# Homological theory of Serre quotient

Ramin Ebrahimi  Department of Pure Mathematics
Faculty of Mathematics and Statistics
University of Isfahan
P.O. Box: 81746-73441, Isfahan, Iran
and School of Mathematics, Institute for Research in Fundamental Sciences
(IPM), P.O. Box: 19395-5746, Tehran, Iran<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>
###### Abstract.

Inspired by the work of C. Psaroudakis, for an abelian category $\mathcal{A}$
and a Serre subcategory $\mathcal{C}$, we investigate homological aspects of
the quotient category $\frac{\mathcal{A}}{\mathcal{C}}$. The existence of
section functor and injective objects are not assumed.

###### Key words and phrases:

abelian category, Serre subcategory

###### 2010 Mathematics Subject Classification:

18E10, 18E20, 18E99

##  1\. Introduction

Let $\mathcal{A}$ be an abelian category. A Serre subcategory
$\mathcal{C}\subseteq\mathcal{A}$ is a full subcategory that is closed under
subobject, extension and quotients. In this case there exist an abelian
category $\dfrac{\mathcal{A}}{\mathcal{C}}$ and an exact functor

$q:\mathcal{A}\rightarrow\dfrac{\mathcal{A}}{\mathcal{C}}$

that is universal among exact functors from $\mathcal{A}$ to an abelian
category that annihilate $\mathcal{C}$. the quotient category
$\dfrac{\mathcal{A}}{\mathcal{C}}$ was defined by Grothendieck and studied
extensively by Gabriel.

The quotients functor $q$ is not full or faithful, but by a result of Gabriel
for each object $A$ in the left perpendicular of $\mathcal{C}$, the natural
map
$q_{X,A}:\operatorname{Hom}_{\mathcal{A}}(X,A)\rightarrow\operatorname{Hom}_{\frac{\mathcal{A}}{\mathcal{C}}}(q(X),q(A))$
is invertible, for every $X\in\mathcal{A}$.

In many situations $\mathcal{A}$ is a known and well behaved category like a
module category or a functor category and we can compute
$\operatorname{Ext}$-groups for any two objects in $\mathcal{A}$. But the
quotient category $\dfrac{\mathcal{A}}{\mathcal{C}}$ has a complicated
structure, for instance if $\mathcal{C}$ is a localizing subcategory (i.e. $q$
has a right adjoint ) then $\dfrac{\mathcal{A}}{\mathcal{C}}$ is equivalence
to a subcategory of $\mathcal{A}$, but not an exact subcategory. Thus it is
hard to relate the $\operatorname{Ext}$-groups of this two category.

The main theorem of this paper is the following theorem.

######  Theorem 1.1.

Let $\mathcal{A}$ be an abelian category with enough projective, $\mathcal{C}$
a Serre subcategory of $\mathcal{A}$, $k$ be a positive integer, and
$q:\mathcal{A}\rightarrow\frac{\mathcal{A}}{\mathcal{C}}$ be the quotient
functor. For an object $A\in\mathcal{A}$ the following conditions are
equivalence.

  * (i)

$A\in\mathcal{C}^{{\bot}_{k+1}}$. i.e
$\operatorname{Ext}_{\mathcal{A}}^{i}(\mathcal{C},A)=0$ for every $0\leq i\leq
k+1$.

  * (ii)

The natural map
$q^{i}_{X,A}:\operatorname{Ext}_{\mathcal{A}}^{i}(X,A)\rightarrow\operatorname{Ext}_{\frac{\mathcal{A}}{\mathcal{C}}}^{i}(q(X),q(A))$
is invertible, for every $X\in\mathcal{A}$ and $0\leq i\leq k$.

Note that C. Psaroudakis prove this result for a recollement of abelian
categories such that have enough projective or injective objects [13]. In
other word we prove that the existence of section(cosection) functor and
injective objects is not necessary.

##  2\. preliminaries

In this section we recall some basic results about localization of abelian
categories and locally finitely presented categories.

###  2.1. Localization of abelian categories

In the following we recall localisation theory of abelian categories, for
reader can find proof in textbooks or Gabriel thesis [3]. Let $\mathcal{A}$ be
an abelian category. A subcategory $\mathcal{C}$ of $\mathcal{A}$ is called a
Serre subcategory if for any exact sequence

$0\rightarrow A_{1}\rightarrow A_{2}\rightarrow A_{3}\rightarrow 0$

we have that $A_{2}\in\mathcal{C}$ if and only if $A_{1}\in\mathcal{C}$ and
$A_{3}\in\mathcal{C}$. In this case we have the quotient category
$\dfrac{\mathcal{A}}{\mathcal{C}}$ that is by definition localisation of
$\mathcal{A}$ with respect to the class of all morphisms $f:X\rightarrow Y$
such that $\rm Ker(f),\rm Coker(f)\in\mathcal{C}$.
$\dfrac{\mathcal{A}}{\mathcal{C}}$ is an abelian category and there exist a
universal exact functor

$q:\mathcal{A}\longrightarrow\dfrac{\mathcal{A}}{\mathcal{C}}$

such that $q(C)=0$ for all $C\in\mathcal{C}$, and any exact functor
$F:\mathcal{A}\rightarrow\mathcal{D}$ annihilating $\mathcal{C}$ where
$\mathcal{D}$ is abelian must factor uniquely through $q$.

A Serre subcategory $\mathcal{C}\subseteq\mathcal{A}$ is called a localizing
subcategory if the canonical functor
$e:\mathcal{A}\rightarrow\dfrac{\mathcal{A}}{\mathcal{C}}$ admits a right
adjoint $r:\dfrac{\mathcal{A}}{\mathcal{C}}\rightarrow\mathcal{A}$. The right
adjoint $r$ is called the section functor, which always is fully faithful.
Note that a localising subcategory is closed under all coproducts which exist
in $\mathcal{A}$. Conversely a Serre subcategory $\mathcal{C}$ of a
Grothendieck category $\mathcal{A}$ wich is closed under coproducts is a
localizing subcategory, and in this case the quotient category
$\dfrac{\mathcal{A}}{\mathcal{C}}$ is also a Grothendieck category.

Let $\mathcal{C}$ be a Serre subcategory of an abelian category $\mathcal{A}$.
Recall that an object $A\in\mathcal{A}$ is called $\mathcal{C}$-closed if for
every morphism $f:X\rightarrow Y$ with $\operatorname{Ker}(f)\in\mathcal{C}$
and $\rm Coker(f)\in\mathcal{C}$ we have that
$\operatorname{Hom}_{\mathcal{A}}(f,A)$ is bijective. Denote by
$\mathcal{C}^{\bot}$ the full subcategory of all $\mathcal{C}$-closed objects,
the following result is well known.

######  Theorem 2.1.

Let $\mathcal{C}$ be a Serre subcategory of an abelian category $\mathcal{A}$.
The following statements hold:

  * $(i)$

We have

$\mathcal{C}^{\bot}=\\{A\in\mathcal{A}\mid\operatorname{Hom}(\mathcal{C},A)=0=\operatorname{Ext}^{1}(\mathcal{C},A)\\}.$

  * $(ii)$

For $B\in\mathcal{A}$ and $A\in\mathcal{C}^{\bot}$, the natural map
$q_{B,A}:\operatorname{Hom}_{\mathcal{A}}(B,A)\rightarrow\operatorname{Hom}_{\frac{\mathcal{A}}{\mathcal{C}}}(q(B),q(A))$
is invertible.

  * $(iii)$

If $\mathcal{C}$ is a localizing subcategory, the restriction
$q:\mathcal{C}^{\bot}\rightarrow\dfrac{\mathcal{A}}{\mathcal{C}}$ is an
equivalence of categories.

  * $(iv)$

If $\mathcal{C}$ is localising and $\mathcal{A}$ has injective envelopes, then
$\mathcal{C}^{\bot}$ has injective envelopes and the inclusion functor
$\mathcal{C}^{\bot}\hookrightarrow\mathcal{A}$ preserves injective envelopes.

The following definition is borrowed from [13].

######  Definition 2.2.

Let $\mathcal{A}$ be an abelian category and $\mathcal{X}$ be a subcategory of
$\mathcal{A}$. For a positive integer $k$ we denote by
$\mathcal{X}^{{\bot}_{k}}$ the full subcategory of $\mathcal{A}$ defined by
$\mathcal{X}^{{\bot}_{k}}=\\{A\in\mathcal{A}|\operatorname{Ext}^{0,...,k}(\mathcal{X},A)=0\\}$.
${}^{{\bot}_{k}}\mathcal{X}$ is defined similarly [13]. Note that by Theorem
2.1 for a Serre subcategory $\mathcal{C}$ of $\mathcal{A}$ we have
$\mathcal{C}^{{\bot}_{1}}=\mathcal{C}^{\bot}$.

Let $\mathcal{B}$ be an abelian category and $\mathcal{S}$ be a localizing
subcategory of $\mathcal{A}$. Denote the localization functor and the section
functor with $e$ and $r$ respectively, we have the following diagram of
functors.

$\mathcal{B}$$\frac{\mathcal{B}}{\mathcal{S}}$$r$$e$

######  Proposition 2.3.

Let $\mathcal{B}$ be a Grothendieck category, $\mathcal{S}$ a localizing
subcategory of $\mathcal{B}$, and $k$ be a positive integer. For an object
$A\in\mathcal{A}$ consider the following statements.

  * (i)

$A\in\mathcal{S}^{{\bot}_{k+1}}$.

  * (ii)

There exist an injective resolution

(2.1) $0\rightarrow A\rightarrow r(I^{0})\rightarrow
r(I^{1})\rightarrow\cdots\rightarrow r(I^{k+1})$

for $A$, where $I^{i}$’s are injective object in
$\frac{\mathcal{B}}{\mathcal{S}}$.

  * (iii)

The natural map
$e_{X,A}:\operatorname{Ext}_{\mathcal{B}}^{i}(X,A)\rightarrow\operatorname{Ext}_{\frac{\mathcal{B}}{\mathcal{S}}}^{i}(e(X),e(A))$
is invertible, for every $X\in\mathcal{B}$ and $0\leq i\leq k$.

Then (i)$\Rightarrow$(ii)$\Rightarrow$(iii). And if $\mathcal{B}$ has enough
projective, then (iii)$\Rightarrow$(i).

###### Proof.

(i)$\Rightarrow$(ii) Let $A\in\mathcal{S}^{{\bot}_{k+1}}$, and $A\rightarrow
I^{0}$ be the injective envelope of $A$ in $\mathcal{B}$. By applying the
functor $\operatorname{Hom}_{\mathcal{B}}(S,-)$ for an arbitrary object
$C\in\mathcal{S}$, to the short exact sequence

$0\rightarrow A\rightarrow I^{0}\rightarrow\Omega^{-1}A\rightarrow 0$

we obtain the exact sequence

$\displaystyle 0$
$\displaystyle\rightarrow\operatorname{Hom}_{\mathcal{B}}(S,A)\rightarrow\operatorname{Hom}_{\mathcal{B}}(S,I^{0})\rightarrow\operatorname{Hom}_{\mathcal{B}}(S,\Omega^{-1}A)$
$\displaystyle\rightarrow\operatorname{Ext}_{\mathcal{B}}^{1}(S,A)\rightarrow\operatorname{Ext}_{\mathcal{B}}^{1}(S,I^{0})\rightarrow\operatorname{Ext}_{\mathcal{B}}^{1}(S,\Omega^{-1}A)$
$\displaystyle\rightarrow\operatorname{Ext}_{\mathcal{B}}^{2}(S,A).$

By assumption
$\operatorname{Hom}_{\mathcal{B}}(S,A)=\operatorname{Ext}_{\mathcal{B}}^{1}(S,A)=\operatorname{Ext}_{\mathcal{B}}^{2}(S,A)=0$.
Because $I^{0}$ is an essential extension of $A$ we have that
$\operatorname{Hom}_{\mathcal{B}}(C,I^{0})=0$. Thus
$I^{0},\Omega^{-1}A\in\mathcal{S}^{\bot}$. By repeating this argument for
$\Omega^{-1}A$ and using the dimension shifting argument we obtain an
injective coresolution as (2.1).

(ii)$\Rightarrow$(iii) Apply the functor
$\operatorname{Hom}_{\mathcal{B}}(S,-)$ to the injective coresolution (2.1).
and use the fact that $(e,r)$ is a adjoint pair

(iii)$\Rightarrow$(i) Obviously $A\in\mathcal{S}^{\bot_{k}}$, so it remains to
show that $\operatorname{Ext}_{\mathcal{B}}^{k+1}(\mathcal{S},A)=0$. For an
arbitrary object $S\in\mathcal{S}$ consider exact sequence $0\rightarrow
B\rightarrow P\rightarrow S\rightarrow 0$ where $P$ is a projective object.
Because $P\cong B$ is $\dfrac{\mathcal{B}}{\mathcal{S}}$ we have

$\displaystyle\operatorname{Ext}_{\mathcal{B}}^{k+1}(S,A)$
$\displaystyle\cong\operatorname{Ext}_{\mathcal{B}}^{k}(B,A)$
$\displaystyle\cong\operatorname{Ext}_{\frac{\mathcal{B}}{\mathcal{S}}}^{k}(B,A)$
$\displaystyle\cong\operatorname{Ext}_{\frac{\mathcal{B}}{\mathcal{S}}}^{k}(P,A)$
$\displaystyle\cong\operatorname{Ext}_{\mathcal{B}}^{k}(P,A)=0.$

∎

Let $\mathcal{A}$ be an abelian category. Recall that
$\operatorname{Mod}\mathcal{A}$ is the category of all additive contravariant
functors from $\mathcal{A}$ to the category of all abelian groups. It is an
abelian category with all limits and colimits, which are defined point-wise. A
functor $F\in\operatorname{Mod}\mathcal{A}$ is called finitely presented(or
coherent) if there exist an exact sequence

$\operatorname{Hom}(-,X)\rightarrow\operatorname{Hom}(-,Y)\rightarrow
F\rightarrow 0$

in $\operatorname{Mod}\mathcal{A}$. We denote by
$\operatorname{mod}\mathcal{A}$ the full subcategory of
$\operatorname{Mod}\mathcal{A}$ consist of finitely presented functors. It is
a well known result that $\operatorname{mod}\mathcal{A}$ is an abelian
category because $\mathcal{A}$ has kernels. And the inclusion
$\operatorname{mod}\mathcal{A}\hookrightarrow\operatorname{Mod}\mathcal{A}$ is
an exact functor. By the Yoneda’s lemma, representable functors are projective
and the direct sum of all representable functors
$\bigoplus_{X\in\mathcal{M}}\operatorname{Hom}(-,X)$, is a generator for
$\operatorname{Mod}\mathcal{A}$. Thus $\operatorname{Mod}\mathcal{A}$ is a
Grothendieck category.

######  Definition 2.4.

  * (i)

A functor $F\in\operatorname{Mod}\mathcal{A}$ is called weakly effaceable, if
for each object $X\in\mathcal{A}$ and $x\in F(X)$ there exists an epimorphism
$f:Y\rightarrow X$ such that $F(f)(x)=0$. We denote by
$\operatorname{Eff}(\mathcal{A})$ the full subcategory of all weakly
effaceable functors.

  * (ii)

A functor $F\in\operatorname{mod}\mathcal{A}$ is called effaceable, if there
exist an exact sequence

$\operatorname{Hom}(-,Y)\rightarrow\operatorname{Hom}(-,X)\rightarrow
F\rightarrow 0$

such that $Y\rightarrow X$ is an epimorphism. We denote by
$\operatorname{eff}(\mathcal{A})$ the full subcategory of all effaceable
functors.

  * (i)

A functor $F\in\operatorname{Mod}\mathcal{A}$ is called a left exact functor,
if for each short exact sequence $0\rightarrow X\rightarrow Y\rightarrow
Z\rightarrow 0$ in $\mathcal{A}$ the sequence of abelian groups $0\rightarrow
F(Z)\rightarrow F(Y)\rightarrow F(X)$ is exact. We denote by
$\mathcal{L}(\mathcal{A})$ the subcategory of all left exact functors.

######  Proposition 2.5.

Let $\mathcal{A}$ be a small abelian category.

  * (i)

$\rm Eff(\mathcal{A})$ is a localizing subcategory of $\rm{Mod}\mathcal{A}$.

  * (ii)

We have

$\displaystyle\mathcal{L}(\mathcal{A})$
$\displaystyle=\operatorname{Eff}(\mathcal{A})^{\bot}\simeq\dfrac{\operatorname{Mod}\mathcal{A}}{\operatorname{Eff}(\mathcal{A})}.$

###### Proof.

This is an standard result, see for example [5]. ∎

We denote by $\mathsf{i}:\mathcal{A}\rightarrow\mathcal{L}(\mathcal{A})$ the
composition of the Yoneda functor $\mathcal{A}\rightarrow\rm Mod\mathcal{A}$
and the localization functor $\rm Mod\mathcal{A}\rightarrow\frac{\rm
Mod\mathcal{A}}{Eff(\mathcal{A})}\simeq
Eff(\mathcal{A})^{\bot}=\mathcal{L}(\mathcal{A})$.

###  2.2. Locally coherent categories

In this subsection we recall some of the basic properties of Locally finitely
presented categories, for details and more information the reader is referred
to [1]. Recall that a non-empty category $\mathcal{I}$ is said to be filtered
provided that for each pair of objects $\lambda_{1},\lambda_{2}\in\mathcal{I}$
there are morphisms $\varphi_{i}:\lambda_{i}\rightarrow\mu$ for some
$\mu\in\mathcal{I}$, and for each pair of morphisms
$\varphi_{1},\varphi_{2}:\lambda\rightarrow\mu$ there is a morphism
$\psi:\mu\rightarrow\nu$ with $\psi\varphi_{1}=\psi\varphi_{2}$. Let
$\mathcal{A}$ be an additive category, $\mathcal{I}$ a small filtered category
and $X:\mathcal{I}\rightarrow\mathcal{A}$ be an additive functor. As usual we
use the term direct limit for colimit of $X$ when $I$ is filtered. And we
denote it by $\underrightarrow{\operatorname{Lim}}_{i\in\mathcal{I}}X_{i}$.

######  Definition 2.6.

Let $\mathcal{B}$ be an additive category with direct limits .

  * (i)

An object $X\in\mathcal{B}$ is called finitely presented (finitely generated)
provided that for every direct limit
$\underrightarrow{\operatorname{Lim}}_{i\in\mathcal{I}}X_{i}$ in $\mathcal{B}$
the natural morphism

$\underrightarrow{\operatorname{Lim}}\operatorname{Hom}_{\mathcal{B}}(X,Y_{i})\rightarrow\operatorname{Hom}_{\mathcal{B}}(X,\underrightarrow{\operatorname{Lim}}Y_{i})$

is an isomorphism (a monomorphism). The full subcategory of finitely presented
objects of $\mathcal{A}$ is denoted by $\operatorname{fp}(\mathcal{A})$.

  * (ii)

$\mathcal{B}$ is called a locally finitely presented category if
$\operatorname{fp}(\mathcal{B})$ is skeletally small and every object in
$\mathcal{B}$ is a direct limit of objects in
$\operatorname{fp}(\mathcal{B})$.

  * (iii)

Let $\mathcal{B}$ be a locally finitely presented abelian category.
$\mathcal{B}$ is said to be locally coherent provided that finitely generated
subobjects of finitely presented objects are finitely presented.

We want to recall a concrete description of locally finitely presented
additive categories, which is due to Crawley-Boevey. Let $\mathcal{C}$ be a
small additive category, recall that there is a tensor product bifunctor

$\displaystyle\operatorname{Mod}\mathcal{C}\otimes_{\mathcal{C}}$
$\displaystyle\operatorname{Mod}\mathcal{C}^{op}\longrightarrow\rm Ab.$
$\displaystyle(F,G)$ $\displaystyle\longmapsto F\otimes_{\mathcal{C}}G$

A functor $F\in\operatorname{Mod}\mathcal{C}$ is said to be flat provided that
$F\otimes_{\mathcal{C}}-$ is an exact functor. We denote by
$\operatorname{Flat}(\mathcal{C})$ the full subcategory of
$\operatorname{Mod}\mathcal{C}$ consist of flat functors. A theorem of Lazard
state that an $R$-module is flat if and only if it is a direct limit of
finitely generated free modules. Lazard theorem has been generalized to
functors by Oberst and Rohrl [9]. Indeed a functor
$F\in\operatorname{Mod}\mathcal{C}$ is flat if and only if $F$ is a direct
limit of representable functors.

######  Lemma 2.7.

Let $\mathcal{A}$ be an abelian category and
$F\in\operatorname{Mod}\mathcal{A}$. The following statements are equivalence.

  * (a)

$F$ is a flat functor.

  * (b)

$F$ is a left exact functor.

In other words $\operatorname{Flat}(\mathcal{A})=\mathcal{L}(\mathcal{A})$.

###### Proof.

See [5]. ∎

######  Theorem 2.8.

$($[1, Theorem 1.4]$)$

  * (a)

If $\mathcal{C}$ is a skeletally small additive category, then
$\operatorname{Flat}(\mathcal{C})$ is a locally finitely presented category,
and $\operatorname{fp}(\operatorname{Flat}(\mathcal{C}))$ consists of the
direct summands of representable functors. if $\mathcal{C}$ has split
idempotents then the Yoneda functor
$h:\mathcal{C}\rightarrow\operatorname{fp}(\operatorname{Flat}(\mathcal{C}))$
is an equivalence.

  * (b)

If $\mathcal{B}$ is a locally finitely presented category then
$\operatorname{fp}(\mathcal{B})$ is a skeletally small additive category with
split idempotents, and the functor

$\displaystyle g:\mathcal{B}$
$\displaystyle\longrightarrow\operatorname{Flat}(\operatorname{fp}(\mathcal{B}))$
$\displaystyle M$
$\displaystyle\longmapsto\operatorname{Hom}(-,M)|_{\operatorname{fp}(\mathcal{B})}$

is an equivalence.

######  Proposition 2.9.

  * (i)

Every locally finitely presented abelian category is a Grothendieck category.

  * (ii)

A locally finitely presented abelian category $\mathcal{B}$ is locally
coherent if and only if $\operatorname{fp}(\mathcal{B})$ is an abelian
category.

###### Proof.

We refer the reader to [1] for the statement (i) and to [14] for (ii). ∎

In the proof of the main theorem we need the following proposition.

######  Proposition 2.10.

Let $\mathcal{A}$ be an abelian category with enough projective. Then
$\mathcal{L}(\mathcal{A})$ is a locally coherent category with enough
projective.

###### Proof.

By Proposition 2.9 and Lemma 2.7 $\mathcal{L}(\mathcal{A})$ is a locally
coherent category. so it remains to show that it has enough projective. First
note that if $P$ be a projective object in $\mathcal{A}$ then $H_{P}$ is a
projective object in $\mathcal{L}(\mathcal{A})$. Indeed for each epimorphism
$F\rightarrow H_{P}$ in $\mathcal{L}(\mathcal{A})$, by Proposition 2.5 it’s
cokernel is weakly effaceable. Thus there exists an epimorphism $X\rightarrow
P$ and the commutative diagram

$H_{X}$$H_{P}$$F$$H_{P}$

Because $H_{X}\rightarrow H_{P}$ is a retraction, $F\rightarrow H_{P}$ is also
a retraction. Now for an arbitrary object $F\in\mathcal{L}(\mathcal{A})$,
there is an epimorphism $\bigoplus_{i\in I}H_{X_{i}}\twoheadrightarrow F$, and
if we choose an epimorphism $P_{i}\twoheadrightarrow X_{i}$ in $\mathcal{A}$
for each $i\in I$ the result follows. ∎

##  3\. Proof of the main theorem

Let $\mathcal{A}$ be a small abelian category and $\mathcal{C}$ be a Serre
subcategory of $\mathcal{A}$. By Theorem 2.8
$\mathcal{L}(\mathcal{A})\simeq\dfrac{\operatorname{Mod}\mathcal{A}}{\operatorname{Eff}(\mathcal{A})}$
is a locally coherent category and the essential image of the canoonical
functor

$\mathsf{i}:\mathcal{A}\longrightarrow\mathcal{L}(\mathcal{A})$

denoted by $\mathsf{i}(\mathcal{C})$ is the subcategory of finitely presented
objects. Thus by [7, Theorem 2.8] $\overrightarrow{\mathsf{i}(\mathcal{C})}$,
the subcategory of $\mathcal{L}(\mathcal{A})$ consist of all direct limits of
objects in $\mathsf{i}(\mathcal{C})$ is a localizing subcategory of
$\mathcal{L}(\mathcal{A})$. Also by [6, Proposition A5]
$\dfrac{\mathcal{L}(\mathcal{A})}{\overrightarrow{{\mathsf{i}(\mathcal{C})}}}$
is a locally coherent category and it’s subcategory of finitely presented
objects is equivalent to $\dfrac{\mathcal{A}}{\mathcal{C}}$. Thus we have the
diagram of functors

(3.1)
$\mathcal{A}$$\dfrac{\mathcal{A}}{\mathcal{C}}$$\mathcal{L}(\mathcal{A})$$\dfrac{\mathcal{L}(\mathcal{A})}{\overrightarrow{\mathsf{i}(\mathcal{C})}}\simeq\mathcal{L}(\dfrac{\mathcal{A}}{\mathcal{C}})$$r$$\mathsf{i}$$\mathsf{j}$$q$$e$

satisfies

  * (i)

$q$ and $e$ are localization functors.

  * (ii)

$(e,r)$ is an adjoint pair and $r$ is fully faithful.

  * (iii)

Both of vertical functors $\mathsf{i}$ and $\mathsf{j}$ are the canonical
functor from an abelian category to the abelian category of left exact
functors.

######  Lemma 3.1.

Let $\mathcal{A}$ be a small abelian category. The canonical functor
$\mathsf{i}:\mathcal{A}\longrightarrow\mathcal{L}(\mathcal{A})$ is an
$\operatorname{Ext}$-preserving functor. i.e for every two object
$A,B\in\mathcal{A}$ and every positive integer $i$ the natural map
$\operatorname{Ext}_{\mathcal{A}}^{i}(A,B)\rightarrow\operatorname{Ext}_{\mathcal{L}(\mathcal{A})}^{i}(\mathsf{i}(A),\mathsf{i}(B))$
is invertible.

###### Proof.

See the section Application of [10] on page 367. ∎

The following proposition is a generalization of [7, Corollary 2.11] for
arbitrary positive integer $k$. Krause prove this result for $k=1$.

######  Proposition 3.2.

Let $\mathcal{B}$ be a locally coherent category with enough projective,
$\mathcal{C}$ be a Serre subcategory of
$\mathcal{A}=\operatorname{fp}(\mathcal{B})$ and $k$ be a positive integer.
Then as subcategories of $\mathcal{B}$ we have
$\mathcal{C}^{\bot_{k}}=(\overrightarrow{\mathcal{C}})^{\bot_{k}}$.

###### Proof.

By [7, Corollary 2.11] we have
$\mathcal{C}^{\bot_{1}}=(\overrightarrow{\mathcal{C}})^{\bot_{1}}$. Obviously
$(\overrightarrow{\mathcal{C}})^{\bot_{k}}\subseteq\mathcal{C}^{\bot_{k}}$.
For the converse let $X\in\mathcal{C}^{\bot_{k}}$. We most show that
$X\in(\overrightarrow{\mathcal{C}})^{\bot_{k}}$. To this end by Proposition
3.2 it is enough to show that there is an injective resolution

$0\rightarrow X\rightarrow I^{0}\rightarrow I^{1}\rightarrow\cdots\rightarrow
I^{k}$

such that $\forall i$ we have
$I^{i}\in(\overrightarrow{\mathcal{C}})^{\bot_{1}}=\mathcal{C}^{\bot_{1}}$.
Let $X\rightarrow I^{0}$ be the injective envelope of $X$ in $\mathcal{B}$. By
applying the functor $\operatorname{Hom}_{\mathcal{B}}(C,-)$ for an arbitrary
object $C\in\mathcal{C}$, to the short exact sequence

$0\rightarrow X\rightarrow I^{0}\rightarrow\Omega^{-1}X\rightarrow 0$

we obtain the exact sequence

$\displaystyle 0$
$\displaystyle\rightarrow\operatorname{Hom}_{\mathcal{B}}(C,X)\rightarrow\operatorname{Hom}_{\mathcal{B}}(C,I^{0})\rightarrow\operatorname{Hom}_{\mathcal{B}}(C,\Omega^{-1}X)$
$\displaystyle\rightarrow\operatorname{Ext}_{\mathcal{B}}^{1}(C,X)\rightarrow\operatorname{Ext}^{1}_{\mathcal{B}}(C,I^{0})\rightarrow\operatorname{Ext}_{\mathcal{B}}^{1}(C,\Omega^{-1}X)$
$\displaystyle\rightarrow\operatorname{Ext}_{\mathcal{B}}^{2}(C,X).$

By assumption
$\operatorname{Hom}_{\mathcal{B}}(C,X)=\operatorname{Ext}_{\mathcal{B}}^{1}(C,X)=\operatorname{Ext}_{\mathcal{B}}^{2}(C,X)=0$.
Because $I^{0}$ is an essential extension of $X$ we have that
$\operatorname{Hom}_{\mathcal{B}}(C,I^{0})=0$. Thus
$I^{0},\Omega^{-1}X\in\mathcal{C}^{\bot}=(\overrightarrow{\mathcal{C}})^{\bot}$.
By repeating this argument for $\Omega^{-1}X$ and using the dimension shifting
argument, similar to the proof of Proposition we obtain the desire injective
coresolution. ∎

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1:Let $A\in\mathcal{C}^{{\bot}_{k+1}}$ and $X$ be an
arbitrary object in $\mathcal{A}$. Having the Diagram (3.1) in mind we see
that

$\displaystyle\operatorname{Ext}_{\mathcal{A}}^{i}(X,A)$
$\displaystyle\overset{\mathsf{i}_{X,A}}{\cong}\operatorname{Ext}_{\mathcal{L}(\mathcal{A})}^{i}(i(X),i(A))$
$\displaystyle\overset{e_{X,A}}{\cong}\operatorname{Ext}_{\frac{\mathcal{L}(\mathcal{A})}{\overrightarrow{\mathcal{C}}}}^{i}(ei(X),ei(A))$

because $\mathsf{j}_{X,A}oq_{X,A}=e_{X,A}o\mathsf{i}_{X,A}$ we have that
$\mathsf{j}_{X,A}oq_{X,A}$ is invertible, and because by Lemma 3.1
$\mathsf{j}_{X,A}$ is invertible, $q_{X,A}$ is also invertible.

Conversely assume that $q_{X,A}$ is invertible for every $X\in\mathcal{A}$ and
$i\in\\{0,1,\cdots,k\\}$. By a similar argument like the proof
(iii)$\Rightarrow$(i) of Proposition 3.2 the result follows.

We state the dual of Theorem 1.1.

######  Theorem 3.3.

Let $\mathcal{A}$ be an abelian category with enough injective, $\mathcal{C}$
a Serre subcategory of $\mathcal{A}$, $k$ be a positive integer, and
$q:\mathcal{A}\rightarrow\frac{\mathcal{A}}{\mathcal{C}}$ be the quotient
functor. For an object $A\in\mathcal{A}$ the following conditions are
equivalent.

  * (i)

$A\in^{{\bot}_{k+1}}\mathcal{C}$.

  * (ii)

The natural map
$q_{A,X}:\operatorname{Ext}_{\mathcal{A}}^{i}(A,X)\rightarrow\operatorname{Ext}_{\frac{\mathcal{A}}{\mathcal{C}}}^{i}(q(A),q(X))$
is invertible, for every $X\in\mathcal{A}$ and $0\leq i\leq k$.

###### Proof.

Apply Theorem 1.1 to the opposite category $\mathcal{A}^{op}$. ∎

##  4\. Applications

Let $n$ be a positive integer and $\mathcal{M}$ be a small $n$-abelian
category in the sense of [4]. Denote by $\operatorname{eff}(\mathcal{M})$ the
full subcategory of $\operatorname{mod}\mathcal{M}$ consist of effaceable
functors, i.e the functors $F\in\operatorname{mod}\mathcal{M}$ with a
projective presentation

(4.1) $(-,Y^{n})\overset{(-,f)}{\longrightarrow}(-,Y^{n+1})\longrightarrow
F\rightarrow 0$

for some epimorphism $f:Y^{n}\rightarrow Y^{n+1}$. In [2, 8] it has been shown
that $\operatorname{eff}(\mathcal{M})$ is a Serre subcategory of
$\operatorname{mod}\mathcal{M}$, the composition of functors

$\displaystyle\mathcal{M}$
$\displaystyle\longrightarrow\operatorname{mod}\mathcal{M}\longrightarrow\dfrac{\operatorname{mod}\mathcal{M}}{\operatorname{eff}(\mathcal{M})}$
$\displaystyle X$ $\displaystyle\longmapsto(-,X)\longmapsto(-,X)$

is fully faithful and it’s essential image is an $n$-cluster tilting
subcategory of
$\dfrac{\operatorname{mod}\mathcal{M}}{\operatorname{eff}(\mathcal{M})}$. The
hardest step is maybe proving that the essential image is $n$-rigid, i.e for
every $k\in\\{1,\cdots,n-1\\}$ and $X,Y\in\mathcal{M}$ we have
$\operatorname{Ext}^{k}_{\frac{\operatorname{mod}\mathcal{M}}{\operatorname{eff}(\mathcal{M})}}(H_{X},H_{Y})=0$.
In the sequel we give a different proof using Theorem 1.1.

Let $F\in\operatorname{eff}(\mathcal{M})$ and consider a projective
presentation (4.1). Because $f$ is an epimorphism, by the axioms of
$n$-abelian categories $f$ fits into an $n$-exact sequence

$Y^{0}\rightarrow Y^{1}\rightarrow\cdots\rightarrow
Y^{n}\overset{f}{\rightarrow}Y^{n+1}.$

By the definition of $n$-exact sequences we have the exact sequence

$0\rightarrow H_{Y^{0}}\rightarrow H_{Y^{1}}\rightarrow\cdots\rightarrow
H_{Y^{n}}\rightarrow H_{Y^{n+1}}\rightarrow F\rightarrow 0.$

which is a projective resolution of $F$. Now let $X\in\mathcal{M}$. Applying
$\operatorname{Hom}_{\rm{mod}\text{-}\mathcal{M}}(-,H_{X})$ to the projective
resolution of $F$, we get the complex

$\displaystyle
0\rightarrow\operatorname{Hom}(H_{Y^{n+1}},H_{X})\rightarrow\operatorname{Hom}(H_{Y^{n}},H_{X})\rightarrow\cdots$
$\displaystyle\rightarrow\operatorname{Hom}(H_{Y^{1}},H_{X})\rightarrow\operatorname{Hom}(H_{Y^{0}},H_{X})\rightarrow
0$

which is by Yoneda’s lemma isomorphic to the complex

$\displaystyle
0\rightarrow\operatorname{Hom}(Y^{n+1},X)\rightarrow\operatorname{Hom}(Y^{n},X)\rightarrow\cdots\rightarrow\operatorname{Hom}(Y^{1},X)\rightarrow\operatorname{Hom}(Y^{0},X)$

that by the definition of $n$-exact sequences is an exact sequence of abelian
groups. Thus we see that for every $k\in\\{0,1,\cdots,n\\}$ we have
$\operatorname{Ext}^{k}_{\frac{\operatorname{mod}\mathcal{M}}{\operatorname{eff}(\mathcal{M})}}(F,H_{X})=0$.
Because $F$ was an arbitrary effaceable functor, for every $X\in\mathcal{M}$,
$H_{X}\in\operatorname{eff}(\mathcal{M})^{\bot_{n}}$. Therefore by Theorem 1.1
for every $k\in\\{1,\cdots,n-1\\}$ and $X,Y\in\mathcal{M}$ we have

$\operatorname{Ext}^{k}_{\frac{\operatorname{mod}\mathcal{M}}{\operatorname{eff}(\mathcal{M})}}(H_{X},H_{Y})\cong\operatorname{Ext}^{k}_{\operatorname{mod}\mathcal{M}}(H_{X},H_{Y})=0$.

## acknowledgements

This research was in part supported by a grant from IPM (No. 1400180047).

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