# Stratifications of abelian categories

Giulian Wiggins  School of Mathematical and Physical Sciences
Macquarie University
Australia

###### Abstract.

This paper is a study of abelian categories that can be decomposed into
smaller abelian categories via iterated recollements - such a decomposition we
call a stratification. Examples include the categories of (equivariant)
perverse sheaves and $\varepsilon$-stratified categories (in particular
highest weight categories) in the sense of Brundan-Stroppel [BS18].

We give necessary and sufficient conditions for an abelian category with a
stratification to be equivalent to a category of finite dimensional modules of
a finite dimensional algebra - this generalizes the main result of Cipriani-
Woolf [CW22]. Furthermore, we give necessary and sufficient conditions for
such a category to be $\varepsilon$-stratified - this generalizes the
characterisation of highest weight categories given by Krause [Kra17].

###### Contents

  1. 1 Introduction
    1. 1.1 Outline and explanation of main results
  2. 2 Preliminaries
  3. 3 The intermediate-extension functor
  4. 4 Recollements with enough projectives/injectives
    1. 4.1 Projective covers
    2. 4.2 Ext-finiteness
    3. 4.3 Main results
  5. 5 Standard and costandard objects
  6. 6 Brundan and Stroppel’s $\varepsilon$-stratified categories
    1. 6.1 Highest weight categories

##  1\. Introduction

A recollement of abelian categories is a short exact sequence of abelian
categories

${{0}}$${{{\mathcal{A}}_{Z}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{U}}}$${{0}}$$\scriptstyle{i_{*}}$$\scriptstyle{j^{*}}$

in which ${\mathcal{A}}_{Z}$ is a Serre subcategory of ${\mathcal{A}}$ (with
Serre quotient ${\mathcal{A}}_{U}$), $i_{*}$ has both a left and right
adjoint, and $j^{*}$ has fully-faithful left and right adjoints. We usually
denote such a recollement of abelian categories by the diagram

${{{\mathcal{A}}_{Z}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{U}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

where $(i^{*},i_{*},i^{!})$ and $(j_{!},j^{*},j_{*})$ are adjoint triples.

This definition is motivated by recollements of triangulated categories as
defined by Beilinson, Bernstein and Deligne [BBD82] to generalize
Grothendieck’s six functors relating the constructible derived category,
${\mathcal{D}}(X,\Bbbk)$, of sheaves on a variety $X$ (with coefficients in a
field $\Bbbk$) and the constructible derived categories,
${\mathcal{D}}(Z,\Bbbk)$ and ${\mathcal{D}}(U,\Bbbk)$, of sheaves on a closed
subvariety $Z\subset X$ and open complement $U:=X\backslash Z$ i.e. the
situation:

${{{\mathcal{D}}(Z,\Bbbk)}}$${{\mathcal{D}}(X,\Bbbk)}$${{{\mathcal{D}}(U,\Bbbk)}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

Here, $i:Z\hookrightarrow X$ is the closed embedding and $j:U\hookrightarrow
X$ is the complimentary open embedding. Note that if ${\mathcal{S}}h(X,\Bbbk)$
is the category of sheaves on $X$, and
${{\mathcal{H}}}^{i}:{\mathcal{D}}(X,\Bbbk)\rightarrow{\mathcal{S}}h(X,\Bbbk)$
is the $i$-th cohomology functor, then the following is an example of a
recollement of abelian categories.

${{{\mathcal{S}}h(Z,\Bbbk)}}$${{{\mathcal{S}}h(X,\Bbbk)}}$${{{\mathcal{S}}h(U,\Bbbk)}}$$\scriptstyle{i_{*}}$$\scriptstyle{{{\mathcal{H}}}^{0}i^{*}}$$\scriptstyle{{{\mathcal{H}}}^{0}i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{{{\mathcal{H}}}^{0}j_{!}}$$\scriptstyle{{{\mathcal{H}}}^{0}j_{*}}$

More generally, given a recollement of triangulated categories with
$t$-structure, there is a canonical recollement of abelian categories on the
hearts of the $t$-structure [BBD82, Proposition 1.4.16] defined using the
zero-th cohomology functor arising from the $t$-structure.

In representation theory, recollements of abelian categories arise from
modules of algebras $A$ with an idempotent $e$. In this situation there is a
recollement

${{0}}$${{A/AeA\text{-}\mathrm{Mod}}}$${A\text{-}\mathrm{Mod}}$${{eAe\text{-}\mathrm{Mod}}}$${{0}}$$\scriptstyle{i_{*}}$$\scriptstyle{j^{*}}$

where $i_{*}:A/AeA\text{-}\mathrm{Mod}\rightarrow A\text{-}\mathrm{Mod}$ is
the restriction functor, and $j^{*}:A\text{-}\mathrm{Mod}\rightarrow
eAe\text{-}\mathrm{Mod}$ is the functor sending an $A$-module $M$ to the
$eAe$-module $eM$.

In applications, recollements often arise as part of an iterated collection of
recollements, giving a kind of filtration of a larger category by smaller
categories. Consider, for example, the category, $P_{\Lambda}(X,\Bbbk)$, of
perverse sheaves that are constructible with respect to a stratification,
$X:=\bigcup_{\lambda\in\Lambda}X_{\lambda}$, of a variety $X$. For each strata
$X_{\lambda}$, there is a recollement

${{0}}$${{P_{\Lambda}(\overline{X_{\lambda}}\setminus
X_{\lambda},\Bbbk)}}$${{P_{\Lambda}(\overline{X_{\lambda}},\Bbbk)}}$${{\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)}}$${{0}}$

where $\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)$ is the category of local
systems on $X_{\lambda}$ of finite type. More generally, if
$\Lambda^{\prime}\subset\Lambda$ is such that
$X_{\Lambda^{\prime}}:=\bigcup_{\lambda\in\Lambda^{\prime}}X_{\lambda}$ is a
closed subvariety of $X$, and $X_{\lambda}$ is open in $X_{\Lambda^{\prime}}$,
then the following is a recollement of abelian categories.

${{0}}$${{P_{\Lambda}(X_{\Lambda^{\prime}\setminus\\{\lambda\\}},\Bbbk)}}$${{P_{\Lambda}(X_{\Lambda^{\prime}},\Bbbk)}}$${{\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)}}$${{0}}$

Similar iterations of recollement occur in highest weight categories (as
defined in [CPS88]). Indeed, let ${\mathcal{A}}$ be a highest weight category
with respect to a poset $\Lambda$, and let
$\\{\Delta_{\lambda}~{}|~{}\lambda\in\Lambda\\}$ be the collection of standard
objects in ${\mathcal{A}}$. For each lower111A subposet
$\Lambda^{\prime}\subset\Lambda$ is lower if for any $\lambda,\mu\in\Lambda$,
$\mu\leq\lambda$ and $\lambda\in\Lambda^{\prime}$ implies
$\mu\in\Lambda^{\prime}$. subposet $\Lambda^{\prime}\subset\Lambda$, let
${\mathcal{A}}_{\Lambda^{\prime}}$ be the Serre subcategory generated by
objects $\\{\Delta_{\lambda}~{}|~{}\lambda\in\Lambda^{\prime}\\}$. For each
lower subpost $\Lambda^{\prime}\subset\Lambda$ and maximal
$\lambda\in\Lambda^{\prime}$ there is a recollement of abelian categories

${{0}}$${{{\mathcal{A}}_{\Lambda^{\prime}\setminus\\{\lambda\\}}}}$${{{\mathcal{A}}_{\Lambda^{\prime}}}}$${{\operatorname{End}_{{\mathcal{A}}}(\Delta_{\lambda})^{op}\text{-}\mathrm{Mod}}}$${{0}}$$\scriptstyle{j^{*}}$

where
$j^{*}:=\operatorname{Hom}(\Delta_{\lambda},-):{\mathcal{A}}_{\Lambda^{\prime}}\rightarrow\operatorname{End}_{{\mathcal{A}}}(\Delta_{\lambda})^{op}\text{-}\mathrm{Mod}$.

These examples are unified by the following definition.

######  Definition 2.4.

A stratification of an abelian category ${\mathcal{A}}$ by a non-empty poset
$\Lambda$ consists of the following data

  1. (i)

For each lower subposet $\Lambda^{\prime}\subset\Lambda$, define a Serre
subcategory, ${\mathcal{A}}_{\Lambda^{\prime}}$, of ${\mathcal{A}}$.

  2. (ii)

For each $\lambda\in\Lambda$, define an abelian category
${\mathcal{A}}_{\lambda}$. These we call strata categories.

This data must satisfy the conditions

  1. (1)

${\mathcal{A}}_{\emptyset}=0$ and ${\mathcal{A}}_{\Lambda}={\mathcal{A}}$.

  2. (2)

For each pair of lower subposets
$\Lambda_{1}^{\prime}\subset\Lambda_{2}^{\prime}\subset\Lambda$, there are
inclusions of Serre categories
${\mathcal{A}}_{\Lambda_{1}^{\prime}}\hookrightarrow{\mathcal{A}}_{\Lambda_{2}^{\prime}}$.

  3. (3)

For each lower subposet $\Lambda^{\prime}\subset\Lambda$, and maximal
$\lambda\in\Lambda^{\prime}$ there is a recollement

${{0}}$${{{\mathcal{A}}_{\Lambda^{\prime}\setminus\\{\lambda\\}}}}$${{\mathcal{A}}_{\Lambda^{\prime}}}$${{{\mathcal{A}}_{\lambda}}}$${{0}}$

For example, the category $P_{\Lambda}(X,\Bbbk)$ has a stratification by the
poset $\Lambda$ with closure order: $\lambda\leq\mu$ if
$X_{\lambda}\subset\overline{X_{\mu}}$. Further examples include equivariant
perverse sheaves, and $\varepsilon$-stratified cateories in the sense of
Brundan and Stroppel [BS18].

This paper is inspired by a result of Cipriani and Woolf [CW22, Corollary 5.2]
that says that a category of perverse sheaves (with coefficients in a field)
on a space stratified by finitely many strata is finite222For any field
$\Bbbk$, a $\Bbbk$-linear abelian category ${\mathcal{A}}$ is finite (over
$\Bbbk$) if ${\mathcal{A}}$ is equivalent to a category of finite dimensional
modules of a finite dimensional $\Bbbk$-algebra. if and only if the same is
true for each category of finite type local systems on each stratum. We extend
this result by showing that an abelian category with a stratification by a
finite poset is finite if and only if the same if true of all strata
categories (Corollary 4.10). Moreover, we give new necessary and sufficient
conditions for a finite abelian category to be $\varepsilon$-stratified
(Theorem 6.4). This result specializes to give necessary and sufficient
conditions for a finite abelian category to be standardly stratified in the
sense of Cline, Parshall, Scott [CPS96], and further specialises to recover a
characterisation of highest weight categories due to Krause [Kra17].

This article revises and extends part of the author’s PhD thesis [Wig23]. This
paper has benefited from discussions with Oded Yacobi, Kevin Coulembier, and
feedback from a referee of [Wig23]. This work was supported by Australian
Research Council grant DP190102432.

###  1.1. Outline and explanation of main results

Let ${\mathcal{A}}$ be an abelian category with a stratification by a poset
$\Lambda$. For each $\lambda\in\Lambda$, define the Serre quotient functor
$j^{\lambda}:{\mathcal{A}}_{\\{\mu\in\Lambda~{}|~{}\mu\leq\lambda\\}}\rightarrow{\mathcal{A}}_{\lambda}$,
and let
$j_{!}^{\lambda}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}_{\\{\mu\in\Lambda~{}|~{}\mu\leq\lambda\\}}$
and
$j_{*}^{\lambda}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}_{\\{\mu\in\Lambda~{}|~{}\mu\leq\lambda\\}}$
be the left and right adjoints of $j^{\lambda}$. By a slight abuse of
notation, write
$j_{!}^{\lambda}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ and
$j_{*}^{\lambda}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ for the
functors obtained by postcomposing with the inclusion functor
${\mathcal{A}}_{\\{\mu\in\Lambda~{}|~{}\mu\leq\lambda\\}}\hookrightarrow{\mathcal{A}}$.

In Section 2 we recall some basic features of recollements and
stratifications, and give examples of stratifications of abelian categories.

In Section 3 we define, for each $\lambda\in\Lambda$, the intermediate-
extension functor
$j^{\lambda}_{!*}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$:

$j_{!*}^{\lambda}X:=\operatorname{im}(\overline{1_{X}}:j_{!}^{\lambda}X\rightarrow
j_{*}^{\lambda}X),$

where $\overline{1_{X}}$ is the morphism corresponding to the identity $1_{X}$
under the natural isomorphism

$\operatorname{Hom}_{{\mathcal{A}}}(j_{!}^{\lambda}X,j_{*}^{\lambda}X)\simeq\operatorname{Hom}_{{\mathcal{A}}_{\lambda}}(X,j^{\lambda}j_{*}^{\lambda}X)\simeq\operatorname{Hom}_{{\mathcal{A}}_{\lambda}}(X,X).$

Proposition 3.4 says that every simple object $L\in{\mathcal{A}}$ is of the
form $j^{\lambda}_{!*}L_{\lambda}$, for a unique (up to isomorphism) simple
object $L_{\lambda}\in{\mathcal{A}}_{\lambda}$ and unique $\lambda\in\Lambda$.
Proposition 3.6 says that if ${\mathcal{A}}$ is an abelian category with a
stratification by a finite poset, then every object in ${\mathcal{A}}$ has a
finite filtration by simple objects if and only if the same is true of all the
strata categories. The proofs here are almost identical to the standard proofs
of these results in the theory of perverse sheaves (see e.g. [Ach21, Chapter
3]).

In Section 4 we prove the following condition for a category to have enough
projectives.

######  Theorem 4.9.

Consider a recollement:

${{0}}$${{{\mathcal{A}}_{Z}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{U}}}$${{0}}$$\scriptstyle{i_{*}}$$\scriptstyle{j^{*}}$

Suppose ${\mathcal{A}}_{U}$ and ${\mathcal{A}}_{Z}$ have finitely many simple
objects and every object has a finite filtration by simple objects. Suppose
moreover that for any simple objects $A,B$ in ${\mathcal{A}}$,

$\dim_{\operatorname{End}_{{\mathcal{A}}}(B)}\operatorname{Ext}^{1}_{{\mathcal{A}}}(A,B)<\infty.$

Then ${\mathcal{A}}$ has enough projectives if and only if both
${\mathcal{A}}_{U}$ and ${\mathcal{A}}_{Z}$ have enough projectives.333Since
$\operatorname{End}_{{\mathcal{A}}}(B)$ is a division ring, any
$\operatorname{End}_{{\mathcal{A}}}(B)$-module is free. For an
$\operatorname{End}_{{\mathcal{A}}}(B)$-module $M$,
$\dim_{\operatorname{End}_{{\mathcal{A}}}(B)}M$ is the rank of $M$ as a free
$\operatorname{End}_{{\mathcal{A}}}(B)$-module.

Theorem 4.9 has the following important consequence.

######  Corollary 4.11.

For any field $\Bbbk$, a $\Bbbk$-linear abelian category with a stratification
by a finite poset is finite if and only if the same is true for all strata
categories.

To state the results in Sections 5 and 6, let ${\mathcal{A}}$ have enough
projectives and injectives, finitely many simple objects, and suppose every
object in ${\mathcal{A}}$ has finite length. Let $B$ be a set indexing the
simple objects in ${\mathcal{A}}$ (up to isomorphism) and write $L(b)$
(respectively $P(b)$, $I(b)$) for the simple (respectively projective
indecomposable, injective indecomposable) object corresponding to $b\in B$.

Define the stratification function

$\rho:B\rightarrow\Lambda$

that maps each $b\in B$ to the corresponding $\lambda\in\Lambda$ in which
$L(b)=j^{\lambda}_{!*}L_{\lambda}(b)$ for some simple object
$L_{\lambda}(b)\in{\mathcal{A}}_{\lambda}$. Write $P_{\lambda}(b)$ and
$I_{\lambda}(b)$ for the projective cover and injective envelope of
$L_{\lambda}(b)$ in ${\mathcal{A}}_{\lambda}$.

For $b\in B$ and $\lambda=\rho(b)$, define the standard and costandard objects

$\Delta(b):=j^{\lambda}_{!}P_{\lambda}(b),\qquad\nabla(b):=j^{\lambda}_{*}I_{\lambda}(b).$

Porism 5.1 says that every projective indecomposable, $P(b)$, in
${\mathcal{A}}$ has a filtration by quotients of standard objects,
$\Delta(b^{\prime})$, in which $\rho(b^{\prime})\geq\rho(b)$. Dually, every
injective indecomposable, $I(b)$, of ${\mathcal{A}}$ has a filtration by
subobjects of costandard objects, $\nabla(b^{\prime})$, in which
$\rho(b^{\prime})\geq\rho(b)$.

Standard and costandard objects play a crucial role in representation
theoretic applications of stratifications of abelian categories, where one
often requires that projective and/or injective indecomposable objects have
filtrations by standard and/or costandard objects. For example, both of these
conditions are required in Cline-Parshall-Scott’s definition of highest weight
category [CPS88].

Categories whose projective indecomposables have filtrations by standard
objects have been widely studied - beginning with the work of Dlab [Dla96] and
Cline, Parshall and Scott [CPS96]. Categories in which both projective objects
have a filtration by standard objects and injective objects have a filtration
by costandard objects have been studied by various authors (see e.g. [Fri07],
[CZ19], [LW15]). Brundan and Stroppel [BS18] (based on the work of [ADL98])
define a general framework called an $\varepsilon$-stratified category that
includes these situations as special cases. We recall this definition now.

For a sign function $\varepsilon:\Lambda\rightarrow\\{+,-\\}$, define the
$\varepsilon$-standard and $\varepsilon$-costandard objects

$\displaystyle\Delta_{\varepsilon}(b)$
$\displaystyle:=\left\\{\begin{array}[]{ll}j^{\lambda}_{!}P_{\lambda}(b)&\text{if
$\varepsilon(\lambda)=+$}\\\ j^{\lambda}_{!}L_{\lambda}(b)&\text{if
$\varepsilon(\lambda)=-$}\end{array}\right.,$
$\displaystyle\nabla_{\varepsilon}(b)$
$\displaystyle:=\left\\{\begin{array}[]{ll}j^{\lambda}_{*}L_{\lambda}(b)&\text{if
$\varepsilon(\lambda)=+$}\\\ j^{\lambda}_{*}I_{\lambda}(b)&\text{if
$\varepsilon(\lambda)=-$}\end{array}\right.,$

where $\lambda=\rho(b)$.

Brundan and Stroppel [BS18] say that ${\mathcal{A}}$ is
$\varepsilon$-stratified if the following equivalent conditions hold:

  1. (1)

Every projective indecomposable, $P(b)$, has a filtration by
$\varepsilon$-standard objects, $\Delta_{\varepsilon}(b^{\prime})$, in which
$\rho(b^{\prime})\geq\rho(b)$.

  2. (2)

Every injective indecomposable, $I(b)$, has a filtration by
$\varepsilon$-costandard objects, $\nabla_{\varepsilon}(b^{\prime})$, in which
$\rho(b^{\prime})\geq\rho(b)$.

Note that ${\mathcal{A}}$ is a highest weight category if and only if each
${\mathcal{A}}_{\lambda}$ is semisimple and ${\mathcal{A}}$ is
$\varepsilon$-stratified for any (and all)
$\varepsilon:\Lambda\rightarrow\\{+,-\\}$.

The following result gives a new criterion for a finite abelian category to be
$\varepsilon$-stratified.

######  Theorem 6.4.

A finite abelian category ${\mathcal{A}}$ with a stratification by a finite
poset $\Lambda$ is $\varepsilon$-stratified (for a function
$\varepsilon:\Lambda\rightarrow\\{+,-\\}$) if and only if the following
conditions hold:

  1. (1)

For each inclusion of Serre subcategories
$i_{*}:{\mathcal{A}}_{\Lambda^{\prime}}\rightarrow{\mathcal{A}}_{\Lambda}$,
and objects $X,Y\in{\mathcal{A}}_{\Lambda^{\prime}}$, there is a natural
isomorphism
$\operatorname{Ext}^{2}_{{\mathcal{A}}_{\Lambda^{\prime}}}(X,Y)\simeq\operatorname{Ext}^{2}_{{\mathcal{A}}}(i_{*}X,i_{*}Y).$

  2. (2)

For each $\lambda\in\Lambda$,

    1. (a)

If $\varepsilon(\lambda)=+$ then
$j^{\lambda}_{*}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ is exact.

    2. (b)

If $\varepsilon(\lambda)=-$ then
$j^{\lambda}_{!}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ is exact.

One application of Theorem 6.4 is a criterion for a category of (equivariant)
perverse sheaves to be $\varepsilon$-stratified - although we remark that
checking these conditions (particularly the first condition) is difficult in
general.

##  2\. Preliminaries

We begin with an axiomatic definition of recollement. The notation used in
this definition will be used throughout the paper.

######  Definition 2.1.

A recollement of abelian categories consists of three abelian categories
${\mathcal{A}}_{Z}$, ${\mathcal{A}}$ and ${\mathcal{A}}_{U}$ and functors:

(2.1)
${{{\mathcal{A}}_{Z}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{U}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

satisfying the conditions:

  1. (R1)

$(i^{*},i_{*},i^{!})$ and $(j_{!},j^{*},j_{*})$ are adjoint triples.

  2. (R2)

The functors $i_{*}$, $j_{!}$, $j_{*}$ are fully-faithful. Equivalently the
adjunction maps $i^{*}i_{*}\rightarrow\operatorname{Id}\rightarrow i^{!}i_{*}$
and $j^{*}j_{*}\rightarrow\operatorname{Id}\rightarrow j^{*}j_{!}$ are
isomorphisms.

  3. (R3)

The functors satisfy $j^{*}i_{*}=0$ (and so by adjunction
$i^{*}j_{!}=0=i^{!}j_{*}$).

  4. (R4)

The adjunction maps produce exact sequences for each object
$X\in{\mathcal{A}}$:

(2.2) $\displaystyle j_{!}j^{*}X\rightarrow$ $\displaystyle X\rightarrow
i_{*}i^{*}X\rightarrow 0$ (2.3) $\displaystyle 0\rightarrow
i_{*}i^{!}X\rightarrow$ $\displaystyle X\rightarrow j_{*}j^{*}X$

Alternatively, condition (R4) can be replaced by the condition

  1. (R4’)

For any object $X\in{\mathcal{A}}$, if $j^{*}X=0$ then X is in the essential
image of $i_{*}$.

A recollement of triangulated categories is defined in the same way as a
recollement of abelian categories except that condition (R4) is replaced by
the existence of the triangles:

(2.4) $\displaystyle j_{!}j^{*}X\rightarrow$ $\displaystyle X\rightarrow
i_{*}i^{*}X\rightarrow$ (2.5) $\displaystyle i_{*}i^{!}X\rightarrow$
$\displaystyle X\rightarrow j_{*}j^{*}X\rightarrow$

for each object $X$.

######  Remark 2.2.

The interchangibility of (R4) and (R4’) follows from the following argument.
If $j^{*}X=0$ then (R4) implies that $i_{*}i^{!}X\simeq X\simeq i_{*}i^{*}X$
and so $X$ is in the essential image of $i_{*}$. Conversely let
$\mu:j_{!}j^{*}\rightarrow\operatorname{Id}$ and
$\eta:\operatorname{Id}\rightarrow i_{*}i^{*}$ be the adjunction natural
transformations. Then there is a commutative diagram

${{j_{!}j^{*}X}}$${X}$${{\operatorname{cok}\mu_{X}}}$${0}$${0}$${{i_{*}i^{*}X}}$${{i_{*}i^{*}(\operatorname{cok}\mu_{X})}}$${0}$$\scriptstyle{\mu_{X}}$$\scriptstyle{\eta_{j_{!}j^{*}X}}$$\scriptstyle{\eta_{X}}$$\scriptstyle{\eta_{\operatorname{cok}\mu_{X}}}$

in which the rows are exact. By applying $j^{*}$ to the top row we see that
$j^{*}\operatorname{cok}\mu_{X}=0$ and so (R4’) implies that
$\operatorname{cok}\mu_{X}\simeq i_{*}i^{*}(\operatorname{cok}\mu_{X})\simeq
i_{*}i^{*}X$. Equation (2.3) holds by a similar argument.

Write ${\mathcal{A}}^{Z}$ for the essential image of $i_{*}$. To reconcile
Definition 2.1 with the definition of recollement in Section 1, note that by
(R2), ${\mathcal{A}}^{Z}\simeq{\mathcal{A}}_{Z}$, and by (R4’),
${\mathcal{A}}^{Z}$ is the kernel of the exact functor $j^{*}$ and is hence a
Serre subcategory of ${\mathcal{A}}$.

It will be useful to note that if we extend the sequences (2.2) and (2.3) to
exact sequences

(2.6) $\displaystyle 0\rightarrow K\rightarrow j_{!}j^{*}X\rightarrow$
$\displaystyle X\rightarrow i_{*}i^{*}X\rightarrow 0$ (2.7) $\displaystyle
0\rightarrow i_{*}i^{!}X\rightarrow$ $\displaystyle X\rightarrow
j_{*}j^{*}X\rightarrow K^{\prime}\rightarrow 0$

then $K$ and $K^{\prime}$ are in ${\mathcal{A}}^{Z}$. Indeed, by applying the
exact functor $j^{*}$ to (2.6) we get that $j^{*}K=0$ and so
$i_{*}i^{!}K\simeq K\simeq i_{*}i^{*}K$. Likewise by applying $j^{*}$ to (2.7)
we get that $K^{\prime}\in{\mathcal{A}}^{Z}$.

Given a recollement of abelian or triangulated categories with objects and
morphisms as in (2.1), the opposite categories form the following recollement

${{{\mathcal{A}}_{Z}^{op}}}$${{\mathcal{A}}^{op}}$${{{\mathcal{A}}_{U}^{op}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{*}}$$\scriptstyle{j_{!}}$

which we call the opposite recollement.

The following proposition describes a useful way to characterise the functors
$i^{*}$ and $i^{!}$ in any recollement.

######  Proposition 2.3.

Let ${\mathcal{A}}$ be an abelian category with a recollement as in (2.1).
Then for any object $X\in{\mathcal{A}}$:

  1. (i)

$i_{*}i^{*}X$ is the largest quotient object of $X$ in ${\mathcal{A}}^{Z}$.

  2. (ii)

$i_{*}i^{!}X$ is the largest subobject of $X$ in ${\mathcal{A}}^{Z}$.

###### Proof.

By the adjunction $(i_{*},i^{*})$ and since $i_{*}$ is fully-faithful we have
natural isomorphisms for $X\in{\mathcal{A}}$, $Y\in{\mathcal{A}}_{Z}$:

$\operatorname{Hom}_{{\mathcal{A}}}(i_{*}i^{*}X,i_{*}Y)\simeq\operatorname{Hom}_{{\mathcal{A}}_{Z}}(i^{*}X,Y)\simeq\operatorname{Hom}_{{\mathcal{A}}}(X,i_{*}Y)$

sending $f$ to $f\circ\eta$ where $\eta:X\rightarrow i_{*}i^{*}X$ is the
adjunction unit. In particular any morphism $X\rightarrow i_{*}Y$ factors
through $i_{*}i^{*}X$. Statement (i) follows. Statement (ii) follows by taking
the opposite recollement. ∎

Say that a subset $\Lambda^{\prime}$ of a poset $\Lambda$ is lower if for any
$\lambda\in\Lambda^{\prime}$, if $\mu\leq\lambda$ then
$\mu\in\Lambda^{\prime}$.

######  Definition 2.4.

A stratification of an abelian/triangulated category ${\mathcal{A}}$ by a non-
empty poset $\Lambda$ consists of the following data:

  1. (i)

An assignment of an abelian/triangulated category
${\mathcal{A}}_{\Lambda^{\prime}}$ for every lower
$\Lambda^{\prime}\subset\Lambda$, and for lower subsets
$\Lambda^{\prime\prime}\subset\Lambda^{\prime}\subset\Lambda$ an embedding
$i_{\Lambda^{\prime\prime},\Lambda^{\prime}*}:{\mathcal{A}}_{\Lambda^{\prime\prime}}\hookrightarrow{\mathcal{A}}_{\Lambda^{\prime}}$.

  2. (ii)

For each $\lambda\in\Lambda$ an abelian/triangulated category
${\mathcal{A}}_{\lambda}$. We call these strata categories.

This data must satisfy the following conditions

  1. (S1)

${\mathcal{A}}_{\emptyset}=0$ and ${\mathcal{A}}_{\Lambda}={\mathcal{A}}$.

  2. (S2)

For each $\lambda\in\Lambda$ and lower subset $\Lambda^{\prime}\subset\Lambda$
in which $\lambda\in\Lambda^{\prime}$ is maximal, the functor
$i_{*}=i_{\Lambda^{\prime}\backslash\\{\lambda\\},\Lambda^{\prime}*}$ fits
into a recollement

${{{\mathcal{A}}_{\Lambda^{\prime}\backslash\\{\lambda\\}}}}$${{\mathcal{A}}_{\Lambda^{\prime}}}$${{{\mathcal{A}}_{\lambda}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

  3. (S3)

If $\Lambda^{\prime\prime}\subset\Lambda^{\prime}$ are lower subsets of
$\Lambda$, and $\lambda\in\Lambda$ is a maximal element of both
$\Lambda^{\prime\prime}$ and $\Lambda^{\prime}$, then the following diagram of
functors commutes

${{{\mathcal{A}}_{\Lambda^{\prime\prime}}}}$${{{\mathcal{A}}_{\Lambda^{\prime}}}}$${{{\mathcal{A}}_{\lambda}}}$$\scriptstyle{i_{\Lambda^{\prime},\Lambda^{\prime\prime}*}}$

We proceed with some important examples of recollements and stratifications.

######  Example 2.5 (Constructible sheaves with respect to a stratification).

A stratification of a quasiprojective complex variety $X$ is a finite
collection, $\\{X_{\lambda}\\}_{\lambda\in\Lambda}$, of disjoint, smooth,
connected, locally closed subvarieties, called strata, in which
$X=\bigcup_{\lambda\in\Lambda}X_{\lambda}$ and for each $\lambda\in\Lambda$,
$\overline{X_{\lambda}}$ is a union of strata. In this case we equip $\Lambda$
with the partial order

$\mu\leq\lambda\text{ if $X_{\mu}\subset\overline{X_{\lambda}}$.}$

We use $\Lambda$ to refer to the stratification of $X$.

For a variety $X$, let $\operatorname{Loc}^{ft}(X,\Bbbk)$ be the category of
local systems on $X$ of finite type with coefficients in a field $\Bbbk$.
Recall that, by taking monodromy, $\operatorname{Loc}^{ft}(X,\Bbbk)$ is
equivalent to the category,
$\Bbbk[\pi_{1}(X_{\lambda})]\text{-}\mathrm{mod}_{fg}$, of finitely generated
$\Bbbk[\pi_{1}(X_{\lambda})]$-modules (see e.g. [Ach21, Theorem 1.7.9]).

Say that a sheaf ${\mathcal{F}}$ on $X$ is constructible with respect to a
stratification, $\Lambda$, of $X$ if ${\mathcal{F}}|_{X_{\lambda}}$ is a local
system of finite type for each $\lambda\in\Lambda$. Write
${\mathcal{D}}^{b}_{\Lambda}(X,\Bbbk)$ for the full triangulated subcategory
of ${\mathcal{D}}^{b}(X,\Bbbk)$ consisting of objects ${\mathcal{F}}$ in which
$H^{k}({\mathcal{F}})$ is constructible with respect to $\Lambda$.

Say that a stratification, $\Lambda$, of $X$ is good if for any
$\lambda\in\Lambda$ and any object
${\mathcal{L}}\in\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)$, we have
$j_{\lambda*}{\mathcal{L}}\in{\mathcal{D}}^{b}_{\Lambda}(X,\Bbbk)$, where
$j_{\lambda}:X_{\lambda}\hookrightarrow X$ is the embedding, and
$j_{\lambda*}$ is the derived pushforward. It is difficult to tell in general
whether a stratification is good (see [Ach21, Remark 2.3.21] for a discussion
of these dificulties). A stratification satisfying the Whitney regularity
conditions [Wit65] is good. In particular, if an algebraic group $G$ acts on
$X$ with finitely many orbits (each connected), then the stratification of $X$
by $G$-orbits is good (see e.g. [Ach21, Exercise 6.5.2]).

Given a good stratification $\Lambda$ on $X$, the triangulated category
${\mathcal{D}}^{b}_{\Lambda}(X,\Bbbk)$ has a stratification by $\Lambda$ with
strata categories

${\mathcal{D}}_{\lambda}:={\mathcal{D}}^{b}(\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk))\simeq{\mathcal{D}}^{b}(\Bbbk[\pi_{1}(X_{\lambda})]\text{-}\mathrm{mod}_{fg})$

and Serre subcategories
${\mathcal{D}}_{\Lambda^{\prime}}:={\mathcal{D}}^{b}_{\Lambda^{\prime}}(\bigcup_{\lambda\in\Lambda^{\prime}}X_{\lambda})$
for each lower $\Lambda^{\prime}\subset\Lambda$.

For a perversity function $p:\Lambda\rightarrow{\mathbb{Z}}$, the category
${}^{p}P_{\Lambda}(X,\Bbbk)$ of perverse sheaves on $X$ with respect to the
stratification $\Lambda$ (and perversity function $p$) is the full subcategory
of ${\mathcal{D}}^{b}_{\Lambda}(X,\Bbbk)$ consisting of complexes
${\mathcal{F}}$ in which for any strata
$h_{\lambda}:X_{\lambda}\hookrightarrow X$:

  * (i)

${\mathcal{H}}^{d}(h_{\lambda}^{*}{\mathcal{F}})=0$ if $d>p(\lambda)$,

  * (ii)

${\mathcal{H}}^{d}(h_{\lambda}^{!}{\mathcal{F}})=0$ if $d<p(\lambda)$,

where ${\mathcal{H}}^{d}({\mathcal{F}})$ refers to the $d$-th cohomology sheaf
of ${\mathcal{F}}$. The category ${\mathcal{A}}={}^{p}P_{\Lambda}(X,\Bbbk)$ is
abelian and has a stratification by $\Lambda$, with strata categories

${\mathcal{A}}_{\lambda}=\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)[p(\lambda)]\simeq\Bbbk[\pi_{1}(X_{\lambda})]\text{-}\mathrm{mod}_{fg}.$

######  Example 2.6 ($G$-equivariant perverse sheaves).

Another example of a stratification arises in the theory of equivariant
perverse sheaves as defined in [BL94].

For a complex algebraic group $G$ and quasiprojective complex $G$-variety $X$,
a $G$-equivariant perverse sheaf on $X$ is, roughly speaking, a perverse sheaf
on $X$ with a $G$-action compatible with the $G$-action on $X$ (see e.g.
[Ach21, Definition 6.2.3] for a precise definition). The category,
$P_{G}(X,\Bbbk)$ of $G$-equivariant perverse sheaves is the heart of a
$t$-structure on the $G$-equivariant derived category,
${\mathcal{D}}_{G}(X,\Bbbk)$ defined by Bernstein-Lunts [BL94]. For a
$G$-equivariant map of $G$-varieties $h:X\rightarrow Y$, there are equivariant
versions of the (proper) pushforward and (proper) pullback functors:
$h_{*},h_{!},h^{!},h^{*}$. If $i:Z\hookrightarrow X$ is the inclusion of a
$G$-invariant closed subvariety with open complement $j:U\hookrightarrow X$,
then there is a recollement of triangulated categories

${{{\mathcal{D}}_{G}^{b}(Z,\Bbbk)}}$${{\mathcal{D}}_{G}^{b}(X,\Bbbk)}$${{{\mathcal{D}}_{G}^{b}(U,\Bbbk)}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

If $X$ is a homogeneous $G$-variety, then every $G$-equivariant perverse sheaf
is a finite type local system (shifted by $\dim_{{\mathbb{C}}}X$). Moreover,
in this case,

(2.8)
$P_{G}(X,\Bbbk)\simeq\Bbbk[G^{x}/(G^{x})^{\circ}]\text{-}\mathrm{mod}_{fg},$

where $G^{x}\subset G$ is the stabilizer of a point $x\in X$, and
$(G^{x})^{\circ}$ is the connected component of $G^{x}$ containing the
identity element (see e.g. [Ach21, Proposition 6.2.13] for a proof of this
statement).

Suppose $G$ acts on $X$ with finitely many orbits (each connected). Let
$\Lambda$ be a set indexing the set of $G$-orbits in $X$, and write
${\mathcal{O}}_{\lambda}$ for the orbit corresponding to $\lambda\in\Lambda$.
Consider $\Lambda$ as a poset with the closure order: $\lambda\leq\mu$ if
${\mathcal{O}}_{\lambda}\subset\overline{{\mathcal{O}}_{\mu}}$. Then the
category ${\mathcal{A}}=P_{G}(X,\Bbbk)$ has a stratification with strata
categories

${\mathcal{A}}_{\lambda}=P_{G}({\mathcal{O}}_{\lambda},\Bbbk)\simeq\Bbbk[G^{x_{\lambda}}/(G^{x_{\lambda}})^{\circ}]\text{-}\mathrm{mod}_{fg},$

where $x_{\lambda}\in{\mathcal{O}}_{\lambda}$.

######  Example 2.7 (Modules with idempotents).

For a ring $A$, let $\mathrm{Mod}\text{-}A$ be the category of all right
$A$-modules, and $\mathrm{mod}\text{-}A$ be the category of finitely presented
right $A$-modules. Let $e$ be an idempotent in $A$, and define the inclusion
functor $i_{*}:\mathrm{Mod}\text{-}A/AeA\rightarrow\mathrm{Mod}\text{-}A$.
Note that $\mathrm{Mod}\text{-}A/AeA$ is equivalent to the Serre subcategory
of $\mathrm{Mod}\text{-}A$ consisting of modules annihilated by $e$. There is
a corresponding Serre quotient
$j^{*}:\mathrm{Mod}\text{-}A\rightarrow\mathrm{Mod}\text{-}eAe$ defined

$j^{*}:=\operatorname{Hom}_{A}(eA,-)\simeq-\otimes_{A}Ae.$

i.e. $j^{*}M=Me$ for any object $M\in\mathrm{Mod}\text{-}A$. These functors
fit into a recollement of abelian categories

${{\mathrm{Mod}\text{-}A/AeA}}$${\mathrm{Mod}\text{-}A}$${{\mathrm{Mod}\text{-}eAe}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

where for any right $A$-module $M$:

  1. (i)

$i^{*}M$ is the largest quotient, $N$, of $M$ in which $Ne=0$.

  2. (ii)

$i^{!}M$ is the largest subobject, $N$, of $M$ in which $Ne=0$.

Moreover $j_{!}:=-\otimes_{eAe}eA$ and
$j_{*}:=\operatorname{Hom}_{eAe}(Ae,-)$.

If $A$ is right artinian and has enough injectives then the inclusion
$i_{*}:\mathrm{mod}\text{-}A/AeA\rightarrow\mathrm{mod}\text{-}A$ fits into a
recollement

${{\mathrm{mod}\text{-}A/AeA}}$${\mathrm{mod}\text{-}A}$${{\mathrm{mod}\text{-}eAe}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

in which $j^{*}$ has left adjoint $j_{!}=-\otimes_{eAe}eA$ (see e.g [Kra17,
Lemma 2.5]).

######  Example 2.8 (Macpherson-Vilonen construction).

Let ${\mathcal{A}}_{Z}$, ${\mathcal{A}}_{U}$ be abelian categories,
$F:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}_{Z}$ be a right exact functor,
$G:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}_{Z}$ be a left exact functor, and
let $\varepsilon:F\rightarrow G$ be a natural transformation.

Macpherson and Vilonen [MV86] define a category, ${\mathcal{A}}(\varepsilon)$,
whose objects are tuples $(X_{U},X_{Z},\alpha,\beta)$, where
$(X_{U},X_{Z})\in\operatorname{Obj}{\mathcal{A}}_{U}\times\operatorname{Obj}{\mathcal{A}}_{Z}$
and $\alpha:F(X_{U})\rightarrow X_{Z}$, $\beta:X_{Z}\rightarrow G(X_{U})$ are
morphisms in ${\mathcal{A}}_{Z}$ in which the following diagram commutes

${{F(X_{U})}}$${{G(X_{U})}}$${{X_{Z}}}$$\scriptstyle{\varepsilon_{X_{U}}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

A morphism
$(X_{U},X_{Z},\alpha,\beta)\rightarrow(X_{U}^{\prime},X_{Z}^{\prime},\alpha^{\prime},\beta^{\prime})$
is a pair

$f=(f_{U}:X_{U}\rightarrow X_{U}^{\prime},f_{Z}:X_{Z}\rightarrow
X_{Z}^{\prime})\in\operatorname{Mor}{\mathcal{A}}_{U}\times\operatorname{Mor}{\mathcal{A}}_{Z}$

in which the following prism commutes:

${{F(X_{U})}}$${{G(X_{U})}}$${{X_{Z}}}$${{F(X_{U}^{\prime})}}$${{G(X_{U}^{\prime})}}$${{X_{Z}^{\prime}}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\varepsilon_{X_{U}}}$$\scriptstyle{F(f_{U})}$$\scriptstyle{G(f_{U})}$$\scriptstyle{\varepsilon_{X_{U}^{\prime}}}$$\scriptstyle{f_{Z}}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$

Macpherson and Vilonen show [MV86, Proposition 1.1] that the category
${\mathcal{A}}(\varepsilon)$ is abelian. Moreover they show that the category
${\mathcal{A}}(\varepsilon)$ fits into a recollement

${{{\mathcal{A}}_{Z}}}$${{\mathcal{A}}(\varepsilon)}$${{{\mathcal{A}}_{U}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

in which

$\displaystyle i_{*}(X_{Z})=(0,X_{Z},0,0),\qquad$ $\displaystyle
j^{*}(X_{U},X_{Z},\alpha,\beta)=X_{U},$ $\displaystyle
i^{*}(X_{U},X_{Z},\alpha,\beta)=\operatorname{cok}\alpha,\qquad$
$\displaystyle
j_{!}(X_{U})=(X_{U},F(X_{U}),1_{F(X_{U})},\varepsilon_{X_{U}}),$
$\displaystyle i^{!}(X_{U},X_{Z},\alpha,\beta)=\ker\beta,\qquad$
$\displaystyle
j_{*}(X_{U})=(X_{U},G(X_{U}),\varepsilon_{X_{U}},1_{G(X_{U})}).$

Macpherson and Vilonen [MV86] use iterations of this formal construction to
construct the category of perverse sheaves on a stratified variety. Franjou
and Pirashvili [FP04, Theorem 8.7] give necessary and sufficient conditions
for a recollement of abelian categories to be equivalent to a recollement
built using the Macpherson-Vilonen construction.

Note that the functor
$i_{*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}(\varepsilon)$ has an exact
retraction $i^{!*}:{\mathcal{A}}(\varepsilon)\rightarrow{\mathcal{A}}_{Z}$
defined $i^{!*}(X_{U},X_{Z},\alpha,\beta)=X_{Z}$. This is a special feature of
recollements built using Macpherson-Vilonen’s construction - the functor
$i_{*}$ does not usually have an exact retraction in a general recollement.

The functor $j_{!*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}(\varepsilon)$
defined
$j_{!*}(X_{U})=(X_{U},\operatorname{im}\varepsilon_{X_{U}},\varepsilon_{X_{U}},1)$
is a special case of an intermediate-extension functor defined in Definition
3.1 below. Note that every simple object in ${\mathcal{A}}(\varepsilon)$ is
either of the form $i_{*}L$ for a simple object $L$ in ${\mathcal{A}}_{Z}$, or
of the form $j_{!*}L$ for a simple object in ${\mathcal{A}}_{U}$. This is a
special case of Proposition 3.4 below.

##  3\. The intermediate-extension functor

Consider again a recollement:

(3.1)
${{{\mathcal{A}}_{Z}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{U}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

In this section we study the full subcategory,
${\mathcal{A}}^{U}\hookrightarrow{\mathcal{A}}$, whose objects have no
subobjects or quotients in ${\mathcal{A}}^{Z}:=\operatorname{im}i_{*}$. The
main result of this section (Proposition 3.3(ii)) is that the restricted
functor $j^{*}:{\mathcal{A}}^{U}\rightarrow{\mathcal{A}}_{U}$ is an
equivalence of categories. The quasi-inverse
$j_{!*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}^{U}$ is defined as follows.

######  Definition 3.1
($j_{!*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}^{U}$).

For an object $X\in{\mathcal{A}}_{U}$, let $\overline{1_{X}}:j_{!}X\rightarrow
j_{*}X$ be the morphism corresponding to the identity on $X$ under the
isomorphism

$\operatorname{Hom}_{{\mathcal{A}}}(j_{!}X,j_{*}X)\simeq\operatorname{Hom}_{{\mathcal{A}}_{U}}(X,j^{*}j_{*}X)\simeq\operatorname{Hom}_{{\mathcal{A}}_{U}}(X,X).$

Define

$j_{!*}X:=\operatorname{im}(\overline{1_{X}}:j_{!}X\rightarrow
j_{*}X)\in{\mathcal{A}}.$

It is easy to check that if $X\in{\mathcal{A}}_{U}$ then
$j_{!*}X\in{\mathcal{A}}^{U}$. Indeed as $i^{!}j_{*}X=0$, $j_{*}X$ has no
subobjects in ${\mathcal{A}}^{Z}$. In particular, as $j_{!*}X$ is a subobject
of $j_{*}X$ it cannot have any subobjects in ${\mathcal{A}}^{Z}$. Likewise as
$j_{!*}X$ is a quotient of $j_{!}X$ it cannot have any quotients in
${\mathcal{A}}^{Z}$.

We call the functor $j_{!*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}$ an
intermediate-extension functor.

######  Remark 3.2.

Not every subquotient of an object in ${\mathcal{A}}^{U}$ need be in
${\mathcal{A}}^{U}$. In particular, an object in ${\mathcal{A}}^{U}$ may still
have simple composition factors in ${\mathcal{A}}^{Z}$.

######  Proposition 3.3.

Let ${\mathcal{A}}$ be an abelian category with a recollement of abelian
categories as in (3.1). Then

  1. (i)

If $X\in{\mathcal{A}}$ has no nonzero quotient objects in ${\mathcal{A}}^{Z}$,
and $Y\in{\mathcal{A}}$ has no nonzero subobjects in ${\mathcal{A}}^{Z}$ (i.e.
$i^{*}X=0$ and $i^{!}Y=0$), then

$\operatorname{Hom}_{{\mathcal{A}}}(X,Y)\simeq\operatorname{Hom}_{{\mathcal{A}}_{U}}(j^{*}X,j^{*}Y).$

  2. (ii)

$j^{*}:{\mathcal{A}}^{U}\rightarrow{\mathcal{A}}_{U}$ is an equivalence of
categories with quasi-inverse
$j_{!*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}^{U}$.

###### Proof.

If $i^{*}X=0$ then (2.6) gives an exact sequence

$0\rightarrow K\rightarrow j_{!}j^{*}X\rightarrow X\rightarrow 0$

in which $K\simeq i_{*}i^{!}K$. So applying $\operatorname{Hom}(-,Y)$ we get
the exact sequence

$0\rightarrow\operatorname{Hom}(X,Y)\rightarrow\operatorname{Hom}(j_{!}j^{*}X,Y)\rightarrow\operatorname{Hom}(i_{*}i^{!}K,Y).$

Applying adjunctions gives the exact sequence

$0\rightarrow\operatorname{Hom}(X,Y)\rightarrow\operatorname{Hom}(j^{*}X,j^{*}Y)\rightarrow\operatorname{Hom}(i^{!}K,i^{!}Y).$

Statement (i) follows as $i^{!}Y=0$.

A corollary of statement (i) is that
$j^{*}:{\mathcal{A}}^{U}\rightarrow{\mathcal{A}}_{U}$ is fully-faithful. To
show that $j^{*}$ is essentially surjective it suffices to show that for any
object $X\in{\mathcal{A}}_{U}$, $j^{*}j_{!*}X\simeq X$. Now, as $j^{*}$ is
exact:

$j^{*}j_{!*}X=j^{*}\operatorname{im}(j_{!}X\rightarrow
j_{*}X)\simeq\operatorname{im}(j^{*}j_{!}X\rightarrow
j^{*}j_{*}X)\simeq\operatorname{im}(\operatorname{Id}:X\rightarrow X)=X$

and so (ii) follows. ∎

If ${\mathcal{A}}$ has a stratification by a finite poset $\Lambda$, then for
each $\lambda\in\Lambda$, there is a functor
$j^{\lambda}_{!*}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ defined by
the composition

${{{\mathcal{A}}_{\lambda}}}$${{{\mathcal{A}}_{\\{\mu\in\Lambda~{}|~{}\mu\leq\lambda\\}}}}$${{\mathcal{A}}}$$\scriptstyle{j_{!*}}$

######  Proposition 3.4.

Let ${\mathcal{A}}$ be an abelian category with a stratification by a finite
poset $\Lambda$. Every simple object $L\in{\mathcal{A}}$ is of the form
$j^{\lambda}_{!*}L_{\lambda}$, for a unique (up to isomorphism) simple object
$L_{\lambda}\in{\mathcal{A}}_{\lambda}$ and unique $\lambda\in\Lambda$.

###### Proof.

Suppose ${\mathcal{A}}$ fits into a recollement as in (3.1). By Proposition
3.3, if $L\in{\mathcal{A}}_{U}$ is a simple object, then $j_{!*}L$ is a simple
object in ${\mathcal{A}}$. Moreover all the simple objects of ${\mathcal{A}}$
are either of the form $i_{*}L$ for a simple object $L\in{\mathcal{A}}_{Z}$,
or of the form $j_{!*}L$ for a simple object $L\in{\mathcal{A}}_{U}$. The
statement follows via an induction argument on $|\Lambda|$. ∎

The following properties of the intermediate-extension functor will be useful.

######  Proposition 3.5.

Let ${\mathcal{A}}$ be an abelian category with a recollement of abelian
categories as in (3.1). Then

  1. (i)

The functor $j_{!*}:{\mathcal{A}}_{U}\rightarrow{\mathcal{A}}$ maps injective
morphisms to injective morphisms and surjective morphisms to surjective
morphisms.

  2. (ii)

If $X\in{\mathcal{A}}$ has no nonzero quotient objects in ${\mathcal{A}}^{Z}$
then there is a canonical short exact sequence

$0\rightarrow i_{*}i^{!}X\rightarrow X\rightarrow j_{!*}j^{*}X\rightarrow 0$

  3. (iii)

If $X\in{\mathcal{A}}$ has no nonzero subobjects in ${\mathcal{A}}^{Z}$ then
there is a canonical short exact sequence

$0\rightarrow j_{!*}j^{*}X\rightarrow X\rightarrow i_{*}i^{*}X\rightarrow 0$

###### Proof.

Let $f:X\rightarrow Y$ be a map in ${\mathcal{A}}_{U}$ and define objects
$K_{1}$, $K_{2}$ in ${\mathcal{A}}$ by the exact sequence

$0\rightarrow K_{1}\rightarrow j_{!*}X\rightarrow j_{!*}Y\rightarrow
K_{2}\rightarrow 0$

To prove statement (i) it suffices to show that if $j^{*}K_{i}=0$ then
$K_{i}=0$. If $j^{*}K_{i}=0$ then by (R4’), $K_{1}\simeq i_{*}i^{*}K_{1}$ and
$K_{2}\simeq i_{*}i^{!}K_{2}$. Then each $K_{i}=0$ since $j_{!*}X$ and
$j_{!*}Y$ are in ${\mathcal{A}}^{U}$.

To prove statement (ii), let $X\in{\mathcal{A}}$ have no nonzero quotients in
${\mathcal{A}}^{Z}$ and consider the short exact sequence

$0\rightarrow i_{*}i^{!}X\rightarrow X\rightarrow K\rightarrow 0$

Applying $i^{!}$ to the sequence we see that $i^{!}K=0$ and so
$K\in{\mathcal{A}}^{U}$. So $K\simeq j_{!*}j^{*}K$ and (by applying $j^{*}$ to
this sequence) $j^{*}X\simeq K$. Statement (ii) follows immediately. The proof
of statement (iii) is similar. ∎

Say that an abelian category is a length category if every object has a finite
filtration by simple objects.

######  Proposition 3.6.

If ${\mathcal{A}}$ be an abelian category with a stratification by a finite
poset, then ${\mathcal{A}}$ is a length category if and only if all the strata
categories are length categories.

###### Proof.

Let ${\mathcal{A}}$ be an abelian category fitting into a recollement of
abelian categories as in (3.1). It suffices to show that ${\mathcal{A}}$ is a
length category if and only if both ${\mathcal{A}}_{Z}$ and
${\mathcal{A}}_{U}$ are length categories. The result follows from this
statement by induction.

Let $X$ be an object in ${\mathcal{A}}$ and let $K$ be defined by the short
exact sequence

$0\rightarrow i_{*}i^{!}X\rightarrow X\rightarrow K\rightarrow 0$

Then $i^{!}K=0$ and so applying Proposition 3.5(iii) we get the short exact
sequence

$0\rightarrow j_{!*}j^{*}K\rightarrow K\rightarrow i_{*}i^{*}K\rightarrow 0$

In particular if every object in ${\mathcal{A}}_{Z}$ and every object in
${\mathcal{A}}_{U}$ has a finite filtration by simple objects, then so does
$K$ and hence so does $X$. The converse statement is obvious. ∎

##  4\. Recollements with enough projectives/injectives

In this section we study the relationship between projective covers of objects
in the different categories of a recollement. More precisely, let
${\mathcal{A}}$ be a category fitting into a recollement as in (3.1).
Proposition 4.5 says that if ${\mathcal{A}}$ has enough projectives/injectives
then so does ${\mathcal{A}}_{U}$. Proposition 4.6 says that if ${\mathcal{A}}$
is a Krull-Schmidt category then if ${\mathcal{A}}$ has enough
projectives/injectives then so does ${\mathcal{A}}_{Z}$. Proposition 4.7 says
that if $X\in{\mathcal{A}}_{U}$ has a projective cover $P$ in
${\mathcal{A}}_{U}$ then $j_{!}P$ is a projective cover in ${\mathcal{A}}$ of
$j_{!*}X$.

Unfortunately it is not easy to find a projective cover in ${\mathcal{A}}$ of
an object $i_{*}X\in{\mathcal{A}}^{Z}$, even if a projective cover of $X$
exists in ${\mathcal{A}}_{Z}$. Theorem 4.9 gives sufficient conditions for
such a projective cover to exist. A consequence of Theorem 4.9 (Corollary
4.11) is that a category ${\mathcal{A}}$ with a stratification by a finite
poset is equivalent to a category of finite dimensional modules of a finite
dimensional algebra if and only if the same is true of the strata categories.

###  4.1. Projective covers

Recall that a surjection $\phi:X\rightarrow Y$ is essential if for any
morphism $\alpha:X^{\prime}\rightarrow X$, if $\phi\circ\alpha$ is surjective
then $\alpha$ is surjective. Equivalently $\phi:X\rightarrow Y$ is essential
if for any subobject $U\subset X$, if $U+\ker\phi=X$ then $U=X$. If
$P\rightarrow X$ is an essential surjection and $P$ is projective then we call
$P$ (or more accurately the morphism $P\rightarrow X$) a projective cover of
$X$. The projective cover of $X$ (if it exists) factors through every other
essential cover of $X$, and is unique up to isomorphism.

If $L\in{\mathcal{A}}$ is a simple object and $P$ is projective then
$\phi:P\rightarrow L$ is a projective cover if and only if the following
equivalent conditions hold:

  * (i)

$\ker\phi$ is the unique maximal subobject of $P$.

  * (ii)

The endomorphism ring of $P$ is local.

See e.g. [Kra15, Lemma 3.6] for a proof of these facts.

The dual concept of an essential surjection is called an essential extension.
If $X\rightarrow I$ is an essential extension and $I$ is injective then this
extension is called the injective envelope of $X$.

An abelian category has enough projectives (resp. enough injectives) if every
object has a projective cover (resp. injective envelope).

An abelian category ${\mathcal{A}}$ is a Krull-Schmidt category if every
object in ${\mathcal{A}}$ is a finite direct sum of objects with local
endomorphism rings. For example, any abelian length category is a Krull-
Schmidt category.

In a Krull-Schmidt category, the projective covers of simple objects are
exactly the projective indecomposable objects. Moreover a Krull-Schmidt
category ${\mathcal{A}}$ has enough projectives if and only if every simple
object has a projective cover. We will need the following well-known
characterisation of projective covers of simple objects in Krull-Schmidt
categories.

######  Proposition 4.1.

Let ${\mathcal{A}}$ be a Krull-Schmidt category. Let $P\in{\mathcal{A}}$ be a
projective object and $L\in{\mathcal{A}}$ be a simple object. A map
$P\rightarrow L$ is a projective cover if and only if for any simple object
$L^{\prime}$

(4.1)
$\dim_{\operatorname{End}_{{\mathcal{A}}}(L^{\prime})}\operatorname{Hom}_{{\mathcal{A}}}(P,L^{\prime})=\begin{cases}1&\text{
if $L=L^{\prime}$,}\\\ 0&\text{ otherwise.}\end{cases}$

######  Remark 4.2.

Since $\operatorname{End}_{{\mathcal{A}}}(B)$ is a division ring, any
$\operatorname{End}_{{\mathcal{A}}}(B)$-module is free. For an
$\operatorname{End}_{{\mathcal{A}}}(B)$-module $M$, we write
$\dim_{\operatorname{End}_{{\mathcal{A}}}(B)}M$ for the rank of $M$ as a free
$\operatorname{End}_{{\mathcal{A}}}(B)$-module.

###### Proof of Proposition 4.1.

Let $\phi:P\rightarrow L$ be a projective cover of a simple object. Since
$\ker\phi$ is the unique maximal subobject of $\phi$,
$\operatorname{Hom}_{{\mathcal{A}}}(P,L^{\prime})=0$ whenever $L\neq
L^{\prime}$. To show equation (4.1), it remains to show that the
$\operatorname{End}_{{\mathcal{A}}}(L)$-equivariant map

$-\circ\phi:\operatorname{End}_{{\mathcal{A}}}(L)\rightarrow\operatorname{Hom}_{{\mathcal{A}}}(P,L)$

is an isomorphism. Since $\phi$ is a surjection this map is injective. To show
surjectivity, let $f\in\operatorname{Hom}_{{\mathcal{A}}}(P,L)$ be nonzero.
Then as $\ker f$ is a maximal subobject of $P$, $\ker\phi\subset\ker f$, and
so $f$ factors through $\phi$.

Conversely, if (4.1) holds, then if $P=P_{1}\oplus P_{2}$, only one $P_{i}$
can have a simple quotient and the other must be zero. In particular, $P$ is
indecomposable. ∎

###  4.2. Ext-finiteness

To state the main result of this section (Theorem 4.9) we need the concept of
$\operatorname{Ext}$-finiteness. In this section we recall this definition and
give two results about $\operatorname{Ext}$-finiteness (Propositions 4.3 and
4.4) that will be needed in the discussion following Theorem 4.9.

For $k\in{\mathbb{N}}$, say that an abelian category ${\mathcal{A}}$ is
$\operatorname{Ext}^{k}$-finite if for any simple objects $A,B$ in
${\mathcal{A}}$,

$\dim_{\operatorname{End}_{{\mathcal{A}}}(B)}\operatorname{Ext}^{k}_{{\mathcal{A}}}(A,B)<\infty.$

Note that if ${\mathcal{A}}$ is a $\Bbbk$-linear category, for some field
$\Bbbk$, then

$\dim_{\operatorname{End}_{{\mathcal{A}}}(B)}\operatorname{Ext}^{k}_{{\mathcal{A}}}(A,B)=\frac{\dim_{\Bbbk}\operatorname{Ext}^{k}_{{\mathcal{A}}}(A,B)}{\dim_{\Bbbk}\operatorname{End}_{{\mathcal{A}}}(B)}.$

So ${\mathcal{A}}$ is $\operatorname{Ext}^{k}$-finite whenever
$\dim_{\Bbbk}\operatorname{Ext}^{k}_{{\mathcal{A}}}(A,B)<\infty$ for every
simple object $A,B$. The converse is true if the endomorphism ring of every
simple object has finite $\Bbbk$-dimension (e.g. if $\Bbbk$ is algebraically
closed).

The following two propositions give useful criteria for a category to be
$\operatorname{Ext}^{k}$-finite.

######  Proposition 4.3.

Any abelian length category with enough projectives is
$\operatorname{Ext}^{k}$-finite for every $k\in{\mathbb{N}}$.

###### Proof.

Let ${\mathcal{A}}$ be an abelian length category. Let $X$ be an object in
${\mathcal{A}}$, and let $Y$ be a simple object in ${\mathcal{A}}$. Consider a
projective presentation of X:

$0\rightarrow K\rightarrow P\rightarrow X\rightarrow 0$

Since $\operatorname{Ext}^{k}_{{\mathcal{A}}}(P,Y)=0$ there is a
$\operatorname{End}_{{\mathcal{A}}}(Y)$-equivariant surjection of
$\operatorname{Ext}^{k-1}_{{\mathcal{A}}}(K,Y)$ onto
$\operatorname{Ext}^{k}_{{\mathcal{A}}}(X,Y)$ for each $k>0$. Since
${\mathcal{A}}$ is a length category,

$\dim_{\operatorname{End}_{{\mathcal{A}}}(Y)}\operatorname{Hom}_{{\mathcal{A}}}(X,Y)<\infty.$

The result follows by induction. ∎

Say that a $\Bbbk$-linear abelian category is finite over $\Bbbk$ if
${\mathcal{A}}$ is a length category with enough projectives, finitely many
simple objects, and finite dimensional $\operatorname{Hom}$-spaces. It is
well-known that ${\mathcal{A}}$ is finite over $\Bbbk$ if and only if there is
a finite-dimensional $\Bbbk$-algebra $A$ in which ${\mathcal{A}}$ is
equivalent to the category, $A\text{-}\mathrm{mod}$, of modules that are
finite dimensional as $\Bbbk$-vector spaces. Indeed, if
$\\{P_{\lambda}\\}_{\lambda\in\Lambda}$ are the projective indecomposables in
${\mathcal{A}}$ (up to isomorphism), then
$A=\operatorname{End}_{{\mathcal{A}}}(\bigoplus_{\lambda\in\Lambda}P_{\lambda})^{op}$
and
$\operatorname{Hom}_{{\mathcal{A}}}(\bigoplus_{\lambda\in\Lambda}P_{\lambda},-):{\mathcal{A}}\simeq
A\text{-}\mathrm{mod}$. Note that there is a contravariant equivalence
$\operatorname{Hom}_{\Bbbk}(-,\Bbbk):A\text{-}\mathrm{mod}\rightarrow
A^{op}\text{-}\mathrm{mod}$. In particular, any finite abelian category has
enough injectives.

######  Proposition 4.4.

Let $\Bbbk$ be a field and let ${\mathcal{A}}$ be a $\Bbbk$-linear abelian
category with a stratification by a finite poset in which every strata
category is a finite abelian category. Then ${\mathcal{A}}$ is
$\operatorname{Ext}^{1}$-finite.

###### Proof.

By the assumptions on the strata categories, ${\mathcal{A}}$ has finite
dimensional $\operatorname{Hom}$-spaces. Suppose ${\mathcal{A}}$ has a
recollement with objects and morphisms as in (3.1). Suppose
${\mathcal{A}}_{U}$ and ${\mathcal{A}}_{Z}$ have enough projectives, and
${\mathcal{A}}_{U}$ has enough injectives. By Proposition 4.3, both
${\mathcal{A}}_{Z}$ and ${\mathcal{A}}_{U}$ are
$\operatorname{Ext}^{1}$-finite. We show that ${\mathcal{A}}$ is
$\operatorname{Ext}^{1}$-finite. It suffices to show that
$\dim_{\Bbbk}\operatorname{Ext}_{{\mathcal{A}}}^{1}(X,Y)<\infty$ for all
simple objects $X,Y$.

Since ${\mathcal{A}}^{Z}$ is a Serre subcategory of ${\mathcal{A}}$, this is
true whenever $X$ and $Y$ are both in ${\mathcal{A}}^{Z}$. Let
$L\in{\mathcal{A}}_{U}$ be simple and let $j_{!*}L$ have projective and
injective presentations:

$\displaystyle 0\rightarrow K\rightarrow j_{!}P\rightarrow j_{!*}L\rightarrow
0$ $\displaystyle 0\rightarrow j_{!*}L\rightarrow j_{*}I\rightarrow
K^{\prime}\rightarrow 0$

The projective presentation implies that
$\operatorname{Hom}_{{\mathcal{A}}}(K,Y)$ surjects onto
$\operatorname{Ext}_{{\mathcal{A}}}^{1}(j_{!*}L,Y)$. The injective
presentation implies that $\operatorname{Hom}_{{\mathcal{A}}}(Y,K^{\prime})$
surjects onto $\operatorname{Ext}_{{\mathcal{A}}}^{1}(Y,j_{!*}L)$. It follows
that ${\mathcal{A}}$ is $\operatorname{Ext}^{1}$-finite. The result follows by
an induction argument. ∎

###  4.3. Main results

This section includes our original results about recollements and projective
covers.

######  Proposition 4.5.

Let ${\mathcal{A}}$ be an abelian category with a recollement of abelian
categories as in (3.1). Then

  1. (i)

If $X\rightarrow Y$ is an essential surjection in ${\mathcal{A}}$ and
$i^{*}Y=0$ then $i^{*}X=0$ and $j^{*}X\rightarrow j^{*}Y$ is an essential
surjection.

  2. (ii)

If $P\in{\mathcal{A}}$ is projective and $i^{*}P=0$ then
$j^{*}P\in{\mathcal{A}}_{U}$ is projective. In particular if $P\rightarrow X$
is a projective cover in ${\mathcal{A}}$ and $i^{*}X=0$ then
$j^{*}P\rightarrow j^{*}X$ is a projective cover in ${\mathcal{A}}_{U}$.

In particular, if ${\mathcal{A}}$ has enough projectives then so does
${\mathcal{A}}_{U}$.

###### Proof.

Let $\phi:X\rightarrow Y$ be an essential surjection in ${\mathcal{A}}$ and
suppose $i^{*}Y=0$. To show that $i^{*}X=0$ it suffices to show that the
canonical map $\epsilon_{X}:j_{!}j^{*}X\rightarrow X$ is surjective. This
follows from the following commutative diagram since $\phi$ is essential.

${X}$${Y}$${{j_{!}j^{*}X}}$${{j_{!}j^{*}Y}}$$\scriptstyle{j_{!}j^{*}\phi}$$\scriptstyle{\epsilon_{X}}$$\scriptstyle{\epsilon_{Y}}$$\scriptstyle{\phi}$

Let $\alpha:X^{\prime}\rightarrow j^{*}X$ be a morphism in
${\mathcal{A}}_{U}$, in which $j^{*}(\phi)\circ\alpha:X^{\prime}\rightarrow
j^{*}Y$ is surjective. Then $\epsilon_{Y}\circ j_{!}j^{*}(\phi)\circ
j_{!}\alpha:j_{!}X^{\prime}\rightarrow Y$ is surjective and so (since $\phi$
is essential) $\epsilon_{X}\circ j_{!}\alpha:j_{!}X^{\prime}\rightarrow X$ is
surjective. Hence $j^{*}(\epsilon_{X}\circ
j_{!}\alpha)\simeq\alpha:X^{\prime}\rightarrow j^{*}X$ is surjective. This
proves (i).

If $P\in{\mathcal{A}}$ is projective and $i^{*}P=0$ then the functor

$\operatorname{Hom}_{{\mathcal{A}}_{U}}(j^{*}P,-)\simeq\operatorname{Hom}_{{\mathcal{A}}}(j_{!}j^{*}P,j_{!}(-))\simeq\operatorname{Hom}_{{\mathcal{A}}}(P,j_{!}(-)):{\mathcal{A}}_{U}\rightarrow{\mathbb{Z}}\text{-}\mathrm{mod}$

is exact. Here the last isomorphism follows from the sequence (2.6). It
follows that $j^{*}P$ is projective. Statement (ii) follows.

∎

######  Proposition 4.6.

Suppose ${\mathcal{A}}$ is a Krull-Schmidt category with a recollement of
abelian categories as in (3.1). If $P\rightarrow L$ is a projective cover in
${\mathcal{A}}$ of a simple object $L\in{\mathcal{A}}^{Z}$, then
$i^{*}P\rightarrow i^{*}L$ is a projective cover in ${\mathcal{A}}_{Z}$. In
particular, if ${\mathcal{A}}$ has enough projectives then so does
${\mathcal{A}}_{Z}$.

###### Proof.

Since $i^{*}$ is the left adjoint of an exact functor it preserves projective
objects. For any simple object $L^{\prime}\in{\mathcal{A}}^{Z}$,
$\operatorname{Hom}_{{\mathcal{A}}_{Z}}(i^{*}P,i^{*}L^{\prime})=\operatorname{Hom}_{{\mathcal{A}}}(P,i_{*}i^{*}L^{\prime})=\operatorname{Hom}_{{\mathcal{A}}}(P,L^{\prime})$.
The result follows. ∎

######  Proposition 4.7.

Let ${\mathcal{A}}$ be an abelian category with a recollement of abelian
categories as in (3.1). Let $X$ and $Y$ be objects in ${\mathcal{A}}_{U}$. If
$X\rightarrow Y$ is an essential surjection in ${\mathcal{A}}_{U}$ then the
composition $j_{!}X\rightarrow j_{!*}X\rightarrow j_{!*}Y$ is an essential
surjection in ${\mathcal{A}}$. In particular:

  1. (i)

The canonical surjection $j_{!}X\rightarrow j_{!*}X$ is essential.

  2. (ii)

If $P\rightarrow X$ is a projective cover of $X$ in ${\mathcal{A}}_{U}$ then
$j_{!}P\rightarrow j_{!}X\rightarrow j_{!*}X$ is a projective cover of
$j_{!*}X$ in ${\mathcal{A}}$.

###### Proof.

Let $\phi:X\rightarrow Y$ be an essential surjection in ${\mathcal{A}}_{U}$.
The map $\phi^{\prime}:j_{!}X\rightarrow j_{!*}X\rightarrow j_{!*}Y$ is
surjective by Proposition 3.5(i). Let $\alpha:X^{\prime}\rightarrow X$ be a
morphism in which $\phi^{\prime}\circ\alpha$ is surjective. Now,
$j^{*}(\phi^{\prime})=\phi:X\rightarrow Y$ and since $j^{*}$ is exact,
$j^{*}(\phi^{\prime}\circ\alpha)=\phi\circ
j^{*}(\alpha):j^{*}X^{\prime}\rightarrow Y$ is surjective. Since $\phi$ is
essential it follows that $j^{*}(\alpha):j^{*}X^{\prime}\rightarrow X$ is
surjective in ${\mathcal{A}}_{U}$ and so
$j_{!}j^{*}(\alpha):j_{!}j^{*}X^{\prime}\rightarrow j_{!}X$ is surjective in
${\mathcal{A}}$. The surjectivity of $\alpha$ follows from the commutative
triangle

${{j_{!}j^{*}X^{\prime}}}$${{j_{!}X}}$${{X^{\prime}}}$$\scriptstyle{j_{!}j^{*}(\alpha)}$$\scriptstyle{\alpha}$

in which the downward arrow is the adjunction counit. ∎

The following result holds by an almost identical argument.

######  Proposition 4.8.

The intermediate-extension functor preserves essential surjections and
essential extensions.

The following is the main result of this section.

######  Theorem 4.9.

Let ${\mathcal{A}}$ be an abelian length category with finitely many simple
objects, and a recollement of abelian categories as in (3.1). If
${\mathcal{A}}$ is $\operatorname{Ext}^{1}$-finite then ${\mathcal{A}}$ has
enough projectives if and only if both ${\mathcal{A}}_{U}$ and
${\mathcal{A}}_{Z}$ have enough projectives. Dually if ${\mathcal{A}}^{op}$ is
$\operatorname{Ext}^{1}$-finite then ${\mathcal{A}}$ has enough injectives if
and only if both ${\mathcal{A}}_{U}$ and ${\mathcal{A}}_{Z}$ have enough
injectives.

Before giving the proof of this theorem we will explain one important
ingredient: the universal extension. Let $A,B$ be objects in an abelian
category ${\mathcal{A}}$ in which $\operatorname{End}_{{\mathcal{A}}}(B)$ is a
division ring and
$d:=\dim_{\operatorname{End}_{{\mathcal{A}}}(B)}\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,B)<\infty$.
We form the universal extension
${\mathcal{E}}\in\operatorname{Ext}^{1}_{{\mathcal{A}}}(A,B^{\oplus d})$ by
the following process. First let
$E_{1},\ldots,E_{d}\in\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,B)$ be an
$\operatorname{End}_{{\mathcal{A}}}(B)$-basis. The diagonal map
$\Delta:A\rightarrow A^{\oplus d}$ induces a map
$\operatorname{Ext}_{{\mathcal{A}}}^{1}(A^{\oplus d},B^{\oplus
d})\rightarrow\operatorname{Ext}^{1}_{{\mathcal{A}}}(A,B^{\oplus d})$. Let
${\mathcal{E}}$ be the image of $E_{1}\oplus\cdots\oplus E_{d}$ under this
map. Note that the $\operatorname{End}_{{\mathcal{A}}}(B)$-equivariant map
$\operatorname{Hom}_{{\mathcal{A}}}(B^{\oplus
d},B)\rightarrow\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,B)$ induced by the
short exact sequence

$0\rightarrow B^{\oplus d}\rightarrow{\mathcal{E}}\rightarrow A\rightarrow 0$

is surjective (this is easy to check on the basis of
$\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,B)$).

When $B_{1},\ldots,B_{n}$ are objects in ${\mathcal{A}}$ in which each ring
$\operatorname{End}_{{\mathcal{A}}}(B_{i})$ is a division ring and
$d_{i}:=\dim_{\operatorname{End}_{{\mathcal{A}}}(B_{i})}\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,B_{i})<\infty$,
we also talk about a universal extension
${\mathcal{E}}\in\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,\bigoplus_{i}B_{i}^{\oplus
d_{i}})$ constructed in the following way. Let
${\mathcal{E}}_{i}\in\operatorname{Ext}^{1}_{{\mathcal{A}}}(A,B_{i}^{\oplus
d_{i}})$ be a universal extension (as defined in the previous paragraph) and
define ${\mathcal{E}}$ to be the image of
${\mathcal{E}}_{1}\oplus\cdots\oplus{\mathcal{E}}_{n}$ under the map
$\operatorname{Ext}_{{\mathcal{A}}}^{1}(\bigoplus_{i}A,\bigoplus_{i}B^{\oplus
d_{i}})\rightarrow\operatorname{Ext}^{1}_{{\mathcal{A}}}(A,\bigoplus_{i}B^{\oplus
d_{i}})$ induced by the diagonal map $\Delta:A\rightarrow A^{\oplus d}$. Then
${\mathcal{E}}$ has the property that the short exact sequence

$0\rightarrow\bigoplus_{i}B^{\oplus d_{i}}\rightarrow{\mathcal{E}}\rightarrow
A\rightarrow 0$

induces a surjection
$\operatorname{Hom}_{{\mathcal{A}}}(\bigoplus_{i}B_{i}^{\oplus
d_{i}},B_{j})\rightarrow\operatorname{Ext}_{{\mathcal{A}}}^{1}(A,B_{j})$ for
each $j=1,\ldots,n$.

Dually, if $\operatorname{End}_{{\mathcal{A}}}(A)^{op}$ is a division ring and
$\dim_{\operatorname{End}_{{\mathcal{A}}}(A)^{op}}\operatorname{Ext}^{1}(A,B)<\infty$
then one can form a universal extension
${\mathcal{E}}^{\prime}\in\operatorname{Ext}_{{\mathcal{A}}}^{1}(A^{\otimes
d},B)$ using the codiagonal map $\delta:B^{\oplus d}\rightarrow B$ instead of
the diagonal map.

###### Proof of Theorem 4.9.

Suppose that ${\mathcal{A}}_{U}$ and ${\mathcal{A}}_{Z}$ have enough
projectives.

Suppose ${\mathcal{A}}^{Z}$ has simple objects $L_{1},\ldots,L_{m}$ with
projective covers $\bar{P}_{1},\ldots,\bar{P}_{m}$ in ${\mathcal{A}}^{Z}$.
Suppose ${\mathcal{A}}^{U}$ has simple objects $L_{m+1},\ldots,L_{m+n}$. By
Proposition 4.7 every simple object in ${\mathcal{A}}^{U}$ has a projective
cover in ${\mathcal{A}}$. It suffices to construct a projective cover in
${\mathcal{A}}$ of each simple object in ${\mathcal{A}}^{Z}$. This amounts to
finding, for each $1\leq t\leq m$, a projective object, $P_{t}$, whose unique
simple quotient is $L_{t}$.

Fix $1\leq t\leq m$.

Step 1. Define $P_{t}$. For simple object $L_{m+k}\in{\mathcal{A}}^{U}$, let
$P_{m+k}$ denote its projective cover in ${\mathcal{A}}$. Define $Q$ to be a
maximal length quotient of

$P:=\bigoplus_{k=1}^{n}P_{m+k}^{\oplus\dim_{\operatorname{End}_{{\mathcal{A}}}(L_{m+k})}\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L_{m+k})}$

in which there is an extension

(4.2) $0\rightarrow Q\rightarrow{\mathcal{E}}\rightarrow\bar{P}_{t}\rightarrow
0$

inducing an isomorphism
$\operatorname{Hom}_{{\mathcal{A}}}(Q,L)\simeq\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L)$
for each $L\in{\mathcal{A}}^{U}$. That is
$0=\operatorname{Hom}_{{\mathcal{A}}}(\bar{P}_{t},L)\simeq\operatorname{Hom}_{{\mathcal{A}}}({\mathcal{E}},L)$
and $\operatorname{Ext}_{{\mathcal{A}}}^{1}({\mathcal{E}},L)$ injects into
$\operatorname{Ext}_{{\mathcal{A}}}^{1}(Q,L)$.

Let $P_{t}$ be any choice of such ${\mathcal{E}}$.

Step 2. $P_{t}$ is well-defined. To show that the maximal quotient $Q$ exists,
we just need to find one quotient of $P$ with the required property. Then
since ${\mathcal{A}}$ is a length category there exists a maximal length
quotient with the required property. Since ${\mathcal{A}}$ has finite
$\operatorname{Ext}^{1}$-spaces, we can let

$R=\bigoplus_{k=1}^{n}L_{m+k}^{\oplus\dim_{\operatorname{End}_{{\mathcal{A}}}(L_{m+k})}\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L_{m+k})}$

and form the universal extension

$0\rightarrow R\rightarrow{\mathcal{E}}\rightarrow\bar{P}_{t}\rightarrow 0$

Since this is a universal extension it induces a surjection
$\operatorname{Hom}_{{\mathcal{A}}}(R,L_{m+k})\rightarrow\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L_{m+k})$
for each $L_{m+k}\in{\mathcal{A}}^{U}$. This map is an isomorphism since

$\dim_{\operatorname{End}_{{\mathcal{A}}}(L_{m+k})}\operatorname{Hom}_{{\mathcal{A}}}(R,L_{m+k})=\dim_{\operatorname{End}_{{\mathcal{A}}}(L_{m+k})}\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L_{m+k}).$

Step 3. $P_{t}$ has unique simple quotient $L_{t}$. By definition of $P_{t}$
and by the $(i^{*},i_{*})$-adjunction, for any simple module
$L\in{\mathcal{A}}$:

$\operatorname{Hom}_{{\mathcal{A}}}(P_{t},L)\simeq\operatorname{Hom}_{{\mathcal{A}}}(\overline{P}_{t},L)\simeq\operatorname{Hom}_{{\mathcal{A}}}(\overline{P}_{t},i_{*}i^{*}L)$

and so the only simple quotient of $P_{t}$ is $L_{t}$ with multiplicity one.

Step 4. $P_{t}$ is projective. We show that
$\operatorname{Ext}^{1}_{{\mathcal{A}}}(P_{t},L)=0$ for each simple
$L\in{\mathcal{A}}$. For any simple $L\in{\mathcal{A}}$ there is an exact
sequence

(4.3)
$0\rightarrow\operatorname{Ext}^{1}_{{\mathcal{A}}}(P_{t},L)\rightarrow\operatorname{Ext}^{1}_{{\mathcal{A}}}(Q,L)\rightarrow\operatorname{Ext}^{2}_{{\mathcal{A}}}(\bar{P}_{t},L)$

Indeed if $L\in{\mathcal{A}}^{U}$ then this holds because
$\operatorname{Hom}_{{\mathcal{A}}}(Q,L)\simeq\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L)$.
If $L\in{\mathcal{A}}^{Z}$ then (4.3) holds since
$\operatorname{Ext}_{{\mathcal{A}}}^{1}(\bar{P}_{t},L)=0$.

To show that $P_{t}$ is projective it suffices to show that the third map in
(4.3) is injective for any $L\in{\mathcal{A}}$. Suppose for contradiction that
there is a nontrivial extension

(4.4) $0\rightarrow L\rightarrow Q^{\prime}\rightarrow Q\rightarrow 0$

in the kernel of this map. Then there is an object
${\mathcal{E}}\in{\mathcal{A}}$ fitting into the following diagram

(4.5)
${0}$${0}$${0}$${0}$${L}$${{Q^{\prime}}}$${{Q}}$${0}$${0}$${L}$${{\mathcal{E}}}$${{P_{t}}}$${0}$${0}$${0}$${{\bar{P}_{t}}}$${{\bar{P}_{t}}}$${0}$${0}$${0}$${0}$

in which each row and column is exact. For each
$L^{\prime}\in{\mathcal{A}}^{U}$ the sequence (4.4) induces an exact sequence

$0\rightarrow\operatorname{Hom}_{{\mathcal{A}}}(Q,L^{\prime})\rightarrow\operatorname{Hom}_{{\mathcal{A}}}(Q^{\prime},L^{\prime})\rightarrow\operatorname{Hom}_{{\mathcal{A}}}(L,L^{\prime})$

Of course, $\operatorname{Hom}_{{\mathcal{A}}}(L,L^{\prime})=0$ if $L\neq
L^{\prime}$. If $L=L^{\prime}$ the third map must be zero. Indeed if
$f:L\rightarrow L$ factors through the inclusion $\iota:L\rightarrow
Q^{\prime}$ via a map $g:Q^{\prime}\rightarrow L$, then $f^{-1}\circ
g:Q^{\prime}\rightarrow L$ is a retraction of $\iota$. This contradicts the
assumption that (4.4) does not split. Hence, for any
$L^{\prime}\in{\mathcal{A}}^{U}$, there is an isomorphism

(4.6)
$\operatorname{Hom}_{{\mathcal{A}}}(Q^{\prime},L^{\prime})\simeq\operatorname{Hom}_{{\mathcal{A}}}(Q,L^{\prime})\simeq\operatorname{Ext}^{1}_{{\mathcal{A}}}(\bar{P}_{t},L^{\prime}).$

Since $P$ is projective the quotient $P\rightarrow Q$ fits into the diagram

${P}$${{Q^{\prime}}}$${{Q}}$$\scriptstyle{\varphi}$

Now $\varphi$ cannot be surjective, as, by (4.6), this would contradict the
maximality of $Q$. Thus the image of $\varphi$ is isomorphic to $Q$ and so the
sequence (4.4) splits. This is a contradiction. Hence $P_{t}$ is projective. ∎

######  Corollary 4.10.

Let ${\mathcal{A}}$ be an abelian category with a stratification in which
every strata category is a length category, and has finitely many simple
objects.

If ${\mathcal{A}}$ is $\operatorname{Ext}^{1}$-finite (respectively
${\mathcal{A}}^{op}$ is $\operatorname{Ext}^{1}$-finite) then ${\mathcal{A}}$
has enough projectives (respectively injectives) if and only if every strata
category has enough projectives (respectively injectives).

###### Proof.

By Proposition 3.6, every category ${\mathcal{A}}_{\Lambda^{\prime}}$ (for
lower $\Lambda^{\prime}\subset\Lambda$) satisfies the conditions of Theorem
4.9. So we can obtain a projective cover in ${\mathcal{A}}$ of any simple
object $j^{\lambda}_{!*}L$ by repeatedly applying the construction in the
proof of Theorem 4.9 to larger and larger Serre subcategories of
${\mathcal{A}}$. ∎

The following result follows immediately from Proposition 4.4 and Corollary
4.10.

######  Corollary 4.11.

Let ${\mathcal{A}}$ be a $\Bbbk$-linear abelian category with a stratification
by a finite poset. Then ${\mathcal{A}}$ is finite over $\Bbbk$ if and only if
the same is true of every strata category.

From this result we recover the following result of Cipriani-Woolf.

######  Corollary 4.12 (Corollary 5.2 of [CW22]).

Let $P_{\Lambda}(X,\Bbbk)$ be the category of perverse sheaves (with
coefficients in a field $\Bbbk$) on a variety $X$ that is constructible with
respect to a stratification, $X=\bigcup_{\lambda\in\Lambda}X_{\lambda}$, with
finitely many strata. Then $P_{\Lambda}(X,\Bbbk)$ is finite over $\Bbbk$ if
and only if each category
$\Bbbk[\pi_{1}(X_{\lambda})]\text{-}\mathrm{mod}_{fg}$ is finite over $\Bbbk$.

For example, if $X$ has a finite stratification,
$X=\bigcup_{\lambda\in\Lambda}X_{\lambda}$, in which each $X_{\lambda}$ has
finite fundamental group, then the category $P_{\Lambda}(X,\Bbbk)$ is finite
over $\Bbbk$.

######  Corollary 4.13.

Let $G$ be an algebraic group and let $X$ be a $G$-variety with finitely many
orbits, each connected. Let $\Bbbk$ be a field. The category,
$P_{G}(X,\Bbbk)$, of $G$-equivariant perverse sheaves is finite over $\Bbbk$
if and only if for each $G$-orbit ${\mathcal{O}}_{\lambda}$ and
$x\in{\mathcal{O}}_{\lambda}$, the category
$\Bbbk[G^{x}/(G^{x})^{\circ}]\text{-}\mathrm{mod}_{fg}$ is finite over
$\Bbbk$.

##  5\. Standard and costandard objects

In this section we focus on abelian length categories ${\mathcal{A}}$ with
finitely many simples, enough projectives and injectives, and admitting a
stratification by a finite non-empty poset $\Lambda$. Let $B$ be a set
indexing the simple objects of ${\mathcal{A}}$ up to isomorphism. Write $L(b)$
for the simple object corresponding to $b\in B$, and write $P(b)$ and $I(b)$
for the projective cover and injective envelope of $L(b)$.

For each $\lambda\in\Lambda$, write
${\mathcal{A}}_{\leq\lambda}:={\mathcal{A}}_{\\{\mu\in\Lambda~{}|~{}\mu\leq\lambda\\}}$
and let $j^{\lambda}_{!}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ be
the composition

${{{\mathcal{A}}_{\lambda}}}$${{{\mathcal{A}}_{\leq\lambda}}}$${{\mathcal{A}}}$$\scriptstyle{j_{!}}$

Define $j^{\lambda}_{*}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ and
$j^{\lambda}_{!*}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ likewise.
Let
$j^{\lambda}:{\mathcal{A}}_{\leq\lambda}\rightarrow{\mathcal{A}}_{\lambda}$
denote the Serre quotient functor.

Define the stratification function $\rho:B\rightarrow\Lambda$ that assigns to
each $b\in B$ the $\lambda\in\Lambda$ in which
$L(b)=j_{!*}^{\lambda}L_{\lambda}(b)$. Let $P_{\lambda}(b)$ and
$I_{\lambda}(b)$ be the projective cover and injective envelope of the simple
object $L_{\lambda}(b)$ in ${\mathcal{A}}_{\lambda}$.

For $b\in B$, define the standard and costandard objects

$\Delta(b):=j^{\lambda}_{!}P_{\lambda}(b),\qquad\nabla(b):=j^{\lambda}_{*}I_{\lambda}(b),$

where $\lambda=\rho(b)$.

The following original result follows from the proofs of Theorem 4.9 and
Corollary 4.10.

######  Porism 5.1.

Let ${\mathcal{A}}$ be an abelian category with a stratification by a finite
non-empty poset $\Lambda$, in which every strata category is a length category
with finitely many simple objects. Let $\rho:B\rightarrow\Lambda$ be the
stratification function for ${\mathcal{A}}$.

If ${\mathcal{A}}$ has enough projectives then for each $\lambda\in\Lambda$
and $b\in\rho^{-1}(\lambda)$, the projective indecomposable object
$P(b)\in{\mathcal{A}}$ fits into a short exact sequence

$0\rightarrow Q(b)\rightarrow P(b)\rightarrow\Delta(b)\rightarrow 0$

in which $Q(b)$ has a filtration by quotients of $\Delta(b^{\prime})$
satisfying $\rho(b^{\prime})>\rho(b)$.

If ${\mathcal{A}}$ has enough injectives then for each $\lambda\in\Lambda$ and
$b\in\rho^{-1}(\lambda)$, the injective indecomposable object
$I(b)\in{\mathcal{A}}$ fits into a short exact sequence

$0\rightarrow\nabla(b)\rightarrow I(b)\rightarrow Q^{\prime}(b)\rightarrow 0$

in which $Q^{\prime}(b)$ has a filtration by subobjects of
$\nabla(b^{\prime})$ satisfying $\rho(b^{\prime})>\rho(b)$.

###### Proof.

We just prove the first statement by induction on $|\Lambda|$. The base case
$|\Lambda|=1$ is trivial.

Consider the projective cover, $P(b)$, of simple object $L(b)$ in
${\mathcal{A}}$. If $\rho(b)$ is maximal then $P(b)\simeq\Delta(b)$ and the
result holds. Suppose $\rho(b)$ is not maximal, and let $\mu\in\Lambda$ be a
maximal element. Consider the recollement

${{{\mathcal{A}}_{<\mu}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{\mu}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

Let $P_{<\mu}(b)$ and $\Delta_{<\mu}(b)$ be the projective indecomposable and
standard object in ${\mathcal{A}}_{<\mu}$ corresponding to the simple object
$i^{*}L(b)\in{\mathcal{A}}_{<\mu}$. By induction there is a short exact
sequence

$0\rightarrow Q_{<\mu}(b)\rightarrow
P_{<\mu}(b)\rightarrow\Delta_{<\mu}(b)\rightarrow 0$

in which $Q_{<\mu}(b)$ has a filtration by quotients of standard objects
$\Delta_{<\mu}(b^{\prime})$ satisfying $\rho(b^{\prime})>\rho(b)$. Since
$i_{*}$ is exact we get the following short exact sequence in ${\mathcal{A}}$:

(5.1) $0\rightarrow i_{*}Q_{<\mu}(b)\rightarrow
i_{*}P_{<\mu}(b)\rightarrow\Delta(b)\rightarrow 0$

and $i_{*}Q_{<\mu}(b)$ has a filtration by quotients of standard objects
$\Delta(b^{\prime})$ satisfying $\mu>\rho(b^{\prime})>\rho(b)$. By applying
the construction in Step 1 of the proof of Theorem 4.9, $P(b)$ fits into the
short exact sequence in ${\mathcal{A}}$:

(5.2) $0\rightarrow Q_{\mu}(b)\rightarrow P(b)\rightarrow
i_{*}P_{<\mu}(b)\rightarrow 0$

and $Q_{\mu}(b)$ is a quotient of a direct sum of standard objects of the form
$\Delta(b^{\prime})$ in which $\rho(b^{\prime})=\mu$. Combining (5.1) and
(5.2) gives the following diagram with exact rows and columns:

${0}$${0}$${0}$${0}$${Q_{\mu}(b)}$${Q_{\mu}(b)}$${0}$${0}$${0}$${Q(b)}$${P(b)}$${\Delta(b)}$${0}$${0}$${i_{*}Q_{<\mu}(b)}$${i_{*}P_{<\mu}(b)}$${\Delta(b)}$${0}$${0}$${0}$${0}$

The result follows.

∎

##  6\. Brundan and Stroppel’s $\varepsilon$-stratified categories

In this section we give necessary and sufficient conditions for a finite
abelian category with a stratification by a finite poset to be
$\varepsilon$-stratified in the sense of Brundan and Stroppel [BS18] (Theorem
6.4). This result specializes to a characterization of highest weight
categories originally given by Krause [Kra17, Theorem 3.4]. We recall this
characterisation in Corollary 6.6.

Let ${\mathcal{A}}$ be an abelian length category with finitely many simples,
enough projectives and injectives, and with a stratification by a finite non-
empty poset $\Lambda$. Let $\rho:B\rightarrow\Lambda$ be the stratification
function corresponding to this stratification.

For $b\in B$ and $\lambda=\rho(b)\in\Lambda$, define proper standard and
proper costandard objects

$\overline{\Delta}(b):=j^{\lambda}_{!}L_{\lambda}(b),\qquad\overline{\nabla}(b):=j^{\lambda}_{*}L_{\lambda}(b).$

For a sign function $\varepsilon:\Lambda\rightarrow\\{+,-\\}$, define the
$\varepsilon$-standard and $\varepsilon$-costandard objects

$\displaystyle\Delta_{\varepsilon}(b)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\Delta(b)&\text{if
$\varepsilon(\rho(b))=+$}\\\ \overline{\Delta}(b)&\text{if
$\varepsilon(\rho(b))=-$}\end{array}\right.,$
$\displaystyle\nabla_{\varepsilon}(b)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\overline{\nabla}(b)&\text{if
$\varepsilon(\rho(b))=+$}\\\ \nabla(b)&\text{if
$\varepsilon(\rho(b))=-$}\end{array}\right..$

Note that since $j_{!}^{\lambda}$ and $j_{*}^{\lambda}$ are fully-faithful,
these objects have local endomorphism rings and are hence irreducible. Note
also that if $\rho(b)>\rho(b^{\prime})$ then for any
$\varepsilon:\Lambda\rightarrow\\{+,-\\}$,

(6.1)
$\operatorname{Hom}_{{\mathcal{A}}}(\Delta_{\varepsilon}(b),\Delta_{\varepsilon}(b^{\prime}))=0=\operatorname{Hom}_{{\mathcal{A}}}(\nabla_{\varepsilon}(b^{\prime}),\nabla_{\varepsilon}(b)).$

Indeed the only simple quotient of $\Delta_{\varepsilon}(b)$ is $L(b)$, and
all simple subobjects, $L(b^{\prime\prime})$, of
$\Delta_{\varepsilon}(b^{\prime})$ satisfy
$\rho(b^{\prime})\geq\rho(b^{\prime\prime})$. Likewise the only simple
subobject of $\nabla_{\varepsilon}(b)$ is $L(b)$, and all simple quotients,
$L(b^{\prime\prime})$, of $\nabla_{\varepsilon}(b^{\prime})$ satisfy
$\rho(b^{\prime})\geq\rho(b^{\prime\prime})$.

The following definition is due to Brundan and Stroppel [BS18].444In Brundan
and Stoppel’s definition, $\varepsilon$-stratified categories need not be
finite (they satisfy slightly weaker finiteness conditions).

######  Definition 6.1 ($\varepsilon$-stratified category).

Let ${\mathcal{A}}$ be a finite abelian category with a stratification by a
poset $\Lambda$ and stratification function $\rho:B\rightarrow\Lambda$. Let
$\varepsilon:\Lambda\rightarrow\\{+,-\\}$ be a function. Say that
${\mathcal{A}}$ is an $\varepsilon$-stratified category if the following
equivalent conditions are satisfied:

  1. ($\varepsilon$-S1)

For every $b\in B$, the projective indecomposable $P(b)$ has a filtration by
objects of the form $\Delta_{\varepsilon}(b^{\prime})$, where
$\rho(b^{\prime})\geq\rho(b)$.

  2. ($\varepsilon$-S2)

For every $b\in B$, the injective indecomposable $I(b)$ has a filtration by
objects of the form $\nabla_{\varepsilon}(b^{\prime})$, where
$\rho(b^{\prime})\geq\rho(b)$.

The equivalence of statements ($\varepsilon$-S1) and ($\varepsilon$-S2) is
shown in [ADL98, Theorem 2.2]. A proof of this fact can also be found in
[BS18, Theorem 3.5].

######  Remark 6.2.

A finite abelian category $A\text{-}\mathrm{mod}$ is $\varepsilon$-stratified
if and only if $A$ is a stratified algebra in the sense of [ADL98].

The following definition is original.

######  Definition 6.3 ($\varepsilon$-exact stratification).

For a function $\varepsilon:\Lambda\rightarrow\\{+,-\\}$, say that a
stratification of an abelian category ${\mathcal{A}}$ by a finite non-empty
poset $\Lambda$ is $\varepsilon$-exact if for all $\lambda\in\Lambda$ the
following hold:

  1. (i)

If $\varepsilon(\lambda)=+$ then the functor
$j_{*}^{\lambda}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ is exact.

  2. (ii)

If $\varepsilon(\lambda)=-$ then the functor
$j_{!}^{\lambda}:{\mathcal{A}}_{\lambda}\rightarrow{\mathcal{A}}$ is exact.

For $k\in{\mathbb{N}}$, say that a recollement of abelian categories (as in
(3.1)) is $k$-homological if for all $n\leq k$ and $X,Y\in{\mathcal{A}}_{Z}$,

$\operatorname{Ext}^{n}_{{\mathcal{A}}_{Z}}(X,Y)\simeq\operatorname{Ext}^{n}_{{\mathcal{A}}}(i_{*}X,i_{*}Y).$

Say that a recollement of abelian categories is homological if it is
$k$-homological for all $k\in{\mathbb{N}}$. A study of homological
recollements is given in [Psa14].

Say that a stratification of an abelian category is $k$-homological
(respectively homological) if each of the recollements in the data of the
stratification is $k$-homological (respectively homological).

The following theorem is the main result of this section.

######  Theorem 6.4.

Let ${\mathcal{A}}$ be a finite abelian category. Then, for any finite non-
empty poset $\Lambda$ and function $\varepsilon:\Lambda\rightarrow\\{+,-\\}$,
the following statements are equivalent:

  1. (1)

${\mathcal{A}}$ has an $\varepsilon$-exact homological stratification by
$\Lambda$.

  2. (2)

${\mathcal{A}}$ has an $\varepsilon$-exact 2-homological stratification by
$\Lambda$.

  3. (3)

${\mathcal{A}}$ is $\varepsilon$-stratified.

To prove this theorem we need the following lemma.

######  Lemma 6.5.

Let ${\mathcal{A}}$ be an abelian length category with finitely many simple
objects, and fitting into a 2-homological recollement of abelian categories as
in (3.1).

If ${\mathcal{A}}$ has enough projectives then for any projective object
$P\in{\mathcal{A}}$, there is a short exact sequence

$0\rightarrow j_{!}j^{*}P\rightarrow P\rightarrow i_{*}i^{*}P\rightarrow 0$

If ${\mathcal{A}}$ has enough injectives then for any injective object
$I\in{\mathcal{A}}$, there is a short exact sequence

$0\rightarrow i_{*}i^{!}I\rightarrow I\rightarrow j_{*}j^{*}I\rightarrow 0$

###### Proof.

Let $P\in{\mathcal{A}}$ be a projective indecomposable object in
${\mathcal{A}}$. If $i^{*}P=0$ then $j_{!}j^{*}P\simeq P$ and the result
holds.

Suppose that $i^{*}P\neq 0$. Consider the exact sequence

$0\rightarrow K\rightarrow j_{!}j^{*}P\rightarrow P\rightarrow
i_{*}i^{*}P\rightarrow 0$

where $K\in{\mathcal{A}}^{Z}$. Since $i^{*}P$ is projective in
${\mathcal{A}}_{Z}$ and the recollement is 2-homological,
$\operatorname{Ext}^{2}_{{\mathcal{A}}}(i_{*}i^{*}P,K)=0$. In particular,
there is an object ${\mathcal{E}}\in{\mathcal{A}}$ and surjection
$q:{\mathcal{E}}\rightarrow P$ that fits into the following diagram in which
all rows and columns are exact.

(6.2)
${0}$${0}$${0}$${0}$${K}$${{j_{!}j^{!}P}}$${{Q}}$${0}$${0}$${K}$${{\mathcal{E}}}$${P}$${0}$${0}$${0}$${{i_{*}i^{*}P}}$${{i_{*}i^{*}P}}$${0}$${0}$${0}$${0}$$\scriptstyle{q}$

Since $P$ is projective, there is a map $\alpha:P\rightarrow{\mathcal{E}}$
making the following diagram commute.

${P}$${{\mathcal{E}}}$${P}$${{i_{*}i^{*}P}}$$\scriptstyle{\alpha}$$\scriptstyle{q}$

Since $P$ is indecomposable, the endomorphism $q\circ\alpha:P\rightarrow P$ is
either nilpotent or an automorphism. Since $i_{*}i^{*}P$ is nonzero, this map
is not nilpotent, and hence is an automorphism of $P$. In particular, the
first two rows of Diagram (6.2) split. Hence $K$ is a quotient of
$j_{!}j^{!}P$. Since $j_{!}j^{!}P$ has no nonzero quotient in
${\mathcal{A}}^{Z}$, $K=0$. The first result follows.

The second result holds by the dual argument. ∎

###### Proof of Theorem 6.4.

The implication $(1)\implies(2)$ is obvious.

For the remainder of the proof, fix a maximal element $\lambda\in\Lambda$ and
recollement

(6.3)
${{{\mathcal{A}}_{<\lambda}}}$${{\mathcal{A}}}$${{{\mathcal{A}}_{\lambda}}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

$(2)\implies(3)$. Let $P(b)\in{\mathcal{A}}$ be a projective indecomposable
object. Suppose that every projective indecomposable, $P(b^{\prime})$, in
${\mathcal{A}}_{<\lambda}$ has a filtration by $\varepsilon$-standard objects,
$\Delta_{\varepsilon}(b^{\prime\prime})$, in which
$\rho(b^{\prime\prime})\geq\rho(b^{\prime})$. In particular $i_{*}i^{*}P(b)$
has a filtration by $\varepsilon$-standard objects,
$\Delta_{\varepsilon}(b^{\prime})$, in which $\rho(b^{\prime})\geq\rho(b)$.

By Lemma 6.5, if the recollement (6.3) is 2-homological then $P(b)$ fits into
a short exact sequence

$0\rightarrow j_{!}j^{*}P(b)\rightarrow P(b)\rightarrow
i_{*}i^{*}P(b)\rightarrow 0$

If $j_{*}$ is exact then $j^{*}P(b)$ is projective (since $j^{*}$ is the left
adjoint of an exact functor) and so $j_{!}j^{*}P(b)$ has a filtration by
standard objects. If $j_{!}$ is exact then $j_{!}j^{*}P(b)$ has a filtration
by proper standard objects.

In particular, in either case $\varepsilon(\lambda)=\pm$, $P(b)$ has a
filtration by $\varepsilon$-standard objects,
$\Delta_{\varepsilon}(b^{\prime})$, in which $\rho(b^{\prime})\geq\rho(b)$.
The result follows by induction on $|\Lambda|$.

$(3)\implies(1)$. Brundan-Stroppel show that if ${\mathcal{A}}$ is
$\varepsilon$-stratified then ${\mathcal{A}}$ has an $\varepsilon$-exact
stratification [BS18, Theorem 3.5] and

(6.4)
$\operatorname{Ext}^{n}_{{\mathcal{A}}}(\Delta_{\varepsilon}(b),\nabla_{\varepsilon}(b^{\prime}))=0$

for all $b,b^{\prime}\in B$ and $n\in{\mathbb{N}}$ [BS18, Theorem 3.14].

Let ${\mathcal{A}}$ be $\varepsilon$-stratified abelian length category. Let
$P$ be a projective object in ${\mathcal{A}}_{<\lambda}$, and $I$ be an
injective object in ${\mathcal{A}}_{<\lambda}$. Then $P$ and $i_{*}P$ have a
filtration by $\varepsilon$-standard objects, and $I$ and $i_{*}I$ have a
filtration by $\varepsilon$-costandard objects. Since ${\mathcal{A}}$ is a
length category, then it follows from (6.4) that
$\operatorname{Ext}^{n}_{{\mathcal{A}}}(i_{*}P,i_{*}I)=0$. Since this is true
for any projective object $P$ and injective object $I$ in
${\mathcal{A}}_{<\lambda}$, the recollement (6.3) is homological (see e.g.
[Psa14, Theorem 3.9]). The result follows by induction on $|\Lambda|$.

∎

###  6.1. Highest weight categories

Let $\Bbbk$ be a field. Say that a $\Bbbk$-linear abelian category
${\mathcal{A}}$ is a highest weight category with respect to a finite poset
$\Lambda$ if ${\mathcal{A}}$ is finite over $\Bbbk$, and for every
$\lambda\in\Lambda$ there is a projective indecomposable, $P_{\lambda}$, that
fits into a short exact sequence in ${\mathcal{A}}$:

$0\rightarrow U_{\lambda}\rightarrow
P_{\lambda}\rightarrow\Delta_{\lambda}\rightarrow 0$

in which the following hold:

  1. (HW1)

$\operatorname{End}_{{\mathcal{A}}}(\Delta_{\lambda})$ is a division ring for
all $\lambda\in\Lambda$.

  2. (HW2)

$\operatorname{Hom}_{{\mathcal{A}}}(\Delta_{\lambda},\Delta_{\mu})=0$ whenever
$\lambda>\mu$.

  3. (HW3)

$U_{\lambda}$ has a filtration by objects $\Delta_{\mu}$ in which
$\lambda<\mu$.

  4. (HW4)

$\bigoplus_{\lambda\in\Lambda}P_{\lambda}$ is a projective generator of
${\mathcal{A}}$.

The following characterisation of highest weight categories is shown by Krause
[Kra17, Theorem 3.4]. We give a new proof using Theorem 6.4.

######  Corollary 6.6.

Let $\Bbbk$ be a field, and let ${\mathcal{A}}$ be a $\Bbbk$-linear abelian
category. The following statements are equivalent.

  1. (1)

${\mathcal{A}}$ is a highest weight category.

  2. (2)

${\mathcal{A}}$ has a homological stratification with respect to $\Lambda$ in
which every strata category is equivalent to
$\mathrm{mod}\text{-}\Gamma_{\lambda}$ for some finite dimensional division
algebra $\Gamma_{\lambda}$.

  3. (3)

${\mathcal{A}}$ has a 2-homological stratification with respect to $\Lambda$
in which every strata category is equivalent to
$\mathrm{mod}\text{-}\Gamma_{\lambda}$ for some finite dimensional division
algebra $\Gamma_{\lambda}$.

###### Proof.

$(1)\implies(2)$. If ${\mathcal{A}}$ is a highest weight category with respect
to $\Lambda$, then a homological stratification of ${\mathcal{A}}$ by
$\Lambda$ is constructed as follows: For a lower subposet
$\Lambda^{\prime}\subset\Lambda$ define ${\mathcal{A}}_{\Lambda^{\prime}}$ to
be the Serre subcategory of ${\mathcal{A}}$ generated by the standard objects
$\Delta_{\lambda}$ in which $\lambda\in\Lambda^{\prime}$. Then for any maximal
$\mu\in\Lambda^{\prime}$ there is a homological recollement of abelian
categories

${{{\mathcal{A}}_{\Lambda^{\prime}\setminus\\{\mu\\}}}}$${{\mathcal{A}}_{\Lambda^{\prime}}}$${{\mathrm{mod}\text{-}\operatorname{End}_{{\mathcal{A}}}(\Delta_{\lambda})}}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{j^{*}}$$\scriptstyle{j_{!}}$$\scriptstyle{j_{*}}$

in which
$j^{*}=\operatorname{Hom}_{{\mathcal{A}}_{\Lambda^{\prime}}}(\Delta_{\mu},-)$
(see the proof of [Kra17, Theorem 3.4]).

$(2)\implies(3).$ This is obvious.

$(3)\implies(1).$ Suppose ${\mathcal{A}}$ has a 2-homological stratification
with strata categories of the form
${\mathcal{A}}_{\lambda}=\mathrm{mod}\text{-}\Gamma_{\lambda}$ for a finite
dimensional division ring $\Gamma_{\lambda}$. Then ${\mathcal{A}}$ is finite
(by Corollary 4.11). Let $L_{\lambda}$ denote the unique simple object in
${\mathcal{A}}_{\lambda}$. Define
$\Delta_{\lambda}:=j_{!}^{\lambda}L_{\lambda}$ and let $P_{\lambda}$ be the
projective cover of $j_{!*}^{\lambda}L_{\lambda}$ in ${\mathcal{A}}$.
Statement (HW1) holds since $j_{!}$ is fully-faithful, Statement (HW2) is
exactly equation (6.1), Statement (HW3) follows from Theorem 6.4 (since each
$j_{!}^{\lambda}$ is exact), and Statement (HW4) is obvious. ∎

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