space of local
cohomology defined as the fiber of
$\mathbf{Map}(X_{x},G)\to\mathbf{Map}(X_{x}-\\{x\\},G)$ with
$X_{x}=\mathbf{Spec}\,\mathcal{O}_{X,x}$. Equivalently, we have a canonical
identification

$Gr_{d}\mathbb{H}^{alg}(X)\simeq\prod_{x\in
X^{(d)}}\mathbb{H}_{x}^{alg}(X_{x})$

where $\mathbb{H}_{x}^{alg}(X_{x})$ is local algebraic homology at $x$,
defined as the cofiber of the natural morphism
$\mathbb{H}^{alg}(X_{x}-\\{x\\})\to\mathbb{H}^{alg}(X_{x})$.

Proof. For any inclusion of open $V\subset U$, with $U\in Op_{d}(X)$ and $V\in
Op_{d+1}(X)$, the closed complement $U-V$ only has a finite number of points
of dimension $d$. Indeed, if $(U-V)^{(d)}$ is infinite, then $U-V$ must be of
dimension at least $d+1$ and thus must contain at least one point of dimension
$d+1$, which contradicts $V\in Op_{d+1}(X)$. Note also that for
$x\in(U-V)^{(d)}$, we have $X_{x}\cap V=X_{x}-\\{x\\}$, as $x$ is the only
point of $X_{x}$ of dimension $d$ in $X$. We thus get this way a commutative
square of schemes

$\textstyle{\coprod_{x\in(U-V)^{(d)}}X_{x}-\\{x\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{x\in(U-V)^{(d)}}X_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U,}$

and thus an induced commutative square on the vertical cofibers on algebraic
homologies

$\prod_{x\in(U-V)^{(d)}}\mathbb{H}_{x}^{alg}(X_{x})\longrightarrow\mathbb{H}^{alg}(U)/\mathbb{H}^{alg}(V).$

By passing to the limit over the pairs $V\subset U$ with $U\in Op_{d}(X)$ and
$V\in Op_{d+1}(X)$ we find the required morphism

$\phi:\prod_{x\in
X^{(d)}}\mathbb{H}_{x}^{alg}(X_{x})\longrightarrow\lim_{V\subset
U}\mathbb{H}^{alg}(U)/\mathbb{H}^{alg}(V)=Gr_{d}\mathbb{H}^{alg}(X).$

It remains to prove that this morphism $\phi$ is an equivalence. For this, by
functoriality in $X$, we localize the construction on the small Zariski site
of $X$. Both constructions

$U\mapsto\prod_{x\in U^{(d)}}\mathbb{H}_{x}^{alg}(U_{x})\qquad U\mapsto
Gr_{d}\mathbb{H}^{alg}(U)$

are costacks with values in $Pro(\mathbf{AbSt}_{k})$ on the small Zariski site
$X_{Zar}$. It is therefore enough to check that the above morphism induces an
equivalence on the fibers at a given point $x\in X$. This fiber is clearly $0$
if $x$ is not of codimension $d$. If $x$ is of codimension $d$ the fiber of
the right hand side is equivalent to

$\lim_{x\in U\in
Op_{d}(X)}\mathbb{H}^{alg}(U)/\mathbb{H}^{alg}(U-\overline{\\{x\\}}).$

The fiber at $x$ of the left hand side is clearly
$\mathbb{H}_{x}^{alg}(X_{x})$. We are thus reduced to show that for any
commutative algebraic group stack $G\in\mathbf{AbSt}_{k}^{alg}$, the canonical
morphisms

$co\lim_{x\in U}\mathbf{Map}(U,G)\to\mathbf{Map}(X_{x},G)\qquad co\lim_{x\in
U}\mathbf{Map}(U-Z_{x},G)\to\mathbf{Map}(X_{x}-\\{x\\},G)$

are equivalences. But this is true because, by definition of commutative group
stacks, $G$ is an Artin stack of finite presentation, and thus its functor of
points sends filtered colimits of commutative rings to filtered colimits.
$\Box$

######  Remark 3.3.

_As we already saw, the commutative algebraic group stacks are not stable by
taking cofibers. So, a priory $Gr_{d}\mathbb{H}^{alg}(X)$ merely belongs to
$Pro(\mathbf{AbSt}_{k}^{lalg})$, the $\infty$-category of pro-objects in
commutative locally algebraic groups stacks. However, it is easy to see by
universal property that for any connected smooth variety $U$ over $k$, the
group of connected components of $\mathbb{H}_{0}^{alg}(U)$ is canonically
isomorphic to $\mathbb{Z}$, as any morphism from $U$ to a discrete group $G$
must be constant. As a result, for any morphism $U\to V$ of such varieties,
the induced morphism $\mathbb{H}_{0}^{alg}(U)\to\mathbb{H}_{0}^{alg}(V)$ is an
isomorphism on the group of connected components. This easily implies that
$Gr_{d}\mathbb{H}^{alg}(X)$ is a pro-object in $\mathbf{AbSt}_{k}$._

As the objects $Gr_{d}\mathbb{H}^{alg}(X)$ are pro-objects in
$\mathbf{AbSt}_{k}^{alg}$, their homotopy sheaves are pro-sheaves of abelian
groups. In the results below we will simply say that
$Gr_{d}(\mathbb{H}^{alg}(X))$ is _$m$ -connective_ for some $m$, to signify
that the pro-sheaves $\pi_{i}(Gr_{d}\mathbb{H}^{alg}(X))$ vanish for $i<m$.

######  Propositon 3.4.

Let $X$ be a connected and smooth $k$-scheme of dimension $n$. Then, for all
integer $d$ the object $Gr_{d}\mathbb{H}^{alg}(X)$ is $(n-d)$-connective: its
pro-homotopy groups $\pi_{i}(Gr_{d}\mathbb{H}^{alg}(X))$ vanish for $i<n-d$.

Proof. When $d=n$ there is nothing to prove as by construction all
$Gr_{d}\mathbb{H}^{alg}(X)$ are $0$-connective. For $d<n$ we have to show that
$\pi_{i}(Gr_{d}\mathbb{H}^{alg}(X))=0$ for $i<n-d$. Equivalently we have to
prove that any $G\in\mathbf{AbSt}_{k}^{alg}$ which is $(n-d-1)$-truncated,
then $\mathbf{Map}_{\mathbf{AbSt}_{k}}(Gr_{d}\mathbb{H}^{alg}(X),G)\simeq*$.
By proposition 3.2 this is equivalent to say that $\mathbb{H}_{x}(X,G)\simeq
0$ for all point $x\in X$ of dimension $d$. We then proceed by Postnikov
decomposition on $G$, and we thus reduce the corollary to the following two
statements.

  1. (1)

For all $x\in X^{(d)}$, and all abelian group scheme $G$, separated and
locally of finite type, we have $H_{x}^{0}(X,G)=0$.

  2. (2)

For all $x\in X^{(d)}$ and all abelian group scheme $G$, separated and of
finite type, we have $H^{i}_{x}(X,G)=0$ for all $i<n-d$

The first of this statement is obviously true because $d<n$ and thus
$X_{x}-\\{x\\}$ is dense in $X_{x}$. The second statement has already been
reminded in proposition 1.2. $\Box$

###  3.2. Grothendieck’s construction

We will now compare the general notion of algebraic homology with the
construction sketched by Grothendieck in [CS01, Page 109] (which also appears
in a letter to L. Breen, see [Gro, cote 134-2] page 60). As we are working
with the fppf topology, we will have to modify slightly the presentation that
was originally done by Grothendieck in the context of the Zariski topology.
For this, and all along this section, we will assume that $X$ is a smooth
scheme, connected and of dimension $n$ over $k$. Let $x\in X^{(d)}$ be a point
of dimension $d$ on $X$.

We consider the functor $G\mapsto H_{x}^{n-d}(X_{x},G)$, of (fppf) local
cohomology at $x$, defined on the category of abelian group schemes (separated
and of finite type) over $k$. By purity (proposition 1.2) we have
$H_{x}^{n-d-1}(X_{x},G)\simeq 0$ and thus the above functor is left exact in
$G$. It is therefore pro-representable by an abelian pro-gourp scheme
$J_{x}(X)$.

For an integer $d$ we then set

$J_{m}(X):=\prod_{x\in X^{(n-m)}}J_{x}(X)$

which is a pro object in the category of commutative group schemes locally of
finite type over $k$. We denote by $J_{*}(X)$ the graded object
$\oplus_{m}J_{m}(X)$. It comes equipped with a homological differential
$J_{m}(X)\to J_{m-1}(X)$ defined as follows. For a fixed $z\in X^{(n-m+1)}$,
we define a morphism

$d_{z}:\prod_{x\in X^{(n-m)}}J_{x}(X)\to J_{z}(X)$

by defining a natural transformation on the functor they pro-represent. The
right hand side pro-represents the functor $G\mapsto H^{m-1}_{z}(X_{z},G)$,
whereas the left hand side pro-represents the functor $G\mapsto\oplus_{x\in
X^{(n-m)}}H^{m}_{x}(X_{x},G)$. In order to define $d_{z}$ we thus have to
construct a morphism on local cohomologies

$H^{m-1}_{z}(X_{z},G)\longrightarrow\oplus_{x\in
X^{(n-m)}}H^{m}_{x}(X_{x},G),$

functorial in $G$. This morphism is the usual differential in the Cousin
complex (see [CTHK97, §1]). Any element $\alpha\in H^{m-1}_{z}(X_{z},G)$ comes
from an element $\alpha_{U}\in H^{m-1}_{Z}(U,G)$ for an open $U\subset X$
containing $z$, and where $Z$ is the closure of $z$ in $U$. We denote by
$x_{1},\dots,x_{q}$ the points of $X$ of dimension $n-m$ which are
specializations of $z$ but not belonging to $U$, and we let $T=X-U$. We
consider the boundary map $H^{m-1}(U,G)\to H^{m-1}_{T}(X,G)$, together with
the restriction morphism $H^{m}_{T}(X,G)\to H^{m}_{x}(X_{x},G)$ along the
inclusion of pairs $(X_{x},\\{x\\})\subset(X,T)$. We get this way a composed
morphism

$H^{m-1}_{Z}(U,G)\to H^{m-1}(U,G)\to H^{m}_{T}(X,G)\to\bigoplus_{1\leq i\leq
q}H^{m}_{x_{i}}(X_{x_{i}},G).$

The image of $\alpha_{U}$ by the morphism is by definition $d_{z}(\alpha)$. It
is well known that this morphism is independant of the choice of $\alpha_{U}$
and squares to zero.

######  Definition 3.5 (Grothendieck).

Let $X$ be a smooth and connected $k$-scheme of dimension $n$ as above.

  1. (1)

The pro-algebraic group $J_{d}(X)=\prod_{x\in X^{(n-i)}}J_{x}(X)$ is called
_Grothendieck’s local generalized jacobian of $X$ at $x$_.

  2. (2)

The complex of local jacobians, with the differentials define above, will be
denoted by

$J_{*}(X):=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
83.06386pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J_{n}(X)\to J_{n-1}(X)\to\dots\to J_{1}(X)\to
J_{0}(X)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

As a first relation between the complex $J_{*}(X)$ and algebraic homology, we
have the following direct observation, which is a direct consequence of
corollary 3.4, proposition LABEL:p5 and and the definitions of $J_{*}(X)$.

######  Corollary 3.6.

Let $X$ be a smooth $k$-scheme of dimension $n$. Then, the first non-zero
homotopy groups of the graded pieces of $\mathbb{H}^{alg}(X)$ are the
Grothendieck’s local generalized jacobians

$\pi_{n-d}(Gr_{n-d}\mathbb{H}^{alg}(X))\simeq J_{d}(X).$

The above result can be made more precise. Indeed, the filtration by dimension
on $\mathbb{H}^{alg}(X)$ induces a spectral sequence

$\pi_{p}(Gr_{q}\mathbb{H}^{alg}(X))\Rightarrow\pi_{p+q}(\mathbb{H}^{alg}(X)),$

and under the identification of the corollary the differential in the complex
$J_{*}(X)$ corresponds to the differential on the $E_{1}$-term of the spectral
sequence. Therefore, by the purity 3.4, Grothendieck’s local generalized
jacobian $J_{d}(X)$ appear as the lowest non-trivial homotopy group of
$Gr_{(n-d)}\mathbb{H}^{alg}(X)$. However, the graded pieces
$Gr_{(n-d)}\mathbb{H}^{alg}(X)$ are in general not concentrated in a single
degree, and thus $Gr_{(n-d)}\mathbb{H}^{alg}(X)$ and $J_{d}(X)$ differs. We
will now see however that the complex $J_{*}(X)$ is essentially enough to
recover the cohomology of $X$ with coefficients in certain commutative
algebraic group as stated in [CS01, Page 109].

The corollary 3.6 can also be phrased by stating that the complex of pro-
algebraic groups $J_{*}(X)$ is the $0$-truncation of the filtered object
$\mathbb{H}^{alg}(X)$ with respect to the Beilinson’s t-structure on filtered
objects. As a result, we do have a canonical projection of filtered objects

$q:\mathbb{H}^{alg}(X)\to J_{*}(X)$

where the filtration on the right hand side is the naive filtration $J_{*\leq
n-d}(X)\subset J_{*}(X)$. This projection is not an equivalence, as the graded
pieces $Gr_{d}\mathbb{H}^{alg}(X)$ are not purely concentrated in a single
degree in general. However, the projection $q$ induces an isomorphisms on
certain ext groups showing that the complex $J_{*}(X)$ can be used in order to
compute the individual cohomology groups $H^{i}_{fppf}(X,G)$ with coefficient
in a commutative algebraic group $G$ (affine if $i>0$, and unipotent if
$i>1$). In order to explain this, we need to first introduce a comparison
functor, between the naive derived category of commutative group schemes and
our category of commutative derived group stacks.

We know that the category $\mathbf{AbSch}_{k}$, of separated commutative group
schemes of finite type over $k$ is an abelian category. Its category of pro-
objects $Pro(\mathbf{AbSch}_{k})$ is known to be abelian and to possess enough
projective objects (see [Mil70]). As such it possesses a bounded derived
category $\mathcal{D}^{b}(Pro(\mathbf{AbSch}_{k}))$, obtained by localizing
the category of bounded complexes in $Pro(\mathbf{AbSch}_{k})$ with respect to
the quasi-isomorphisms. Because the functor sending $G\in
Pro(\mathbf{AbSch}_{k})$ to the fppf-sheaf it represents is exact, there is a
canonical $\infty$-functor

$h:\mathcal{D}^{b}(Pro(\mathbf{AbSch}_{k}))\longrightarrow\mathbf{AbSt}_{k},$

It is well known that this functor is not fully faithful in general, as this
can be seen on higher ext-groups. The ext-groups defined in
$\mathcal{D}^{b}(Pro(\mathbf{AbSch}_{k}))$ are the _naive extension groups_ of
commutative pro-algebraic groups, and can be described in terms of Yoneda
extensions. On the other side, the extensions groups in $\mathbf{AbSt}_{k}$
are the ext-groups of sheaves. It is know that, when $k$ is algebraically
closed, $Ext^{i}$ vanish when $i>2$ and when computed in
$\mathcal{D}^{b}(Pro(\mathbf{AbSch}_{k}))$ (see [Mil70]), but this is not the
case in $\mathbf{AbSt}_{k}$ (see for instance [Bre69]).

Using the projection $q$ and the $\infty$-functor $h$, we get for any
commutative group scheme $G$ of finite type over $k$, and any integer $i$ a
natural morphism of groups

$\phi^{i}_{G}:[J_{*}(X),G[i]]_{\mathcal{D}^{b}(Pro(\mathbf{AbSch}_{k}))}\longrightarrow[h(J_{*}(X)),h(G)[i]]\longrightarrow[\mathbb{H}^{alg}(X),h(G)[i]].$

By universal property the right hand side can be identified with
$H^{i}_{fppf}(X,G)$. We thus get under this restriction a natural morphism

$\phi^{i}_{G}:[J_{*}(X),G[i]]_{\mathcal{D}^{b}(Pro(\mathbf{AbSch}_{k}))}\to
H^{i}_{fppf}(X,G).$

For simplicity we denote by $Ext^{i}_{naive}(J_{*}(X),G)$ the left hand side
of the above morphism.

######  Propositon 3.7.

For all $i\geq 0$ and all commutative group scheme $G$ which is affine if
$i>0$ and unipotent if $i>1$, the morphism

$\phi_{G}^{i}:Ext^{i}_{naive}(J_{*}(X),G)\to H^{i}_{fppf}(X,G)$

is an isomorphism.

Proof. As in [Mil70] we can use Galois descent in order to reduce to the case
where $k$ is algebraically closed. We start by using that the projection
$q:\mathbb{H}^{alg}(X)\to J_{*}(X)$ is compatible with the filtrations on both
sides: the filtration by dimension on $\mathbb{H}^{alg}(X)$ and the naive
filtration on $J_{*}(X)$. These filtrations induce convergent spectral
sequences

$[Gr_{n-q}\mathbb{H}^{alg}(X),G[p]]\Rightarrow H^{p+q}_{fppf}(X,G)\qquad
Ext^{p}_{naive}(J_{q}(X),G)\Rightarrow Ext^{p+q}_{naive}(J_{*}(X),G).$

The morphism $\phi_{G}$ of the proposition is then compatible with these
spectral sequences, and we are thus reduced to show that, for fixed $p$ and
$q$, the induced morphism

$Ext^{p}_{naive}(J_{q}(X),G)\to[Gr_{n-q}\mathbb{H}^{alg}(X),G[p]]$

is an isomorphism. By proposition 3.2, the right hand side can be identified
with local cohomology $\coprod_{x\in X^{(n-q)}}H^{p}_{x}(X,G)$. We also note
that by construction $J_{q}(X)=\prod_{x\in X^{(n-q)}}J_{x}(X)$, and thus we
are reduced to show that for a given point $x\in X$, of codimension $d$, and
all $i\geq 0$, the natural morphism

$\psi^{i}_{G}:Ext^{i}_{naive}(J_{x}(X),G)\to H^{d+i}_{x}(X,G),$

under the extra condition stated in the proposition: $G$ is affine if $d+i>0$
and unipotent if $d+i>1$. Recall here that $J_{x}(X)$ corepresents the functor
$G\mapsto H^{d}_{x}(X,G)$, where $d$ is the codimension of $x$ in $X$.

We first deal with the two special cases $i+d=0$ and $i+d=1$. When $i+d=0$,
and thus $i=d=0$, we have $Ext^{0}(J_{x}(X),G)=Hom(J_{x}(X),G)\simeq
H^{d}_{x}(X,G)$ and it is then obvious that $\psi^{0}_{G}$ is bijective.
Suppose now that $d+i=1$, and thus $G$ is assumed to be affine. If $x$ is of
codimension $1$, we have $i=0$, and again $\psi_{G}^{0}$ is bijective by
definition of $J_{x}(X)$. If $x$ is of codimension $0$ and $i=1$, we have to
prove that the natural morphism

$Ext^{1}_{naive}(J_{0}(X),G)\to H^{1}_{fppf}(K(X),G),$

where $K(X)$ is the fraction field of $X$. To show this we start by the
following lemma treating the case of $G=\mathbb{G}_{a}$ and
$G=\mathbb{G}_{m}$.

######  Lemma 3.8.

With the notation above we have

$\psi_{G}^{1}:Ext^{1}_{naive}(J_{0}(X),\mathbb{G}_{a})\simeq
Ext^{1}_{naive}(J_{0}(X),\mathbb{G}_{m})\simeq 0.$

Proof of the lemma. Suppose that we are given an extension of commutative pro-
algebraic groups
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
By the universal property of $J_{0}(X)$, construction a section $J_{0}(X)\to
H$ is equivalent to give a point $s\in H(K(X))$, which projects to the
universal element in $J_{0}(X)(K(X))$. But we have an exact sequence in $fppf$
cohomology of $K(X)$ $H(K(X))\to J_{0}(X)(K(X))\to
H^{1}_{fppf}(K(X),\mathbb{G}_{a})=0$, showing that $H(K(X))\to J_{0}(X)(K(X))$
is surjective. Therefore the short exact sequence splits showing that
$Ext^{1}_{naive}(J_{0}(X),\mathbb{G}_{a})=0$. The proof for $\mathbb{G}_{m}$
is similar. $\Box$

For a general commutative affine group scheme $G$ over $k$, we can decompose
$G$ as a direct sum $G^{u}\oplus G^{m}$, of a unipotent part and a
multiplicative part. We can moreover use compositions series on each
components $G^{u}$ and $G^{m}$ to reduce to the case where $G$ is either
$\mathbb{G}_{a}$ or $\mathbb{G}_{m}$, or a subgroup of $\mathbb{G}_{a}$ or
$\mathbb{G}_{m}$ with short exact sequences

$0\to G\to\mathbb{G}_{m}\to\mathbb{G}_{m}\to 0\qquad 0\to
G\to\mathbb{G}_{a}\to\mathbb{G}_{a}\to 0.$

We then consider the action of $\psi_{G}^{1}$ on the associated long exact
sequences, and lemma 3.8 shows that indeed $\psi_{G}^{1}$ is bijective for any
$G$.

We now turn to the case where $i+d>1$ and $G$ is unipotent, which is proven in
a very similar fashion than the case $i=1$ and $d=0$. To begin with, we prove
that $\psi_{G}^{i}$ is an isomorphism for $G=\mathbb{G}_{a}$. Indeed, when
$i=0$ this is obvious by the definition of $J_{x}(X)$. For $i>0$ the domain
and codomain of $\psi_{\mathbb{G}_{a}}^{i}$ vanishes. Indeed,
$H^{d+i}_{x}(X,\mathbb{G}_{a})\simeq H^{d+i}_{x}(X,\mathcal{O})\simeq 0$ for
$i>0$ by local duality for coherent sheaves. On the other hand,
$Ext^{i}_{naive}(J_{x}(X),\mathbb{G}_{a})=0$ by the following lemma.

######  Lemma 3.9.

With the notation above and for all $i>0$

$\psi_{\mathbb{G}_{a}}^{i}:Ext^{i}_{naive}(J_{x}(X),\mathbb{G}_{a})\simeq 0.$

Proof of the lemma. Suppose that we are given an extension of commutative pro-
algebraic groups
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J_{x}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
By the universal property of $J_{0}(X)$, construction a section $J_{0}(X)\to
H$ is equivalent to give an element $s\in H^{d}_{x}(X,H)$, which projects to
the universal element in $H^{d}_{x}(X,J_{x}(X))$. But we have an exact
sequence in $fppf$ cohomology of $X$ local at $x$ $H^{d}_{x}(X,H)\to
H^{d}_{x}(X,J_{x}(X))\to H^{d+1}_{x}(X,\mathbb{G}_{a})=0$, showing that
$H^{d}_{x}(X,H)\to H^{d}_{x}(X,J_{x}(X))$ is surjective. Therefore the short
exact sequence splits showing that
$Ext^{1}_{naive}(J_{x}(X),\mathbb{G}_{a})=0$. Moreover, it is well known, and
can be proven using Dieudonné theory, that
$Ext^{i}_{naive}(A,\mathbb{G}_{a})=0$ for all $i>1$ and all commutative
unipotent pro-algebraic group $A$ over $k$ (see [DG70] proposition V-1 5.1 and
5.2). $\Box$

We thus know that $\psi_{\mathbb{G}_{a}}^{i}$ is an isomorphism for all $i$.
It remains to treat the case of a general commutative unipotent group $G$. Any
such group possesses a decomposition series whose layers $G_{\alpha}$ can be
realized in a short exact sequence $0\to
G_{\alpha}\to\mathbb{G}_{a}\to\mathbb{G}_{a}$. The associated long exact
sequences and the lemma easily implies that $\psi_{G}^{i}$ is also an
isomorphism for all $i$ (note that the domains and codomains of $\psi_{G}^{i}$
can be non-zero for $i=1$). $\Box$

######  Remark 3.10.

  1. (1)

_The previous proposition provides a close relation between our algebraic
homology $\mathbb{H}^{alg}(X)$ and the original Grothendieck’s construction
$J_{*}(X)$ of. The former is a truncated version of the later, but both can be
used in order to express the fppf cohomology of $X$ with coefficients in a
commutative group scheme. _

  2. (2)

_The extra conditions on $G$, being affine for $i>0$ and unipotent for $i>1$,
does not appear in Grothendieck’s original construction. This is because his
construction is originaly made using the Zariski topology instead othe fppf
topology, and higher Zariski cohomology groups with coefficients in group
schemes vanish outside of the unipotent context._

##  4\. Partial generalization to more general schemes

In this section we partially extend the results of the previous sections to
the more general case of schemes over arbitrary bases. For this, we restrict
ourselves to the unipotent part of algebraic homology and use the formalism of
commutative groups in affine stacks (in the sense of [Toe06]) as replacement
for (pro) commutative group stacks. On the way, we develop the general
framework of _affine homology_ , the natural homology theory associated to
affine stacks.

We fix a base commutative ring $k$. We will work from now with the fpqc
topology on affine $k$-schemes, so $\mathbf{St}_{k}$ will denote the
$\infty$-category of hypercomplete fpqc-stacks on $\mathbf{Aff}_{k}$.

###  4.1. Affine homology

When $C=\mathbf{St}_{k}$ is the $\infty$-category of stacks over $k$,
$\mathbf{Ab}C$ is naturally equivalent to $\mathbf{AbSt}_{k}$ (already used
previously in this work, but in the setting of the fppf topology), and is
identified with the $\infty$-category of connective complexes of sheaves of
abelian groups on the site of affine $k$-schemes with the fppf topology.

######  Definition 4.1.

The $\infty$-category of _commutative affine group stacks over $k$_ is defined
to be the full sub-$\infty$-category of $\mathbf{AbSt}_{k}$ consisting of all
commutative group stacks $E$ such that for all $n\geq 0$ the underlying stack
$|E[n]|\in\mathbf{St}_{k}$ is an affine stack in the sense of [Toe06]. The
$\infty$-category of all commutative affine group stacks will be denoted by
$\mathbf{AbChAff}_{k}$.

We warn the reader that the notation $\mathbf{AbChAff}_{k}$ can be misleading.
Indeed, the $\infty$-category $\mathbf{ChAff}_{k}$ defines another
$\infty$-category $Ab(\mathbf{ChAff}_{k})$, of abelian group objects in
$\mathbf{ChAff}_{k}$. Recall that this is formally defined as the full
sub-$\infty$-category of $Fun^{\infty}(T^{op},\mathbf{ChAff}_{k})$, consisting
of all $\infty$-functors sending finite sums in $T$ to finite products in $C$,
and where $T$ is the algebraic theory of abelian group (i.e. the opposite of
the category of free abelian group of finite rank). The $\infty$-category
$\mathbf{AbChAff}_{k}$ is smaller than $Ab(\mathbf{ChAff}_{k})$. Indeed, any
affine commutative group scheme $G$ defines an object in
$Ab(\mathbf{ChAff}_{k})$. However, we will see below (see proposition 4.2)
that $\mathbb{G}_{m}$ is not an object in $\mathbf{AbChAff}_{k}$. We thus have
canonical fully faithful $\infty$-functors
$\mathbf{AbChAff}_{k}\hookrightarrow
Ab(\mathbf{ChAff}_{k})\hookrightarrow\mathbf{AbSt}_{k}$, but these three
$\infty$-categories are all distinct.

Note also that because affine stacks are stable by small limits in
$\mathbf{St}_{k}$, $\mathbf{AbChAff}_{k}\subset\mathbf{AbSt}_{k}$ is clearly
stable by small limits.

An important first observation is the following statement, which identifies
the $\infty$-category $\mathbf{AbChAff}_{k}$ when $k$ is a field. It is the
commutative group version of the characterization of affine stacks by means of
their homotopy sheaves which is proven in [Toe06].

######  Propositon 4.2.

If $k$ is a field, then the natural inclusion

$\mathbf{AbChAff}_{k}\subset\mathbf{AbSt}_{k}$

identifies $\mathbf{AbChAff}_{k}$ with the sub-$\infty$-category consisting of
all $E$ such that for all $i\geq 0$ the sheaf $\pi_{i}(E)$ is representable by
a commutative and unipotent affine group scheme over $k$.

Proof. We start by the following lemma.

######  Lemma 4.3.

Let $E$ be an affine stack over $k$ which is endowed with a group structure.
Then, for all $i>0$ the sheaves $\pi_{i}(E)$ are representable by affine
unipotent group schemes.

Proof of the lemma. This is already contained in the proof of [Toe06, Cor.
3.2.7] which we reproduce here for simplicity. Let $A=\mathcal{O}(E)$ be the
commutative cosimplicial algebra corresponding to $E$, which is endowed with a
$H_{\infty}$-Hopf algebra structure in the sense of [Toe06, Def. 3.1.4], or in
other words which is a cogroup in the $\infty$-category of commutative
cosimplicial $k$-algebras. As a consequence, $H^{*}(A)$ is a graded
commutative Hopf $k$-algebra and therefore each $H^{0}(A)$-module $H^{i}(A)$
is free (because it corresponds to an equivariant quasi-coherent sheaf on the
group scheme $\mathbf{Spec}\,H^{0}(A)$), and in particular $A$ is flat as a
complex of $H^{0}(A)$-modules. Let $G=\mathbf{Spec}\,H^{0}(A)$ and $p:E\to G$
the natural projection induced by the canonical morphism $H^{0}(A)\to A$. The
fiber of $p$ is $\mathbf{Spec}\,A^{\prime}$, where
$A^{\prime}=A\otimes_{H^{0}(A)}k$, which, by the flatness of $A$ over
$H^{0}(A)$, is reduced $H^{0}(A^{\prime})\simeq k$. By [Toe06, Thm. 2.4.5] the
corresponding affine stack $E^{\prime}=\mathbf{Spec}\,A^{\prime}$ is pointed
and connected, and thus all the sheaves $\pi_{i}(E^{\prime})$ are
representable by unipotent group schemes. The long exact sequence in homotopy
therefore implies the lemma, as $\pi_{i}(E)\simeq\pi_{i}(E^{\prime})$ for all
$i>0$. $\Box$

The lemma 4.3 as the following important consequence: for any object
$E\in\mathbf{AbChAff}_{k}$, the homotopy groups $\pi_{i}(E)$ are all
representable by unipotent group schemes, for any $i\geq 0$. Indeed, by the
lemma we already know that this is the case for $i>0$. We now consider $E[1]$,
and by definition we know that $E[1]$ is again a group object in affine
stacks. In particular the lemma implies that $\pi_{1}(E[1])\simeq\pi_{0}(E)$
is representable by a unipotent group scheme.

Conversely, let $E\in\mathbf{AbSt}_{k}$ be an object such that the sheaves
$\pi_{i}(E)$ are representable by unipotent group schemes. We have to show
that each stack $|E[n]|\in\mathbf{St}_{k}$ is affine. But, the classifying
stack $B(|E[n]|)$ is affine because of [Toe06, Thm. 2.4.1], and thus so is
$\Omega_{*}B(|E[n]|)\simeq|E[n]|$ as affine stacks are stable by limits.
$\Box$

As a direct consequence of the proposition 4.2 we have the following
corollary, which is the abelian analogue of the characterizations of affine
stacks as being limits of stacks of the form $K(\mathbb{G}_{a},n)$.

######  Corollary 4.4.

We continue to assume that $k$ is a field. The sub-$\infty$-category
$\mathbf{AbChAff}_{k}\subset\mathbf{AbSt}_{k}$ is the smallest
sub-$\infty$-category which is stable by small limits and contains all the
objects $\mathbb{G}_{a}[n]$ for all $n\geq 0$.

Proof of the corollary. Let $C\subset\mathbf{AbSt}_{k}$ be the smallest
sub-$\infty$-category containing $\mathbb{G}_{a}[n]$ and stable by small
limits. Clearly $C$ is contained in $\mathbf{AbChAff}_{k}$, and we know that
$\mathbf{AbChAff}_{k}$ is stable by limits and that it contains all the
objects $\mathbb{G}_{a}[n]$. The converse is a direct consequence of Postnikov
decomposition and the proposition. By [Toe20, Prop. B1], Postnikov towers
always converge for the fpqc topology (see also [MR22] for a more general
statement), and thus any $E\in\mathbf{AbChAff}_{k}$ can be written as
$E\simeq\lim_{n}\tau_{\leq n}E$. Moreover, the proposition implies that each
$\tau_{n}(E)$ belongs to $\mathbf{AbChAff}_{k}$. Finally, we proceed by
induction on $n$ to show that each $\tau_{n}(E)$ belongs to $C$, by means of
the cartesian squares

$\textstyle{\tau_{n}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tau_{n-1}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{n}(E)[n+1].}$

As $\pi_{n}(E)$ is a unipotent group scheme by proposition 4.2, it is itself
contains in $C$, and thus so is $\pi_{n}(E)[n+1]$. $\Box$

We do not know if the corollary 4.2 remains valid over an arbitrary base ring
$k$. We thus restrict our notion of commutative affine group stacks to a
notion of _affine homology types_ , by considering all the objects generated
by limits of $\mathbb{G}_{a}[n]$.

######  Definition 4.5.

The _$\infty$ -category of affine homology types over $k$_ is the smallest
sub-$\infty$-category of $\mathbf{AbSt}_{k}$ which is stable by small limits
and contains the objects $\mathbb{G}_{a}[n]$ for all $n\geq 0$. It is denoted
by

$\widehat{\mathbf{AbChAff}_{k}}\subset\mathbf{AbSt}_{k}.$

As a first observation, all the objects $\mathbb{G}_{a}[n]$ are commutative
affine group stacks, and thus we clearly have
$\widehat{\mathbf{AbChAff}_{k}}\subset\mathbf{AbChAff}_{k}$. However, outside
of the case of a base field we do not know if this inclusion is strict. In any
case, we will see below that $\widehat{\mathbf{AbChAff}_{k}}$ is better
behaved than $\mathbf{AbChAff}_{k}$, thanks to the following proposition. We
do not know if the similar statement holds true for $\mathbf{AbChAff}_{k}$.

######  Propositon 4.6.

The forgetful $\infty$-functor of $\infty$-categories
$\widehat{\mathbf{AbChAff}_{k}}\hookrightarrow\mathbf{ChAff}_{k}$, sending $E$
to $|E|$, possesses a left adjoint.

Proof. We consider the inclusion of opposite $\infty$-categories. We know that
$(\mathbf{ChAff}_{k})^{op}$ is equivalent to the $\infty$-category associated
with the combinatorial model category of commutative cosimplicial $k$-algebras
by means of the $\mathbf{Spec}$ functor (see [Toe06]). Therefore,
$(\mathbf{ChAff}_{k})^{op}$ is a presentable $\infty$-category. Moreover,
$(\widehat{\mathbf{AbChAff}_{k}})^{op}$ is generated by colimits by the small
set of objects $\mathbb{G}_{a}[n]$. Moreover, when considered as objects in
$(\widehat{\mathbf{AbChAff}_{k}})^{op}$, the objects $\mathbb{G}_{a}[n]$ are
small, or more precisely $\mathbb{G}_{a}[n]$ are cosmall objects in
$\mathbf{AbChAff}_{k}$ as this is implied by the following lemma.

######  Lemma 4.7.

Let $C$ be a complete $\infty$-category and $Ab(C)$ be the $\infty$-category
of abelian group objects in $C$. Let $G\in Ab(C)$ be an object satisfying the
following two conditions.

  1. (1)

The underlying object $G$ is cocompact in $C$: $Map_{C}(-,G)$ sends filtered
limits to filtered colimits.

  2. (2)

The object $G$ is truncated in $C$: there is an $n$ such that $Map(E,C)$ is
$n$-truncated for all $E\in c$.

Then, $G$ is cocompact as an object in $Ab(C)$.

Proof of the lemma. The mapping spaces between two abelian group objects $H$
and $G$ in $C$ can be computed as a certain homotopy limit over $\Delta$ of
the form

$\mathbf{Map}_{Ab(C)}(H,G)\simeq\lim_{i\in\Delta}\mathbf{Map}_{C}(H^{n(i)},G^{m(i)}),$

where $i\mapsto n(i)$ and $i\mapsto m(i)$ are certain functions. The existence
of such a limit decomposition can be seen for instance using MacLane
resolutions of abelian group objects as presented in [Bre70, §2]. Because $G$
is $n$-truncated this homotopy limit reduces to a finite homotopy limit, which
in turn will commute with filtered colimit. The result follows easily as all
objects $G^{m(i)}$ are cocompact in $C$. $\Box$

The lemma implies that $(\widehat{\mathbf{AbChAff}_{k}})^{op}$ is a
presentable $\infty$-category It is therefore a formal consequence of Freyd
representability theorem (see [NRS20, Thm. 4.1.3]) that the forgetful
$\infty$-functor
$(\mathbf{AbChAff}_{k})^{op}\rightarrow(\mathbf{ChAff}_{k})^{op}$, which
commutes with small colimits, admits a right adjoint. $\Box$

We warn the reader that the forgetful $\infty$-functor
$\mathbf{AbSt}_{k}\rightarrow\mathbf{St}_{k}$ also possesses a left adjoint,
given by $F\mapsto\mathbb{Z}\otimes F$, but that the square

$\textstyle{\mathbf{ChAff}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{St}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbf{AbChAff}_{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{AbSt}_{k}}$

of course does not commute, even when $k$ is a field. Indeed, the construction
$\mathbf{ChAff}_{k}\to\widehat{\mathbf{AbChAff}_{k}}$ involves considering
some free abelian group objects in $\mathbf{ChAff}_{k}$, and thus involves
certain colimits, which are computed in $\mathbf{ChAff}_{k}$ (and the
inclusion $\mathbf{ChAff}_{k}\subset\mathbf{AbSt}_{k}$ does not preserve
colimits).

We also remind that there are set theory issues in the situation as the site
of affine schemes $Aff_{k}$ is not a small site. As a result, the inclusion
$\mathbf{ChAff}_{k}\subset\mathbf{St}_{k}$ does not possess a left adjoint as
objects in $\mathbf{St}_{k}$ might not be small. However, we can restrict
ourselves to small stacks $\mathbf{St}_{k}^{sm}\subset\mathbf{St}_{k}$, which
are all objects that can be written as small colimits of affine schemes. By
definition affine stacks are small, as they are spectrum of small commutative
cosimplicial $k$-algebras, so we have an inclusion
$\mathbf{ChAff}_{k}\hookrightarrow\mathbf{St}_{k}^{sm}$. This last inclusion
does have a left adjoint, called the _affinization_ and denote by
$F\mapsto(F\otimes k)^{u}$. It is explicitely given by considering
$\mathcal{O}(F)$, the commutative cosimplicial $k$-algebra of functions on
$F$, which is small as $F$ is assumed to be small, and setting $(F\otimes
k)^{u}\simeq\mathbf{Spec}\,\mathcal{O}(F)$.

As a result, and including our proposition 4.6, we get an $\infty$-functor

$\mathbb{H}^{u}(-):\mathbf{St}_{k}^{sm}\longrightarrow\widehat{\mathbf{AbChAff}_{k}},$

which is a left adjoint to the forgetful $\infty$-functor
$|-|:\widehat{\mathbf{AbChAff}_{k}}\rightarrow\mathbf{St}_{k}^{sm}$.

######  Definition 4.8.

For a small stack $X\in\mathbf{St}_{k}$, its _affine homology_ is defined as
$\mathbb{H}^{u}(X)$.

Affine homology can be hard to compute. We give here the simplest example
namely the affine homology of a constant stack with finite $1$-connected
fibers. Let $K$ be a $1$-connected finite simplicial set, which is considered
as a constant stack $\underline{K}\in\mathbf{St}_{k}$. As such it is a colimit
of the punctual stack $*$ taken over the category of simplices in $K$.
Therefore, its affine homology is given as

$\mathbb{H}^{u}(\underline{K})\simeq co\lim_{\Delta(K)}\mathbb{H}^{u}(*),$

and we are thus reduced to compute the affine homology of the point. For this,
we recall the existence of the commutative group scheme $\mathbb{H}$ of
[Toe20], which can be defined as the spectrum of the Hopf algebra of integral
valued polynomials. Equivalently, $\mathbb{H}$ sits inside the group scheme of
big Witt vectors $\mathbb{W}$ as the sub-functor of simultaneously fixed
points by all Frobenius $F_{p}$. The group scheme $\mathbb{H}$ is the additive
part of ring scheme structure, for which the multiplication is induced by the
multiplication of Witt vectors.

We claim that $\mathbb{H}^{u}(*)\simeq\mathbb{H}$, considered as a commutative
group for the additive structure. First of all, as shown in [Toe20] all the
$K(\mathbb{H},n)$ are abelian group objects in affine stacks, and thus
$\mathbb{H}$ is indeed an object of $\mathbf{AbChAff}_{k}$. Using the
presentation as fixed points of Frobenius on $\mathbb{W}$, it can be checked
that $\mathbb{H}$ lies furthermore in $\widehat{\mathbf{AbChAff}_{k}}$ and
thus is an affine homology type. In [Toe20] is also known that when $n>0$
$K(\mathbb{H},n)$ is the affinization of the space $K(\mathbb{Z},n)$, and thus
the natural morphism $K(\mathbb{Z},n)\to K(\mathbb{H},n)$, which is a morphism
of abelian group objects induced by the canonical map
$\mathbb{Z}\to\mathbb{H}$, makes $K(\mathbb{H},n)$ into the affinization of
$K(\mathbb{Z},n)$. Indeed, for any abelian group object $E$ in
$\mathbf{ChAff}_{k}$, the mapping space
$Map_{\mathbf{AbSt}_{k}}(K(\mathbb{Z},n),E)$ can be computed by a certain
homotopy end

$Map_{\mathbf{AbSt}_{k}}(K(\mathbb{Z},n),E)\simeq hoend\left((i,j)\mapsto
Map_{\mathbf{St}_{k}}(K(\mathbb{Z},n)^{i},E^{j})\right),$

where $i$ and $j$ lives in the category of free abelian groups of finite rank.
Similarly, we have

$Map_{\mathbf{AbChAff}_{k}}(K(\mathbb{H},n),E)\simeq hoend\left((i,j)\mapsto
Map_{\mathbf{ChAff}_{k}}(K(\mathbb{H},n)^{i},E^{j})\right).$

The affinization of $K(\mathbb{Z},n)^{i}\simeq K(\mathbb{Z}^{i},n)$ being
$K(\mathbb{H}^{i},n)$ (see [Toe20]), we deduce easily that the natural
morphism

$Map_{\mathbf{AbChAff}_{k}}(K(\mathbb{H},n),E)\longrightarrow
Map_{\mathbf{AbSt}_{k}}(K(\mathbb{Z},n),E)\simeq\Omega_{*}^{n}(E(k))$

is an equivalence for any $E\in\mathbf{AbChAff}_{k}$. In other words, the
reduced affine homology of $S^{n}$ is $K(\mathbb{H},n)$, for any $n>0$. But we
then have $Map_{\mathbf{AbSt}_{k}}(\mathbb{H},E)\simeq
Map_{\mathbf{AbSt}_{k}}(K(\mathbb{H},1),K(E,1))\simeq\Omega_{*}(K(E,1)(k))\simeq
E(k)$, and thus $\mathbb{H}$ is indeed the affine homology of $*$ as wanted.

To put things in a more concise manner, we can write, with the notations and
conditions above

$\mathbb{H}^{u}(\underline{K})\simeq K\otimes\mathbb{H},$

where the tensored structure is computed in the $\infty$-category
$\mathbf{AbChAff}_{k}$ of commutative affine group stacks over $k$.

For our purpose, two general properties of affine homology are of some
importance, namely base change and filtration by dimension.

Base change. As a start, affine homology is base sensitive, and should rather
be written as $\mathbb{H}^{u}(X/k)$ in order to mention the base ring. Let
$k\to k^{\prime}$ be a morphism of commutative rings, and
$X\in\mathbf{St}_{k}^{sm}$ a small stack over $k$. We denote by
$\otimes_{k}k^{\prime}$ the base change from stacks over $k$ to stacks over
$k^{\prime}$, and similarly for affine stacks. The base change commutes with
limits and thus induces a well defined base change
$\mathbf{AbChAff}_{k}\to\mathbf{AbChAff}_{k^{\prime}}$. As this sends
$\mathbb{G}_{a}$ to $\mathbb{G}_{a}$, base change restricts to an
$\infty$-functor

$\otimes_{k}k^{\prime}:\widehat{\mathbf{AbChAff}_{k}}\to\widehat{\mathbf{AbChAff}_{k^{\prime}}}.$

Therefore, by universal property we have a canonical morphism of affine
homology types over $k^{\prime}$

$\mathbb{H}^{u}(X\otimes_{k}k^{\prime}/k^{\prime})\to\mathbb{H}^{u}(X/k)\otimes_{k}k^{\prime}.$

This morphism is not an equivalence in general, but we claim it is so when
$k^{\prime}$ is finite and of local complete intersection over $k$. Indeed, by
the very definition of affine homology this would follow formally from the
fact that the direct image $\infty$-functor
$R_{k^{\prime}/k}:\mathbf{AbSt}_{k^{\prime}}\to\mathbf{AbSt}_{k}$, right
adjoint to the base change, preserves affine homology types. As these are
generated by limits from the $\mathbb{G}_{a}[n]$, it is enough to show that
$R_{k^{\prime}/k}(\mathbb{G}_{a}[n])$ is an affine homology type over $k$. But
$R_{k^{\prime}/k}(\mathbb{G}_{a}[n])$ is nothing else that
$|p_{*}(\mathbb{G}_{a})[n]|$, where
$p:\mathbf{Spec}\,k^{\prime}\to\mathbf{Spec}\,k$. Moreover, by assumption on
$k^{\prime}/k$ the stack $p_{*}(\mathbb{G}_{a})[n]$ is a linear stack over $k$
associated to the perfect complex $E:=(k^{\prime})^{\vee}[-n]$, which is the
dual of $k^{\prime}[n]$. More explicitly, $p_{*}(\mathbb{G}_{a})[n]$ can be
written as the spectrum of the cosimplicial commutative $k$-algebras
$Sym_{k}(E)$, with $E$ a perfect complex of $k$-modules. These linear stacks,
as objects in $\mathbf{AbSt}_{k}$ can be decomposed by considering a
projective resolution of $E$, and are easily seen to be expressible by limits
of stacks of the form $V[n]$ where $V$ is a vector bundle on
$\mathbf{Spec}\,k$, which are retracts of $\mathbb{G}_{a}^{r}[n]$ and thus are
affine homology types over $k$.

An important cases of base change is when $k$ is regular, and $k\to k(p)$ is
the quotient corresponding to a closed point $p\in\mathbf{Spec}\,k$. In this
case base change tells us that $\mathbb{H}^{u}(X/k)$ specializes to
$\mathbb{H}^{u}(X_{p}/k(p))$ when pull-backed to the point $p$.

Filtration by dimension. We let $X$ be a $k$-schemes separated and of finite
type over $k$. For simplicity we assume that $k$ is noetherian and $X$ is of
finite dimension $d$. We can proceed as in definition 3.1 and construct a
finite filtration on $\mathbb{H}^{u}(X/k)$

$\textstyle{0=F_{d+1}\mathbb{H}^{u}(X/k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{d}\mathbb{H}^{u}(X/k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\mathbb{H}^{u}(X/k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}\mathbb{H}^{u}(X/k)=\mathbb{H}^{u}(X/k).}$

The associated graded for this filtration, $Gr_{d}\mathbb{H}^{u}(X/k)$,
defined considering cofibers in $\widehat{\mathbf{AbChAff}_{k}}$, are affine
homology types over $k$ which are related to local cohomology in dimension $d$
as follows. We first introduce affine homology types of finite presentation,
as being the objects $E\in\widehat{\mathbf{AbChAff}_{k}}$ being obtained by
finite limits and retracts of objects of the form $\mathbb{G}_{a}[n]$. As we
have already seen in lemma 4.7 the objects $\mathbb{G}_{a}[n]$ are cocompact
in $\mathbf{AbChAff}_{k}$, and thus any affine homology type of finite
presentation are cocompact objects amoung affine homology types.

Let then $E$ be an affine homology type of finite presentation. We then have

$\mathbf{Map}_{\mathbf{AbChAff}_{k}}(Gr_{d}\mathbb{H}^{u}(X),E))\simeq\oplus_{x\in
X^{(d)}}\mathbb{H}^{*}_{x}(X,E).$

Because affine homology type of finite presentation generate the whole
$\infty$-category $\widehat{\mathbf{AbChAff}_{k}}$ by limits, we deduce that
we have we have a general formula in terms of local affine homology

$Gr_{d}\mathbb{H}^{u}(X)\simeq\prod_{x\in X^{(d)}}\mathbb{H}^{u}_{x}(X).$

###  4.2. Affine homology as a dg-module

We consider the object $\mathbb{G}_{a}\in\mathbf{AbSt}_{k}$ and the
corresponding $\mathbb{Z}$-linear dg-algebra of endomorphisms

$C_{\mathbb{G}_{a}}:=\mathbb{R}End(\mathbb{G}_{a}).$

We get this way the usual adjunction of $\infty$-categories

$M:(\mathbf{AbSt}_{k})^{op}\leftrightarrows C_{\mathbb{G}_{a}}-\mathbf{dg}:D,$

where $M(E):=\mathbb{R}Hom(E,\mathbb{G}_{a})$. As $\mathbb{G}_{a}[n]$
generates $\widehat{\mathbf{AbChAff}}_{k}$ by limits, and as
$\mathbb{G}_{a}[n]$ are cocompact objects, it is formal to see that for any
$E\in\widehat{\mathbf{AbChAff}}_{k}$ the adjunction morphism

$E\longrightarrow D(M(E))$

is an equivalence. As a result we get the following corollary.

######  Corollary 4.9.

The $\infty$-functor $M$ restricts to a full embedding

$M:(\widehat{\mathbf{AbChAff}}_{k})^{op}\longrightarrow(C_{\mathbb{G}_{a}}-\mathbf{dg})^{op}.$

Its essential image consists of all $C_{\mathbb{G}_{a}}$-dg-modules which are
colimits of $C_{\mathbb{G}_{a}}$.

The above corollary implies that for a small stack $X$ its affine homology
$\mathbb{H}^{u}(X/k)$ is characterized by
$\mathbb{R}\Gamma(X,\mathbb{G}_{a})$, as a dg-module over
$C_{\mathbb{G}_{a}}$. The dg-algebra $C_{\mathbb{G}_{a}}$ is however quite
complicated in general. It simplifies when $\mathbb{G}_{a}$ is restricted to
sites of perfect schemes (see [Bre81]). In particular, things tend to simplify
when the base ring $k$ is perfect (or at least perfectoid). Another
simplification consists of using the group of big Witt vectors $\mathbb{W}$
instead of the additive group $\mathbb{G}_{a}$, as for instance
$Ext^{1}(\mathbb{W},\mathbb{W})=0$, and again higher ext vanish when computed
over the site of perfect schemes. Moreover, $End(\mathbb{W})$ is the usual
Dieudonné ring so dg-modules over $\mathbb{R}End(\mathbb{W})$ are closed
related to usual Dieudonné modules. However, $\mathbb{W}$ is not a cocompact
object in $\widehat{\mathbf{AbChAff}}_{k}$, and therefore the corresponding
adjunction with dg-modules is not as well behaved and relates dg-modules over
$\mathbb{R}End(\mathbb{W})$ and certain pro-objects inside
$\mathbf{AbChAff}_{k}$.

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Bertrand Toën, IMT, CNRS, Université de Toulouse, Toulouse (France)
<EMAIL_ADDRESS>