Introduction to coherent sheaves on weighted projective lines

TL;DR

This paper explores the structure of coherent sheaves on weighted projective lines, presenting two approaches—axiomatic and expansion-based—to describe their abelian categories, with detailed analysis and tilting theory insights.

Contribution

It introduces a new axiomatic framework and an expansion approach for understanding the abelian categories of coherent sheaves on weighted projective lines, enriching the theoretical foundation.

Findings

Descriptions of abelian categories for weighted projective lines

Development of an axiomatic approach to expansions

Discussion of tilting theory in this context

Abstract

These notes provide a description of the abelian categories that arise as categories of coherent sheaves on weighted projective lines. Two different approaches are presented: one is based on a list of axioms and the other yields a description in terms of expansions of abelian categories. A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process `expansion' and treat it axiomatically. Thus most of these notes are devoted to studying abelian categories, including a brief discussion of tilting theory. We provide many details and have tried to keep the exposition as self-contained as possible.

Equations

$$\mathbf{L}(\mathbf{p})=⟨{\vec{x}}_1,\dots ,{\vec{x}}_n,\vec{c}\mid p_1{\vec{x}}_1=\cdots =p_n{\vec{x}}_n=\vec{c}⟩,$$

$$S(\mathbf{p},\mathbf{\lambda })=k[u,v,x_1,\dots ,x_n]/(x_i^{p_i}+{\lambda }_{i1}u-{\lambda }_{i0}v),$$

$${\mathrm{Ext}}^1(S_x,S_x)=k,{\mathrm{Ext}}^1(S_{ij},S_{i{j}^{\prime }})=k(j^{\prime }\equiv j-1(\mathrm{mod}{p}_i)).$$

$${\mathbf{D}}^b(\mathcal{A})\stackrel{\sim }{⟶}{\mathbf{D}}^b(\mathrm{mod}\Lambda )$$

$$\S 3\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Hom}}_{\mathcal{A}}(A,B)\times {\mathrm{Hom}}_{\mathcal{A}}(B,C)⟶{\mathrm{Hom}}_{\mathcal{A}}(A,C)$$

$$\coprod _{i=1}^r{A}_i\cong \underset{i=1}{\overset{r}{⨁}}{A}_i\cong \prod _{i=1}^r{A}_i.$$

$$\mathrm{Ker}\varphi \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A_1\stackrel{{\varphi }_1}{⟶}{A}_2\stackrel{{\varphi }_2}{⟶}\cdots \stackrel{{\varphi }_n}{⟶}{A}_{n+1}$$

$${\mathrm{Hom}}_{\mathcal{A}/\mathcal{C}}(A,B)=\underset{(A^{\prime },B^{\prime })}{\mathrm{colim}}{\mathrm{Hom}}_{\mathcal{A}}(A^{\prime },B/B^{\prime })$$

$$A\stackrel{(Q\iota {)}^{-1}}{\to }{A}^{\prime }\stackrel{Q\varphi }{\to }B/B^{\prime }\stackrel{(Q\pi {)}^{-1}}{\to }B$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{A}[S(\mathcal{C}{)}^{-1}]=\mathcal{A}/\mathcal{C}.$$

$$A\leftarrowtail A^{\prime }\twoheadrightarrow I\rightarrowtail B/B^{\prime }\twoheadleftarrow B$$

$$\mathcal{A}/{\mathcal{A}}_1\stackrel{\sim }{⟶}(\mathcal{A}/{\mathcal{A}}_2)/({\mathcal{A}}_1/{\mathcal{A}}_2).\text{qed}$$

$${\mathcal{C}}^{\perp }$$

$${}^{\perp }\mathcal{C}$$

$$0⟶A^{\prime }⟶A⟶\bar{A}⟶A^{\prime \prime }⟶0$$

$${\mathrm{Hom}}_{\mathcal{A}}(A,GB)\stackrel{\sim }{\to }{\mathrm{Hom}}_{\mathcal{A}/\mathcal{C}}(FA,F(GB))\stackrel{\sim }{\to }{\mathrm{Hom}}_{\mathcal{A}/\mathcal{C}}(FA,B).$$

$${\mathrm{Hom}}_{\mathcal{A}}(-,A){|}_{\mathcal{C}}\cong {\mathrm{Hom}}_{\mathcal{C}}(-,A^{\prime })\text{and}{\mathrm{Ext}}_{\mathcal{A}}^1(-,A/A^{\prime }){|}_{\mathcal{C}}\cong {\mathrm{Hom}}_{\mathcal{C}}(-,A^{\prime \prime }),$$

$${\mathrm{Ext}}_{\mathcal{A}}^n(A,B)\times {\mathrm{Ext}}_{\mathcal{A}}^m(B,C)⟶{\mathrm{Ext}}_{\mathcal{A}}^{n+m}(A,C)$$

$$proj.dimA=\inf \{n\ge 0\mid {\mathrm{Ext}}_{\mathcal{A}}^{n+1}(A,-)=0\}.$$

$$0⟶B⟶E_{n+1}⟶E_n⟶\cdots ⟶E_1⟶A⟶0.$$

$$0=A_0\subseteq A_1\subseteq \cdots \subseteq A_{n-1}\subseteq A_n=A$$

$$\sum _{S^{\prime }\in {\Sigma }_0}{\dim }_{\Delta (S^{\prime })}{\mathrm{Ext}}_{\mathcal{A}}^1(S^{\prime },S)\le 1\text{ and }\sum _{S^{\prime }\in {\Sigma }_0}{\dim }_{\Delta (S^{\prime }{)}^{\mathrm{op}}}{\mathrm{Ext}}_{\mathcal{A}}^1(S,S^{\prime })\le 1.$$

$${\mathrm{Ext}}_{\mathcal{A}}^1(S,A)\stackrel{\sim }{\to }{\mathrm{Ext}}_{\mathcal{A}}^1(S,B).$$

$$\sum _{S^{\prime }\in {\Sigma }_0}{\dim }_{\Delta (S^{\prime }{)}^{\mathrm{op}}}{\mathrm{Ext}}_{\mathcal{A}}^1(S,S^{\prime })\ge 2.$$

$$S=S_{[1]}\rightarrowtail S_{[2]}\rightarrowtail S_{[3]}\rightarrowtail \cdots$$

$$\cdots \twoheadrightarrow S^{[3]}\twoheadrightarrow S^{[2]}\twoheadrightarrow S^{[1]}=S.$$

$${\mathrm{mod}}_0\Lambda \stackrel{\sim }{⟶}\coprod _{0\ne \mathfrak{p}\in \mathrm{Spec}\Lambda }{\mathrm{mod}}_0({\Lambda }_{\mathfrak{p}}).$$

$$D{\mathrm{Ext}}_{\mathcal{A}}^1(A,B)\stackrel{\sim }{\to }{\mathrm{Hom}}_{\mathcal{A}}(B,\tau A)$$

Full text

A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process ‘expansion’ and treat it axiomatically. Thus most of these notes are devoted to studying abelian categories, including a brief discussion of tilting theory. We provide many details and have tried to keep the exposition as self-contained as possible.

Version from September 18, 2010.

1. 1 Abelian categories
2. 2 Derived categories
3. 3 Tilting theory
4. 4 Expansions of abelian categories
5. 5 Coherent sheaves on the projective line
6. 6 Coherent sheaves on weighted projective lines
7. 7 Canonical algebras
8. 8 Further topics

Introduction

We begin with a brief description of weighted projective lines and their categories of coherent sheaves.

Let $k$ be an algebraically closed field, let $\mathbb{P}^{1}_{k}$ be the projective line over $k$, let $\boldsymbol{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ be a (possibly empty) collection of distinct closed points of $\mathbb{P}^{1}_{k}$, and let $\mathbf{p}=(p_{1},\dots,p_{n})$ be a weight sequence, that is, a sequence of positive integers. The triple $\mathbb{X}=(\mathbb{P}^{1}_{k},\boldsymbol{\lambda},\mathbf{p})$ is called a weighted projective line. Geigle and Lenzing [10] have associated to each weighted projective line a category $\operatorname{coh}\mathbb{X}$ of coherent sheaves on $\mathbb{X}$, which is the quotient category of the category of finitely generated $\mathbf{L}(\mathbf{p})$-graded $S(\mathbf{p},\boldsymbol{\lambda})$-modules, modulo the Serre subcategory of finite length modules. Here $\mathbf{L}(\mathbf{p})$ is the rank 1 additive group

[TABLE]

and

[TABLE]

with grading $\deg u=\deg v=\vec{c}$ and $\deg x_{i}=\vec{x}_{i}$, where $\lambda_{i}=[\lambda_{i0}:\lambda_{i1}]$ in $\mathbb{P}^{1}_{k}$. Geigle and Lenzing showed that $\operatorname{coh}\mathbb{X}$ is a hereditary abelian category with finite dimensional Hom and Ext spaces. The free module $S(\mathbf{p},\boldsymbol{\lambda})$ yields a structure sheaf ${\mathcal{O}}$, and shifting the grading gives twists $E(\vec{x})$ for any sheaf $E$ and $\vec{x}\in\mathbf{L}(\mathbf{p})$.

Every sheaf is the direct sum of a torsion-free sheaf and a finite length sheaf. A torsion-free sheaf has a finite filtration by line bundles, that is, sheaves of the form ${\mathcal{O}}(\vec{x})$. The finite length sheaves are easily described as follows. There are simple sheaves $S_{x}$ ($x\in\mathbb{P}^{1}_{k}\smallsetminus\boldsymbol{\lambda}$) and $S_{ij}$ ($1\leq i\leq n$, $1\leq j\leq p_{i}$) satisfying for any $r\in\mathbb{Z}$ that $\operatorname{Hom}({\mathcal{O}}(r\vec{c}),S_{ij})\neq 0$ if and only if $j=1$, and the only extensions between them are

[TABLE]

For each simple sheaf $S$ and $l>0$ there is a unique sheaf with length $l$ and top $S$, which is uniserial, meaning that it has a unique composition series. These are all the finite length indecomposable sheaves.

Categories of the form $\operatorname{coh}\mathbb{X}$ for some weighted projective line $\mathbb{X}$ play a special role in the study of abelian categories. This follows from a theorem of Happel [13] which we now explain. Consider a connected hereditary abelian category ${\mathcal{A}}$ that is $k$-linear with finite dimensional Hom and Ext spaces. Suppose in addition that ${\mathcal{A}}$ admits a tilting object, that is some object $T$ with $\operatorname{Ext}^{1}_{\mathcal{A}}(T,T)=0$ such that $\operatorname{Hom}_{\mathcal{A}}(T,A)=0$ and $\operatorname{Ext}^{1}_{\mathcal{A}}(T,A)=0$ imply $A=0$. Thus the functor $\operatorname{Hom}_{\mathcal{A}}(T,-)\colon{\mathcal{A}}\to\operatorname{mod}\Lambda$ into the category of modules over the endomorphism algebra $\Lambda=\operatorname{End}_{\mathcal{A}}(T)$ induces an equivalence

[TABLE]

of derived categories. There are two important classes of such hereditary abelian categories admitting a tilting object: module categories over path algebras of finite connected quivers without oriented cycles, and categories of coherent sheaves on weighted projective lines. Happel’s theorem then states that there are no further classes. More precisely, an abelian category ${\mathcal{A}}$ as above is, up to a derived equivalence, either of the form $\operatorname{mod}k\Gamma$ for some finite connected quiver $\Gamma$ without oriented cycles or of the form $\operatorname{coh}\mathbb{X}$ for some weighted projective line $\mathbb{X}$.

The following treatment of coherent sheaves on weighted projective lines is based on a list of axioms (extending the list in Happel’s theorem) which we postpone until §6. Before that we discuss in some detail the necessary background material: abelian categories, derived categories, tilting theory, expansions of abelian categories, and coherent sheaves on $\mathbb{P}^{1}_{k}$.

[TABLE]

Acknowledgements

These notes are based on a seminar and a course held at the University of Paderborn in the first half of 2009. Both authors wish to thank the participants for their interest and for stimulating discussions related to the topic of these notes. In particular, we are grateful to Dirk Kussin and Helmut Lenzing for their advice. It is a pleasure to thank Marco Angel Bertani-Økland and David Ploog for their detailed comments.

Most of the material presented here is taken from the existing literature. An exception is §4, where the concept of an ‘expansion of abelian categories’ is introduced. The previously unpublished proof of Theorem 1.7.1 is due to Yu Ye.

1. Abelian categories

1.1. Additive and abelian categories

A category ${\mathcal{A}}$ is additive if every finite family of objects has a product, each morphism set $\operatorname{Hom}_{\mathcal{A}}(A,B)$ is an abelian group, and the composition maps

[TABLE]

are bilinear. Given a finite number of objects $A_{1},\dots,A_{r}$ of an additive category ${\mathcal{A}}$, there exists a direct sum $A_{1}\oplus\dots\oplus A_{r}$, which is by definition an object $A$ together with morphisms $\iota_{i}\colon A_{i}\to A$ and $\pi_{i}\colon A\to A_{i}$ for $1\leq i\leq r$ such that $\sum_{i=1}^{r}\iota_{i}\pi_{i}=\operatorname{id}_{A}$, $\pi_{i}\iota_{i}=\operatorname{id}_{A_{i}}$, and $\pi_{j}\iota_{i}=0$ for all $i\neq j$. Note that the morphisms $\iota_{i}$ and $\pi_{i}$ induce isomorphisms

[TABLE]

Given any object $A$ in ${\mathcal{A}}$, we denote by $\operatorname{add}A$ the full subcategory of ${\mathcal{A}}$ consisting of all finite direct sums of copies of $A$ and their direct summands.

A decomposition ${\mathcal{A}}={\mathcal{A}}_{1}\amalg{\mathcal{A}}_{2}$ of an additive category ${\mathcal{A}}$ is a pair of full additive subcategories ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ such that each object in ${\mathcal{A}}$ is a direct sum of two objects from ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$, and $\operatorname{Hom}_{\mathcal{A}}(A_{1},A_{2})=0=\operatorname{Hom}_{\mathcal{A}}(A_{2},A_{1})$ for all $A_{1}\in{\mathcal{A}}_{1}$ and $A_{2}\in{\mathcal{A}}_{2}$. An additive category ${\mathcal{A}}$ is connected if it admits no proper decomposition ${\mathcal{A}}={\mathcal{A}}_{1}\amalg{\mathcal{A}}_{2}$.

A functor $F\colon{\mathcal{A}}\to{\mathcal{B}}$ between additive categories is additive if the induced map $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{\mathcal{B}}(FA,FB)$ is linear for each pair of objects $A,B$ in ${\mathcal{A}}$. The kernel $\operatorname{Ker}F$ of an additive functor $F\colon{\mathcal{A}}\to{\mathcal{B}}$ is by definition the full subcategory of ${\mathcal{A}}$ formed by all objects $A$ such that $FA=0$. The essential image $\operatorname{Im}F$ of $F\colon{\mathcal{A}}\to{\mathcal{B}}$ is the full subcategory of ${\mathcal{B}}$ formed by all objects $B$ such that $B$ is isomorphic to $FA$ for some $A$ in ${\mathcal{A}}$.

An additive category ${\mathcal{A}}$ is abelian if every morphism $\phi\colon A\to B$ has a kernel and a cokernel, and if the canonical factorization

[TABLE]

of $\phi$ induces an isomorphism $\bar{\phi}$.

Given an abelian category ${\mathcal{A}}$, a finite sequence of morphisms

[TABLE]

in ${\mathcal{A}}$ is exact if $\operatorname{Im}\phi_{i}=\operatorname{Ker}\phi_{i+1}$ for all $1\leq i<n$. An additive functor $F\colon{\mathcal{A}}\to{\mathcal{B}}$ between abelian categories is exact if $F$ sends each exact sequence in ${\mathcal{A}}$ to an exact sequence in ${\mathcal{B}}$.

Example 1.1.1**.**

(1) Let $\Lambda$ be a right noetherian ring. The category $\operatorname{mod}\Lambda$ of finitely generated right modules over $\Lambda$ is an abelian category.

(2) Let $k$ be a field and $\Gamma$ a quiver. The category $\operatorname{rep}(\Gamma,k)$ of finite dimensional $k$-linear representations of $\Gamma$ is an abelian category.

Conventions

Throughout, all categories are supposed to be skeletally small, unless otherwise stated. This means that the isomorphism classes of objects form a set. Subcategories are usually full subcategories and closed under isomorphisms. Functors between additive categories are always assumed to be additive. The composition of morphisms is written from right to left, and modules over a ring are usually right modules.

1.2. Serre subcategories and quotient categories

Let $\mathcal{A}$ be an abelian category. A non-empty full subcategory ${\mathcal{C}}$ of ${\mathcal{A}}$ is called a Serre subcategory provided that ${\mathcal{C}}$ is closed under taking subobjects, quotients and extensions. This means that for every exact sequence $0\to A^{\prime}\to A\to A^{\prime\prime}\to 0$ in ${\mathcal{A}}$, the object $A$ belongs to ${\mathcal{C}}$ if and only if $A^{\prime}$ and $A^{\prime\prime}$ belong to ${\mathcal{C}}$.

Example 1.2.1**.**

The kernel of an exact functor ${\mathcal{A}}\to{\mathcal{B}}$ between abelian categories is a Serre subcategory of ${\mathcal{A}}$.

Given a Serre subcategory ${\mathcal{C}}$ of ${\mathcal{A}}$, the quotient category ${\mathcal{A}}/{\mathcal{C}}$ of ${\mathcal{A}}$ with respect to ${\mathcal{C}}$ is defined as follows. The objects in ${\mathcal{A}}/{\mathcal{C}}$ are the objects in ${\mathcal{A}}$. Given two objects $A,B$ in ${\mathcal{A}}$, there is for each pair of subobjects $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$ an induced map $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{\mathcal{A}}(A^{\prime},B/B^{\prime})$. The pairs $(A^{\prime},B^{\prime})$ such that both $A/A^{\prime}$ and $B^{\prime}$ lie in ${\mathcal{C}}$ form a directed set, and one obtains a direct system of abelian groups $\operatorname{Hom}_{\mathcal{A}}(A^{\prime},B/B^{\prime})$. We define

[TABLE]

and the composition of morphisms in ${\mathcal{A}}$ induces the composition in ${\mathcal{A}}/{\mathcal{C}}$.111One needs to verify that ${\mathcal{A}}/{\mathcal{C}}$ is a category, in particular that the composition of morphisms is associative. This requires some work; see [8, 9].

The quotient functor $Q\colon{\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ is by definition the identity on objects. The functor takes a morphism in $\operatorname{Hom}_{{\mathcal{A}}}(A,B)$ to its image under the canonical map $\operatorname{Hom}_{{\mathcal{A}}}(A,B)\to\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A,B)$.

Lemma 1.2.2**.**

Each morphism $A\to B$ in ${\mathcal{A}}/{\mathcal{C}}$ is of the form

[TABLE]

for some pair $(A^{\prime},B^{\prime})$ of subobjects with $A/A^{\prime}$ and $B^{\prime}$ in ${\mathcal{C}}$ and some morphism $\phi\colon A^{\prime}\to B/B^{\prime}$ in ${\mathcal{A}}$, where $\iota\colon A^{\prime}\to A$ and $\pi\colon B\to B/B^{\prime}$ denote the canonical morphisms in ${\mathcal{A}}$.

Proof.

For each morphism $A\to B$ in ${\mathcal{A}}/{\mathcal{C}}$, there is by definition a pair $(A^{\prime},B^{\prime})$ of subobjects and a morphism $\phi\colon A^{\prime}\to B/B^{\prime}$ in ${\mathcal{A}}$ such that the following diagram commutes.

[TABLE]

Now observe that for each object $C$ in ${\mathcal{A}}$ the inclusion $\iota\colon A^{\prime}\to A$ induces a bijection $\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A,C)\to\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A^{\prime},C)$. Thus $Q\iota$ is invertible. Analogously, one shows that $Q\pi$ is invertible. ∎

The following result summarizes the basic properties of a quotient category and the corresponding quotient functor.

Proposition 1.2.3**.**

Let ${\mathcal{A}}$ be an abelian category and ${\mathcal{C}}$ a Serre subcategory.

1. (1)

The category $\mathcal{A}/\mathcal{C}$ is abelian and the quotient functor $Q\colon{\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ is exact with kernel $\operatorname{Ker}Q={\mathcal{C}}$. 2. (2)

Let $F\colon{\mathcal{A}}\to{\mathcal{B}}$ be an exact functor between abelian categories. If ${\mathcal{C}}\subseteq\operatorname{Ker}F$, then there is a unique functor $\bar{F}\colon{\mathcal{A}}/{\mathcal{C}}\to{\mathcal{B}}$ such that $F=\bar{F}Q$. Moreover, the functor $\bar{F}$ is exact.

Proof.

(1) It follows from the construction that the morphism sets of the quotient category ${\mathcal{A}}/{\mathcal{C}}$ are abelian groups. Also, the quotient functor induces linear maps between the morphism sets and it preserves finite direct sums. Thus the quotient category and the quotient functor are both additive.

The quotient functor sends a morphism in ${\mathcal{A}}$ to the zero morphism if and only if its image belongs to ${\mathcal{C}}$. Thus $\operatorname{Ker}Q={\mathcal{C}}$.

Let $\psi=(Q\pi)^{-1}Q\phi(Q\iota)^{-1}$ be a morphism in ${\mathcal{A}}/{\mathcal{C}}$ as in (1.2.1). Denote by $\iota^{\prime}\colon\operatorname{Ker}\phi\to A^{\prime}$ the kernel and by $\pi^{\prime}\colon B/B^{\prime}\to\operatorname{Coker}\phi$ the cokernel of $\phi$ in ${\mathcal{A}}$. Then the kernel of $\psi$ is $Q(\iota\iota^{\prime})\colon\operatorname{Ker}\phi\to A$, whereas the cokernel of $\psi$ is $Q(\pi^{\prime}\pi)\colon B\to\operatorname{Coker}\phi$. It follows that the category ${\mathcal{A}}/{\mathcal{C}}$ is abelian and that the quotient functor preserves kernels and cokernels.

(2) The functor $\bar{F}\colon{\mathcal{A}}/{\mathcal{C}}\to{\mathcal{B}}$ takes an object $A$ to $FA$ and a morphism of the form $(Q\pi)^{-1}Q\phi(Q\iota)^{-1}$ as in (1.2.1) to $(F\pi)^{-1}F\phi(F\iota)^{-1}$. Note that $F\iota$ and $F\pi$ are isomorphisms in ${\mathcal{B}}$, since $F$ is exact and ${\mathcal{C}}\subseteq\operatorname{Ker}F$.

The functor $\bar{F}$ is additive and the description of (co)kernels in (1) shows that $\bar{F}$ preserves (co)kernels. Thus $\bar{F}$ is exact. ∎

The quotient functor ${\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ is the universal functor that inverts the class $S({\mathcal{C}})$ of morphisms $\sigma$ in ${\mathcal{A}}$ with $\operatorname{Ker}\sigma$ and $\operatorname{Coker}\sigma$ in ${\mathcal{C}}$. More precisely, for any class $S$ of morphisms in ${\mathcal{A}}$, there exists a universal functor $P\colon{\mathcal{A}}\to{\mathcal{A}}[S^{-1}]$ such that

1. (1)

the morphism $P\sigma$ is invertible for every $\sigma\in S$, and 2. (2)

every functor $F\colon{\mathcal{A}}\to{\mathcal{B}}$ such that $F\sigma$ is invertible for each $\sigma\in S$ admits a unique functor $\bar{F}\colon{\mathcal{A}}[S^{-1}]\to{\mathcal{B}}$ such that $F=\bar{F}P$.

The category ${\mathcal{A}}[S^{-1}]$ is the localization of ${\mathcal{A}}$ with respect to $S$ and is unique up to a unique isomorphism; see [9, I.1].

Lemma 1.2.4**.**

Let ${\mathcal{C}}$ be a Serre subcategory of ${\mathcal{A}}$. The quotient functor $Q\colon{\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ is the universal functor that inverts all morphisms in $S({\mathcal{C}})$. Therefore

[TABLE]

Proof.

From Proposition 1.2.3 it follows that $Q$ inverts all morphisms in $S({\mathcal{C}})$. Now let $F\colon{\mathcal{A}}\to{\mathcal{B}}$ be a functor such that $F\sigma$ is invertible for each $\sigma\in S({\mathcal{C}})$. Then for each pair $A,B$ of objects in ${\mathcal{A}}$ and each pair of subobjects $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$ with $A/A^{\prime}$ and $B^{\prime}$ in ${\mathcal{C}}$, the map $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{\mathcal{B}}(FA,FB)$ factors through the canonical map $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{\mathcal{A}}(A^{\prime},B/B^{\prime})$. Thus there are induced maps $\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A,B)\to\operatorname{Hom}_{\mathcal{B}}(FA,FB)$ which induce a unique functor $\bar{F}\colon A/{\mathcal{C}}\to{\mathcal{B}}$ such that $F=\bar{F}Q$. It follows that ${\mathcal{A}}[S({\mathcal{C}})^{-1}]={\mathcal{A}}/{\mathcal{C}}$. ∎

Example 1.2.5**.**

(1) Let $\Lambda$ be a commutative noetherian ring and $\Lambda_{\mathfrak{p}}$ the localization with respect to a prime ideal $\mathfrak{p}$. The localization functor $T\colon\operatorname{mod}\Lambda\to\operatorname{mod}\Lambda_{\mathfrak{p}}$ sending a $\Lambda$-module $M$ to $M_{\mathfrak{p}}=M\otimes_{\Lambda}\Lambda_{\mathfrak{p}}$ is exact and induces an equivalence $\operatorname{mod}\Lambda/\operatorname{Ker}T\xrightarrow{\sim}\operatorname{mod}\Lambda_{\mathfrak{p}}$. Roughly speaking, restriction of scalars along the morphism $\Lambda\to\Lambda_{\mathfrak{p}}$ yields a quasi-inverse.

(2) Let $\Lambda$ be a right noetherian ring and $e^{2}=e\in\Lambda$ an idempotent. The functor $T\colon\operatorname{mod}\Lambda\to\operatorname{mod}e\Lambda e$ sending a $\Lambda$-module $M$ to $Me=M\otimes_{\Lambda}\Lambda e$ is exact. The kernel $\operatorname{Ker}T$ identifies with $\operatorname{mod}\Lambda/\Lambda e\Lambda$ and $T$ induces an equivalence $\operatorname{mod}\Lambda/\operatorname{Ker}T\xrightarrow{\sim}\operatorname{mod}e\Lambda e$. The functor $\operatorname{Hom}_{e\Lambda e}(\Lambda e,-)$ yields a quasi-inverse.

(3) Let $\Lambda$ be a right artinian ring. Given a set $S_{1},\dots,S_{n}$ of simple $\Lambda$-modules, the $\Lambda$-modules $M$ having a finite filtration $0=M_{0}\subseteq M_{1}\subseteq\dots\subseteq M_{r}=M$ with each factor $M_{i}/M_{i-1}$ isomorphic to one of the simples $S_{1},\dots,S_{n}$ form a Serre subcategory of $\operatorname{mod}\Lambda$. Moreover, each Serre subcategory of $\operatorname{mod}\Lambda$ arises in this way and is therefore of the form $\operatorname{mod}\Lambda/\Lambda e\Lambda$ for some idempotent $e\in\Lambda$.

1.3. Properties of quotient categories

We collect some further properties of abelian quotient categories.

Lemma 1.3.1**.**

Let ${\mathcal{A}}$ be an abelian category that is not supposed to be skeletally small, and let ${\mathcal{C}}$ be a Serre subcategory. Then the following are equivalent:

1. (1)

The category ${\mathcal{A}}$ is skeletally small. 2. (2)

The categories ${\mathcal{C}}$ and ${\mathcal{A}}/{\mathcal{C}}$ are skeletally small. In addition, $\operatorname{Ext}^{1}_{\mathcal{A}}(A,C)$ and $\operatorname{Ext}^{1}_{\mathcal{A}}(C,A)$ are sets for all $A\in{\mathcal{A}}$ and $C\in{\mathcal{C}}$.

Proof.

One direction is clear. So suppose that ${\mathcal{C}}$ and ${\mathcal{A}}/{\mathcal{C}}$ are skeletally small, and that extensions with objects in ${\mathcal{C}}$ form sets. First observe that the morphisms in $\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A,B)$ form a set for each pair of objects $A,B$. Here one uses that the subobjects $A^{\prime}\subseteq A$ with $A^{\prime}$ or $A/A^{\prime}$ in ${\mathcal{C}}$ form, up to isomorphism, a set, since ${\mathcal{C}}$ is skeletally small. Next observe that for each object $A$ in ${\mathcal{A}}$, there is only a set of isomorphism classes of objects $B$ with $A\cong B$ in ${\mathcal{A}}/{\mathcal{C}}$. This follows from the fact that each isomorphism $A\to B$ in ${\mathcal{A}}/{\mathcal{C}}$ is represented by a chain

[TABLE]

of epis and monos in ${\mathcal{A}}$ with kernel and cokernel in ${\mathcal{C}}$; see Lemma 1.2.2. Here one uses that ${\mathcal{C}}$ is skeletally small and that extensions with objects in ${\mathcal{C}}$ form sets. From this it follows that the isomorphism classes of objects in ${\mathcal{A}}$ form a set, since the quotient ${\mathcal{A}}/{\mathcal{C}}$ has this property. ∎

The following example gives an abelian category ${\mathcal{A}}$ with a Serre subcategory ${\mathcal{C}}$ such that ${\mathcal{C}}$ and ${\mathcal{A}}/{\mathcal{C}}$ are skeletally small but ${\mathcal{A}}$ itself is not.

Example 1.3.2**.**

Let $k$ be a field and $\Gamma$ a quiver with set of vertices $\{1,2\}$ and a proper class of arrows $1\to 2$. Each arrow of $\Gamma$ corresponds to a canonical element of $\operatorname{Ext}^{1}(S_{1},S_{2})$, where $S_{i}$ denotes the simple representation supported at the vertex $i$. These extensions are linearly independent and yield pairwise non-isomorphic two-dimensional representations. The functor $T\colon\operatorname{rep}(\Gamma,k)\to\operatorname{mod}k$ sending a representation of $\Gamma$ to the corresponding vector space at vertex $1$ induces an equivalence $\operatorname{rep}(\Gamma,k)/\operatorname{Ker}T\xrightarrow{\sim}\operatorname{mod}k$, and $\operatorname{Ker}T$ is equivalent to $\operatorname{mod}k$.

An abelian category is called noetherian if each of its objects is noetherian (i.e. satisfies the ascending chain condition on subobjects).

Lemma 1.3.3**.**

Let ${\mathcal{A}}$ be an abelian category and ${\mathcal{C}}$ a Serre subcategory. Then the following are equivalent:

1. (1)

The category ${\mathcal{A}}$ is noetherian. 2. (2)

The categories ${\mathcal{C}}$ and ${\mathcal{A}}/{\mathcal{C}}$ are noetherian, and each object in ${\mathcal{A}}$ has a largest subobject that belongs to ${\mathcal{C}}$.

Proof.

(1) $\Rightarrow$ (2): Suppose ${\mathcal{A}}$ is noetherian. Then ${\mathcal{C}}$ is noetherian. Also, ${\mathcal{A}}/{\mathcal{C}}$ is noetherian because each ascending chain of subobjects in ${\mathcal{A}}/{\mathcal{C}}$ can be represented by an ascending chain of subobjects in ${\mathcal{A}}$; see Lemma 1.2.2. Noetherianess implies that each non-empty set of subobjects has a maximal element. In particular, each object has a subobject that is maximal among all subobjects belonging to ${\mathcal{C}}$.

(2) $\Rightarrow$ (1): Let $A_{1}\subseteq A_{2}\subseteq\dots\subseteq A$ be an ascending chain of subobjects in ${\mathcal{A}}$. Using that ${\mathcal{A}}/{\mathcal{C}}$ is noetherian, there exists some integer $n$ such that $A_{m}/A_{n}$ belongs to ${\mathcal{C}}$ for all $m>n$. Let $\bar{A}$ be the maximal subobject of $A/A_{n}$ belonging to ${\mathcal{C}}$. Then the chain $A_{n+1}/A_{n}\subseteq A_{n+2}/A_{n}\subseteq\dots\subseteq\bar{A}$ becomes stationary since ${\mathcal{C}}$ is noetherian. It follows that the original chain of subobjects of $A$ becomes stationary. Thus $A$ is noetherian. ∎

Next we give an example of an abelian category ${\mathcal{A}}$ with a Serre subcategory ${\mathcal{C}}$ such that ${\mathcal{C}}$ and ${\mathcal{A}}/{\mathcal{C}}$ are noetherian but ${\mathcal{A}}$ itself is not.

Example 1.3.4**.**

The ring $\Lambda=\left[\begin{smallmatrix}\mathbb{Q}&\mathbb{Q}\\ 0&\mathbb{Z}\end{smallmatrix}\right]$ is well known to be left but not right noetherian. Consider the abelian category $\operatorname{mod}\Lambda$ of finitely presented (right) $\Lambda$-modules and the functor $T\colon\operatorname{mod}\Lambda\to\operatorname{mod}\mathbb{Q}$ sending a $\Lambda$-module $M$ to $Me$ with $e=\left[\begin{smallmatrix}1&0\\ 0&0\end{smallmatrix}\right]$. Then $\operatorname{Ker}T$ is equivalent to $\operatorname{mod}\mathbb{Z}$ and $T$ induces an equivalence $\operatorname{mod}\Lambda/\operatorname{Ker}T\xrightarrow{\sim}\operatorname{mod}\mathbb{Q}$.

The next lemma provides the analogue of a Noether isomorphism for abelian categories. The proof is straightforward.

Lemma 1.3.5**.**

Let ${\mathcal{A}}$ be an abelian category and ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$ a pair of Serre subcategories such that ${\mathcal{A}}_{2}\subseteq{\mathcal{A}}_{1}$. Then the following holds:

1. (1)

The inclusion ${\mathcal{A}}_{1}\to{\mathcal{A}}$ identifies ${\mathcal{A}}_{1}/{\mathcal{A}}_{2}$ with a Serre subcategory of ${\mathcal{A}}/{\mathcal{A}}_{2}$. 2. (2)

The quotient functor ${\mathcal{A}}\to{\mathcal{A}}/{\mathcal{A}}_{2}$ induces an isomorphism

[TABLE]

Recall that a non-zero object $S$ of an abelian category is simple if $S$ has no proper subobject $0\neq U\subsetneq S$. The next lemma says that a quotient functor preserves this property if the object is not annihilated. The proof is straightforward.

Lemma 1.3.6**.**

Let ${\mathcal{A}}$ be an abelian category and ${\mathcal{C}}$ a Serre subcategory. If $S$ is a simple object not belonging to ${\mathcal{C}}$, then $S$ is simple in ${\mathcal{A}}/{\mathcal{C}}$ and the quotient functor induces an isomorphism $\operatorname{End}_{\mathcal{A}}(S)\xrightarrow{\sim}\operatorname{End}_{{\mathcal{A}}/{\mathcal{C}}}(S)$. ∎

1.4. Perpendicular categories

Let ${\mathcal{A}}$ be an abelian category. In some cases, the quotient functor ${\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ with respect to a Serre subcategory ${\mathcal{C}}$ admits a right adjoint. Then the perpendicular category ${\mathcal{C}}^{\perp}$ provides another description of the quotient category ${\mathcal{A}}/{\mathcal{C}}$.

For any class ${\mathcal{C}}$ of objects in ${\mathcal{A}}$, its perpendicular categories are by definition the full subcategories

[TABLE]

Lemma 1.4.1**.**

Let ${\mathcal{A}}$ be an abelian category and ${\mathcal{C}}$ a Serre subcategory. Then the following are equivalent for an object $B$ in ${\mathcal{A}}$:

1. (1)

The object $B$ belongs to ${\mathcal{C}}^{\perp}$. 2. (2)

The quotient functor induces a bijection $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A,B)$ for every object $A$ in ${\mathcal{A}}$. 3. (3)

The map $\operatorname{Hom}_{\mathcal{A}}(\sigma,B)$ is bijective for every morphism $\sigma$ in ${\mathcal{A}}$ with $\operatorname{Ker}\sigma$ and $\operatorname{Coker}\sigma$ in ${\mathcal{C}}$.

Proof.

(1) $\Rightarrow$ (2): For each pair of subobjects $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$ such that both $A/A^{\prime}$ and $B^{\prime}$ lie in ${\mathcal{C}}$, the map $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{\mathcal{A}}(A^{\prime},B/B^{\prime})$ is bijective. Thus the quotient functor induces a bijection $\operatorname{Hom}_{\mathcal{A}}(A,B)\to\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(A,B)$.

(2) $\Rightarrow$ (3): The quotient functor sends a morphism $\sigma$ in ${\mathcal{A}}$ with $\operatorname{Ker}\sigma$ and $\operatorname{Coker}\sigma$ in ${\mathcal{C}}$ to an isomorphism in ${\mathcal{A}}/{\mathcal{C}}$. Thus $\operatorname{Hom}_{{\mathcal{A}}/{\mathcal{C}}}(\sigma,B)$ is bijective. Using the bijections in (2) it follows that $\operatorname{Hom}_{{\mathcal{A}}}(\sigma,B)$ is bijective.

(3) $\Rightarrow$ (1): Let $C$ be an object in ${\mathcal{C}}$. Then $\sigma\colon 0\to C$ induces a bijection $\operatorname{Hom}_{\mathcal{A}}(\sigma,B)$. Thus $\operatorname{Hom}_{\mathcal{A}}(C,B)=0$. Let $\xi\colon 0\to B\to E\to C\to 0$ an exact sequence in ${\mathcal{A}}$. The morphism $\sigma\colon B\to E$ induces a bijection $\operatorname{Hom}_{\mathcal{A}}(\sigma,B)$, and therefore $\xi$ splits. Thus $\operatorname{Ext}^{1}_{\mathcal{A}}(C,B)=0$. ∎

We need the following elementary lemma about pairs of adjoint functors.

Lemma 1.4.2**.**

Let $F\colon{\mathcal{A}}\to{\mathcal{B}}$ and $G\colon{\mathcal{B}}\to{\mathcal{A}}$ be a pair of functors such that $G$ is a right adjoint of $F$. Denote by $S(F)$ the class of morphisms $\sigma$ in ${\mathcal{A}}$ such that $F\sigma$ is invertible. Then the following are equivalent:

1. (1)

The functor $F$ induces an equivalence ${\mathcal{A}}[S(F)^{-1}]\xrightarrow{\sim}{\mathcal{B}}$. 2. (2)

The functor $G$ is fully faithful. 3. (3)

The adjunction morphism $F(GA)\to A$ is invertible for each $A\in{\mathcal{B}}$.

Proof.

See [9, I.1.3]. ∎

The next result provides a useful criterion for an exact functor to be a quotient functor. Moreover, it describes the right adjoint of a quotient functor.

Proposition 1.4.3**.**

Let $F\colon{\mathcal{A}}\to{\mathcal{B}}$ be an exact functor between abelian categories and suppose that $F$ admits a right adjoint $G\colon{\mathcal{B}}\to{\mathcal{A}}$. Then the following are equivalent:

1. (1)

The functor $F$ induces an equivalence ${\mathcal{A}}/\operatorname{Ker}F\xrightarrow{\sim}{\mathcal{B}}$. 2. (2)

The functor $F$ induces an equivalence $(\operatorname{Ker}F)^{\perp}\xrightarrow{\sim}{\mathcal{B}}$. 3. (3)

The functor $G$ induces an equivalence ${\mathcal{B}}\xrightarrow{\sim}(\operatorname{Ker}F)^{\perp}$. 4. (4)

The functor $G$ is fully faithful.

Moreover, in that case $(\operatorname{Ker}F)^{\perp}=\operatorname{Im}G$ and $\operatorname{Ker}F={{}^{\perp}}(\operatorname{Im}G)$.

Proof.

Let $S(F)$ denote the class of morphisms $\sigma$ in ${\mathcal{A}}$ such that $F\sigma$ is invertible. Then it follows from Lemma 1.2.4 that ${\mathcal{A}}[S(F)^{-1}]={\mathcal{A}}/\operatorname{Ker}F$.

(1) $\Rightarrow$ (2): The functor $F$ induces a full and faithful functor $(\operatorname{Ker}F)^{\perp}\to{\mathcal{B}}$ by Lemma 1.4.1. For each $B\in{\mathcal{B}}$, we have $F(GB)\cong B$ by Lemma 1.4.2, and $GB$ belongs to $(\operatorname{Ker}F)^{\perp}$ by Lemma 1.4.1. Thus $F$ induces an equivalence $(\operatorname{Ker}F)^{\perp}\xrightarrow{\sim}{\mathcal{B}}$.

(2) $\Rightarrow$ (3): Note that $\operatorname{Im}G\subseteq(\operatorname{Ker}F)^{\perp}$. Thus $G$ induces a functor ${\mathcal{B}}\to(\operatorname{Ker}F)^{\perp}$ which is a right adjoint of the equivalence $(\operatorname{Ker}F)^{\perp}\to{\mathcal{B}}$. Now one uses that an adjoint of an equivalence is again an equivalence.

(3) $\Rightarrow$ (4): An equivalence is fully faithful.

(4) $\Rightarrow$ (1): Use Lemma 1.4.2.

Observe that (3) implies $(\operatorname{Ker}F)^{\perp}=\operatorname{Im}G$. In particular, $\operatorname{Ker}F\subseteq{{}^{\perp}}(\operatorname{Im}G)$. The other inclusion follows from the isomorphism $\operatorname{Hom}_{\mathcal{B}}(FA,B)\cong\operatorname{Hom}_{\mathcal{A}}(A,GB)$. ∎

The next result characterizes the fact that the quotient functor admits a right adjoint.

Proposition 1.4.4**.**

Let ${\mathcal{A}}$ be an abelian category and ${\mathcal{C}}$ a Serre subcategory. Then the following are equivalent:

1. (1)

The quotient functor ${\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ admits a right adjoint ${\mathcal{A}}/{\mathcal{C}}\to{\mathcal{A}}$. 2. (2)

Every object $A$ in ${\mathcal{A}}$ fits into an exact sequence

[TABLE]

such that $A^{\prime},A^{\prime\prime}\in{\mathcal{C}}$ and $\bar{A}\in{\mathcal{C}}^{\perp}$. 3. (3)

The quotient functor induces an equivalence ${\mathcal{C}}^{\perp}\xrightarrow{\sim}{\mathcal{A}}/{\mathcal{C}}$.

In that case the functor ${\mathcal{A}}\to{\mathcal{C}}$ sending $A$ to $A^{\prime}$ is a right adjoint of the inclusion ${\mathcal{C}}\to{\mathcal{A}}$, and the functor ${\mathcal{A}}\to{\mathcal{C}}^{\perp}$ sending $A$ to $\bar{A}$ is a left adjoint of the inclusion ${\mathcal{C}}^{\perp}\to{\mathcal{A}}$.

Proof.

(1) $\Rightarrow$ (2): We apply Proposition 1.4.3. Suppose that the quotient functor $F\colon{\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$ admits a right adjoint $G$. The functor $F$ inverts the adjunction morphism $\eta_{A}\colon A\to G(FA)=\bar{A}$, since $FG\cong\operatorname{Id}_{{\mathcal{A}}/{\mathcal{C}}}$ by Lemma 1.4.2. The exactness of $F$ then implies that $A^{\prime}=\operatorname{Ker}\eta_{A}$ and $A^{\prime\prime}=\operatorname{Coker}\eta_{A}$ belong to ${\mathcal{C}}$. The object $\bar{A}$ belongs to $\operatorname{Im}G={\mathcal{C}}^{\perp}$ by construction.

(2) $\Rightarrow$ (3): The quotient functor induces a fully faithful functor ${\mathcal{C}}^{\perp}\to{\mathcal{A}}/{\mathcal{C}}$ by Lemma 1.4.1. This functor is an equivalence, because each object $A$ in ${\mathcal{A}}/{\mathcal{C}}$ is isomorphic to one in its image via the isomorphism $A\xrightarrow{\sim}\bar{A}$.

(3) $\Rightarrow$ (1): Choose a quasi-inverse $G\colon{\mathcal{A}}/{\mathcal{C}}\to{\mathcal{C}}^{\perp}$ of the equivalence ${\mathcal{C}}^{\perp}\to{\mathcal{A}}\xrightarrow{F}{\mathcal{A}}/{\mathcal{C}}$. For each $A$ in ${\mathcal{A}}$ and $B$ in ${\mathcal{A}}/{\mathcal{C}}$, there are bijections

[TABLE]

The first map is bijective by Lemma 1.4.1 and the second is bijective because $FG\cong\operatorname{Id}_{{\mathcal{A}}/{\mathcal{C}}}$. Thus $G$ is right adjoint to the quotient functor ${\mathcal{A}}\to{\mathcal{A}}/{\mathcal{C}}$.

For any $C$ in ${\mathcal{C}}$, the induced map $\operatorname{Hom}_{\mathcal{A}}(C,A^{\prime})\to\operatorname{Hom}_{\mathcal{A}}(C,A)$ is bijective. Therefore sending $A$ to $A^{\prime}$ provides a right adjoint of the inclusion ${\mathcal{C}}\to{\mathcal{A}}$. On the other hand, for any $B$ in ${\mathcal{C}}^{\perp}$, the induced map $\operatorname{Hom}_{\mathcal{A}}(\bar{A},B)\to\operatorname{Hom}_{\mathcal{A}}(A,B)$ is bijective. Therefore sending $A$ to $\bar{A}$ provides a left adjoint of the inclusion ${\mathcal{C}}^{\perp}\to{\mathcal{A}}$. ∎

Remark 1.4.5**.**

The objects $A^{\prime}$ and $A^{\prime\prime}$ occurring in (1.4.1) represent certain functors defined on ${\mathcal{C}}$. We have

[TABLE]

where $A^{\prime}$ is viewed as a subobject of $A$.

1.5. Global dimension

Let ${\mathcal{A}}$ be an abelian category. For a pair of objects $A,B$ and $n\geq 1$, let $\operatorname{Ext}^{n}_{\mathcal{A}}(A,B)$ denote the group of extensions in the sense of Yoneda. Set $\operatorname{Ext}^{0}_{\mathcal{A}}(A,B)=\operatorname{Hom}_{\mathcal{A}}(A,B)$ and $\operatorname{Ext}^{n}_{\mathcal{A}}(A,B)=0$ for $n<0$. Note that there are composition maps

[TABLE]

for all $n,m\in\mathbb{Z}$. The projective dimension of an object $A$ is by definition

[TABLE]

Dually, one defines the injective dimension $\operatorname{inj.dim}A$.

Lemma 1.5.1**.**

Let ${\mathcal{A}}$ be an abelian category. For $A$ in ${\mathcal{A}}$ and $n>0$ the following are equivalent:

1. (1)

$\operatorname{Ext}_{\mathcal{A}}^{n}(A,-)$* is right exact.* 2. (2)

$\operatorname{Ext}_{\mathcal{A}}^{n+1}(A,-)=0$. 3. (3)

$\operatorname{Ext}_{\mathcal{A}}^{m}(A,-)=0$* for all $m>n$.* 4. (4)

$\operatorname{proj.dim}A\leq n$.

Proof.

It suffices to show that (1) and (2) are equivalent; the rest is straightforward. We use the long exact sequence for $\operatorname{Ext}_{\mathcal{A}}^{*}(A,-)$.

(1) $\Rightarrow$ (2): Fix an element $\xi\in\operatorname{Ext}^{n+1}_{\mathcal{A}}(A,B)$ which is represented by an exact sequence

[TABLE]

Let $C$ be the image of $E_{n+1}\to E_{n}$ and write $\xi=\xi^{\prime\prime}\xi^{\prime}$ as the composite of extensions $\xi^{\prime}\in\operatorname{Ext}^{n}_{\mathcal{A}}(A,C)$ and $\xi^{\prime\prime}\in\operatorname{Ext}^{1}_{\mathcal{A}}(C,B)$. For the connecting morphism $\delta\colon\operatorname{Ext}^{n}_{\mathcal{A}}(A,C)\to\operatorname{Ext}^{n+1}_{\mathcal{A}}(A,B)$ induced by $\xi^{\prime\prime}$, we have $\delta=0$ since $\operatorname{Ext}_{\mathcal{A}}^{n}(A,-)$ is right exact. Thus $\xi=\delta(\xi^{\prime})=0$.

(2) $\Rightarrow$ (1): Clear. ∎

The global dimension of ${\mathcal{A}}$ is by definition the smallest integer $n\geq 0$ such that $\operatorname{Ext}_{\mathcal{A}}^{n+1}(-,-)=0$. As usual, the dimension is infinite if such a number $n$ does not exist. We denote this dimension by $\operatorname{gl.dim}{\mathcal{A}}$ and observe that it is equal to $\sup\{\operatorname{proj.dim}A\mid A\in{\mathcal{A}}\}$ and $\sup\{\operatorname{inj.dim}A\mid A\in{\mathcal{A}}\}$. The category ${\mathcal{A}}$ is called hereditary provided that $\operatorname{Ext}_{\mathcal{A}}^{2}(-,-)=0$.

For a right noetherian ring $\Lambda$, the global dimension of the module category $\operatorname{mod}\Lambda$ is called the (right) global dimension of $\Lambda$ and denoted by $\operatorname{gl.dim}\Lambda$.

Example 1.5.2**.**

(1) Let $\Lambda$ be the ring of integers $\mathbb{Z}$ or the polynomial ring $k[x]$ over a field $k$. Then $\operatorname{mod}\Lambda$ is hereditary. More generally, $\operatorname{mod}\Lambda$ is hereditary if $\Lambda$ is a Dedekind domain.

(2) For a field $k$ and a quiver $\Gamma$, the category of representations $\operatorname{rep}(\Gamma,k)$ is hereditary.

(3) Let ${\mathcal{A}}$ be a hereditary abelian category and ${\mathcal{C}}$ a Serre subcategory. Then ${\mathcal{C}}$ and ${\mathcal{A}}/{\mathcal{C}}$ are again hereditary.

1.6. Length categories

Let ${\mathcal{A}}$ be an abelian category. An object $A$ of ${\mathcal{A}}$ has finite length if there exists a finite chain of subobjects

[TABLE]

such that each quotient $A_{i}/A_{i-1}$ is a simple object. Such a chain is called a composition series of $A$. A composition series is not necessarily unique but its length is an invariant of $A$ by the Jordan-Hölder theorem; it is called the length of $A$ and is denoted by $\ell(A)$. Note that an object has finite length if and only if it is both artinian (i.e. satisfies the descending chain condition on subobjects) and noetherian (i.e. satisfies the ascending chain condition on subobjects).

Every object of finite length decomposes essentially uniquely into a finite direct sum of indecomposable objects with local endomorphism rings. This follows from the Krull-Remak-Schmidt theorem.

The objects of finite length form a Serre subcategory of ${\mathcal{A}}$ which is denoted by ${\mathcal{A}}_{0}$. The abelian category ${\mathcal{A}}$ is called a length category if ${\mathcal{A}}={\mathcal{A}}_{0}$.

Let ${\mathcal{A}}$ be a length category. The Ext-quiver or Gabriel quiver of ${\mathcal{A}}$ is a valued quiver $\Sigma=\Sigma({\mathcal{A}})$ which is defined as follows. The set $\Sigma_{0}$ of vertices is a fixed set of representatives of the isomorphism classes of simple objects in ${\mathcal{A}}$. For a simple object $S$, let $\Delta(S)$ denote its endomorphism ring, which is a division ring. Observe that $\operatorname{Ext}_{\mathcal{A}}^{1}(S,T)$ carries a natural $\Delta(T)$-$\Delta(S)$-bimodule structure for each pair $S,T$ in $\Sigma_{0}$. There is an arrow $S\to T$ with valuation $\delta_{S,T}=(s,t)$ in $\Sigma$ if $\operatorname{Ext}_{\mathcal{A}}^{1}(S,T)\neq 0$ with $s=\operatorname{dim}_{\Delta(S)}\operatorname{Ext}_{\mathcal{A}}^{1}(S,T)$ and $t=\operatorname{dim}_{\Delta(T)^{\mathrm{op}}}\operatorname{Ext}_{\mathcal{A}}^{1}(S,T)$. We write $\delta_{S,T}=(0,0)$ if $\operatorname{Ext}_{\mathcal{A}}^{1}(S,T)=0$.

The following observation is easily proved.

Lemma 1.6.1**.**

A length category is connected if and only if its Ext-quiver is connected.∎

Example 1.6.2**.**

(1) Let $\Lambda$ be a commutative noetherian local ring with maximal ideal $\mathfrak{m}$. Then $(\operatorname{mod}\Lambda)_{0}$ equals the category of $\mathfrak{m}$-torsion modules and $\Lambda/\mathfrak{m}$ is the unique simple $\Lambda$-module.

(2) For a field $k$ and a quiver $\Gamma$, the category of representations $\operatorname{rep}(\Gamma,k)$ is a length category. Suppose that $\Gamma$ has no oriented cycle and no pair of parallel arrows. Then the Ext-quiver of $\operatorname{rep}(\Gamma,k)$ is isomorphic to $\Gamma$, with valuation $(1,1)$ for each arrow.

1.7. Uniserial categories

Let $\mathcal{A}$ be a length category. An object $A$ is uniserial provided it has a unique composition series. Note that any non-zero uniserial object is indecomposable. Moreover, subobjects and quotient objects of uniserial objects are uniserial. The length category ${\mathcal{A}}$ is called uniserial provided that each indecomposable object is uniserial.

The following result characterizes uniserial categories in terms of their Ext-quivers.

Theorem 1.7.1** (Gabriel).**

A length category ${\mathcal{A}}$ is uniserial if and only if for each simple object $S$, we have

[TABLE]

The proof of this result requires some preparations and we begin with some notation. Let $A$ be any object in ${\mathcal{A}}$. We denote by $\operatorname{rad}A$ the intersection of all its maximal subobjects and let $\operatorname{top}A=A/\operatorname{rad}A$. Analogously, $\operatorname{soc}A$ denotes the sum of all simple subobjects of $A$.

Lemma 1.7.2**.**

Let ${\mathcal{A}}$ be a length category and suppose that (1.7.1) holds for each simple object $S$. Let $\xi\colon 0\rightarrow A\rightarrow E\rightarrow S\rightarrow 0$ be a non-split extension such that $A$ is uniserial and $S$ is simple. Then we have the following:

1. (1)

Every epimorphism $A\to B\neq 0$ induces an isomorphism

[TABLE] 2. (2)

*The object $E$ is uniserial. * 3. (3)

Given any non-split extension $\xi^{\prime}$ in $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A)$, there exists an isomorphism $\tau\colon A\rightarrow A$ such that $\xi^{\prime}=\tau\xi$.

Proof.

(1) It is sufficient to consider the case $B=\operatorname{top}A$. Moreover, it is sufficient to show that the induced map $\operatorname{Ext}^{1}_{\mathcal{A}}(S,A)\to\operatorname{Ext}^{1}_{\mathcal{A}}(S,\operatorname{top}A)$ is a monomorphism, since $\operatorname{dim}_{\Delta(S)}\operatorname{Ext}^{1}_{\mathcal{A}}(S,\operatorname{top}A)\leq 1$ by (1.7.1). We use the long exact sequence which is obtained from the short exact sequence $0\to A^{\prime}\to A\to\operatorname{top}A\to 0$ by applying $\operatorname{Hom}_{\mathcal{A}}(S,-)$.

If $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A^{\prime})=0$, then the induced map $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A)\rightarrow\operatorname{Ext}_{\mathcal{A}}^{1}(S,\operatorname{top}A)$ is a monomorphism. Now assume that $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A^{\prime})\neq 0$. Using induction on the length, we have that $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A^{\prime})\xrightarrow{\sim}\operatorname{Ext}_{\mathcal{A}}^{1}(S,\operatorname{top}A^{\prime})\neq 0$. Next observe that $\operatorname{Ext}_{\mathcal{A}}^{1}(\operatorname{top}A,\operatorname{top}A^{\prime})\neq 0$ since $A$ is uniserial. Thus (1.7.1) implies $S\cong\operatorname{top}A$, and a dimension argument shows that the connecting morphism $\operatorname{Hom}_{\mathcal{A}}(S,\operatorname{top}A)\rightarrow\operatorname{Ext}_{\mathcal{A}}^{1}(S,A^{\prime})$ is an isomorphism. Thus from the long exact sequence we infer that the natural map $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A)\rightarrow\operatorname{Ext}_{\mathcal{A}}^{1}(S,\operatorname{top}A)$ is a monomorphism.

(2) It suffices to show that every proper subobject $U\subseteq E$ is contained in $A$. Otherwise we have an induced extension $0\rightarrow A\cap U\rightarrow U\rightarrow S\rightarrow 0$. Thus the inclusion $A\cap U\to A$ induces a non-zero map $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A\cap U)\rightarrow\operatorname{Ext}_{\mathcal{A}}^{1}(S,A)$. Composing this with the isomorphism $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A)\xrightarrow{\sim}\operatorname{Ext}_{\mathcal{A}}^{1}(S,A/{A\cap U})$ from (1) gives a non-zero map $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A\cap U)\rightarrow{\rm Ext}_{\mathcal{A}}^{1}(S,A/{A\cap U})$ which is induced by the composite $A\cap U\rightarrow A\rightarrow A/{A\cap U}$. This is impossible.

(3) We use induction on the length of $A$. The case $\ell(A)=1$ follows from the equality $\operatorname{dim}_{\Delta(A)^{\mathrm{op}}}\operatorname{Ext}_{\mathcal{A}}^{1}(S,A)=1$. If $\ell(A)>1$, choose a maximal subobject $A^{\prime}\subseteq A$ and let $\bar{A}=A/A^{\prime}$. It follows from (1) that the canonical morphism $\pi\colon A\to\bar{A}$ induces an isomorphism $\operatorname{Ext}^{1}_{\mathcal{A}}(S,A)\xrightarrow{\sim}\operatorname{Ext}^{1}_{\mathcal{A}}(S,\bar{A})$ taking $\xi$ to $\pi\xi$. There is an isomorphism $\bar{\tau}\colon\bar{A}\to\bar{A}$ such that $\pi\xi^{\prime}=\bar{\tau}(\pi\xi)$ since $\ell(\bar{A})=1$. We claim that $\bar{\tau}$ extends to an isomorphism $\tau\colon A\to A$ satisfying $\pi\tau=\bar{\tau}\pi$. This implies $\pi\xi^{\prime}=\bar{\tau}\pi\xi=\pi\tau\xi$, and therefore $\xi^{\prime}=\tau\xi$. Thus it remains to construct $\tau$. To this end consider the non-split extension $\mu\colon 0\to A^{\prime}\to A\to\bar{A}\to 0$. The induction hypothesis yields an isomorphism $\tau^{\prime}\colon A^{\prime}\to A^{\prime}$ such that $\tau^{\prime}(\mu\bar{\tau})=\mu$ since $\ell(A^{\prime})<\ell(A)$. This gives the isomorphism $\tau$ satisfying $\pi\tau=\bar{\tau}\pi$. ∎

Lemma 1.7.3**.**

Let ${\mathcal{A}}$ be a length category and suppose that (1.7.1) holds for each simple object $S$. For two uniserial objects $A$ and $B$ the following are equivalent:

1. (1)

$A\cong B$. 2. (2)

$\operatorname{top}A\cong\operatorname{top}B$* and $\ell(A)=\ell(B)$.* 3. (3)

$\operatorname{soc}A\cong\operatorname{soc}B$* and $\ell(A)=\ell(B)$.*

Proof.

The condition (1.7.1) is self-dual. Thus it suffices to show the equivalence (1) $\Leftrightarrow$ (2). This equivalence follows from Lemma 1.7.2 using induction on the length $\ell(A)$. ∎

Proof of Theorem 1.7.1.

Suppose first that ${\mathcal{A}}$ is uniserial. Choose a simple object $S$ and assume that

[TABLE]

Then there exists an extension $\xi\colon 0\rightarrow S^{\prime}\oplus S^{\prime\prime}\rightarrow E\rightarrow S\rightarrow 0$ with $S^{\prime},S^{\prime\prime}\in\Sigma_{0}$ such that for each non-zero morphism $\theta\colon S^{\prime}\oplus S^{\prime\prime}\rightarrow T$ with $T\in\Sigma_{0}$, the induced extension $\theta\xi$ does not split. It is not difficult to check that $E$ is indecomposable and has at least two different composition series. Thus $E$ is not uniserial which is a contradiction.

Now assume that (1.7.1) holds for each simple object $S$ and fix an indecomposable object $A$. We show by induction on $\ell(A)$ that $A$ is uniserial. The case $\ell(A)=1$ is clear. Thus we choose an exact sequence $\xi\colon 0\rightarrow A^{\prime}\rightarrow A\rightarrow S\rightarrow 0$ with $S$ simple and fix a decomposition $A^{\prime}=\bigoplus_{i=1}^{l}A_{i}$ into indecomposable objects. Note that each $A_{i}$ is uniserial by our hypothesis. If $l=1$, then $A$ is uniserial by Lemma 1.7.2. Now assume $l>1$. Denote by $\xi_{i}$ the pushout of $\xi$ along the projection $A^{\prime}\rightarrow A_{i}$. Note that $\xi_{i}\neq 0$; otherwise $A_{i}$ is isomorphic to a direct summand of $A$. Therefore $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A_{i})\neq 0$ for all $i$, and Lemma 1.7.2 implies $\operatorname{top}A_{i}\cong\operatorname{top}A_{1}$ for all $i$. Assume that $\ell(A_{1})\geq\ell(A_{2})$. Then we have an epimorphism $\pi\colon A_{1}\rightarrow A_{2}$ by Lemma 1.7.3 which induces an isomorphism $\operatorname{Ext}_{\mathcal{A}}^{1}(S,A_{1})\xrightarrow{\sim}\operatorname{Ext}_{\mathcal{A}}^{1}(S,A_{2})$ by Lemma 1.7.2. Moreover, there exists an isomorphism $\tau\colon A_{2}\to A_{2}$ such that $\pi\xi_{1}=\tau\xi_{2}$. Consider the morphism $\phi\colon A^{\prime}\to A_{2}$ with $\phi_{1}=\pi$, $\phi_{2}=-\tau$ and $\phi_{i}=0$ for $2<i\leq l$. We have $\phi\xi=0$ by construction, and therefore $A_{2}$ is isomorphic to a direct summand of $A$. This is a contradiction. Thus $A$ is uniserial. ∎

Let ${\mathcal{A}}$ be a uniserial category. Choose a complete set of representatives of the isomorphism classes of indecomposable objects in ${\mathcal{A}}$ and denote it by $\operatorname{ind}{\mathcal{A}}$. An object in $\operatorname{ind}{\mathcal{A}}$ with top $S$ and length $n$ is denoted by $S^{[n]}$. Analogously, we write $S_{[n]}$ for an object in $\operatorname{ind}{\mathcal{A}}$ with socle $S$ and length $n$.

For each simple object $S$, we have a chain of monomorphisms

[TABLE]

which is either finite or infinite. Dually, there is a chain of epimorphisms

[TABLE]

A morphism $\phi\colon A\to B$ in an additive category is called irreducible if $\phi$ is neither a split monomorphism nor a split epimorphism and if for any factorisation $\phi=\phi^{\prime\prime}\phi^{\prime}$ the morphism $\phi^{\prime}$ is a split monomorphism or $\phi^{\prime\prime}$ is a split epimorphism.

Lemma 1.7.4**.**

Let ${\mathcal{A}}$ be a uniserial category. For a morphism $\phi\colon A\to B$ between indecomposable objects, the following are equivalent:

1. (1)

The morphism $\phi$ is irreducible. 2. (2)

The object $\operatorname{Ker}\phi\oplus\operatorname{Coker}\phi$ is simple. 3. (3)

There exists a simple object $S$ and an integer $n$ such that $\phi$ is, up to isomorphism, of the form $S_{[n]}\rightarrowtail S_{[n+1]}$ or $S^{[n+1]}\twoheadrightarrow S^{[n]}$.

Proof.

(1) $\Rightarrow$ (2): An irreducible morphism is either a monomorphism or an epimorphism. It suffices to discuss the case that $\phi$ is an epimorphism; the other case is dual. If $\ell(\operatorname{Ker}\phi)>1$ and $S\subseteq\operatorname{Ker}\phi$ is a simple subobject, then $\phi$ can be written as composite $A\to A/S\to B$ of two proper epimorphisms. This is a contradiction, and therefore $\operatorname{Ker}\phi\oplus\operatorname{Coker}\phi$ is simple.

(2) $\Rightarrow$ (3): Clear.

(3) $\Rightarrow$ (1): It suffices to consider the morphism $\phi\colon S^{[n+1]}\twoheadrightarrow S^{[n]}$; the dual argument works for $S_{[n]}\rightarrowtail S_{[n+1]}$. Let $S^{[n+1]}\xrightarrow{\alpha}X\xrightarrow{\beta}S^{[n]}$ be a factorization and fix a decomposition $X=\bigoplus_{i}X_{i}$ into indecomposable objects. Then $\beta_{i_{0}}\alpha_{i_{0}}$ is an epimorphism for at least one index $i_{0}$. It follows from Lemma 1.7.3 that $X_{i_{0}}=S^{[m]}$ for some $m\geq n$. If $m=n$, then $\beta_{i_{0}}$ is an isomorphism, and therefore $\beta$ is a split epimorphism. Otherwise, we obtain a factorization $S^{[n+1]}\xrightarrow{\alpha_{i_{0}}}X_{i_{0}}\xrightarrow{\beta^{\prime}_{i_{0}}}S^{[n+1]}\twoheadrightarrow S^{[n]}$ of the epimorphism $\beta_{i_{0}}\alpha_{i_{0}}$. It follows that $\beta^{\prime}_{i_{0}}\alpha_{i_{0}}$ is an epimorphism and hence an isomorphism. Thus $\alpha$ is a split monomorphism. ∎

Remark 1.7.5**.**

Let ${\mathcal{A}}$ be a uniserial category and $S$ a simple object. Suppose there is a bound $n$ such that $\ell(A)\leq n$ for each indecomposable object with $\operatorname{soc}A\cong S$. Then each indecomposable object $A$ of length $n$ with $\operatorname{soc}A\cong S$ is injective, since $\operatorname{Ext}_{\mathcal{A}}^{1}(T,A)=0$ for every simple object $T$ in ${\mathcal{A}}$ by Lemma 1.7.2.

Example 1.7.6**.**

(1) Let $\Lambda$ be a Dedekind domain. The finitely generated torsion modules over $\Lambda$ form a uniserial category. We denote this category by $\operatorname{mod}_{0}\Lambda$ because it coincides with the category $(\operatorname{mod}\Lambda)_{0}$ of finite length objects of $\operatorname{mod}\Lambda$. Let $\operatorname{Spec}\Lambda$ denote the set of prime ideals. The functor which takes a $\Lambda$-module $M$ to the family of localizations $(M_{\mathfrak{p}})_{\mathfrak{p}\in\operatorname{Spec}\Lambda}$ induces an equivalence

[TABLE]

(2) Let $k$ be a field and $P\in k[x]$ an irreducible polynomial. For each $n>0$, the finitely generated $k[x]/(P^{n})$-modules form a uniserial category with a unique simple object.

(3) Let $k$ be a field and $\Gamma$ a quiver having no oriented cycle. The category of representations $\operatorname{rep}(\Gamma,k)$ is uniserial if and only if for each vertex $x$ of $\Gamma$, there is at most one arrow starting at $x$ and at most one arrow ending at $x$.

1.8. Serre duality

Let $k$ be a commutative ring. A category ${\mathcal{A}}$ is $k$-linear if each morphism set is a $k$-module and the composition maps are $k$-bilinear. A functor between $k$-linear categories is $k$-linear provided that the induced maps between the morphism sets are $k$-linear. A $k$-linear category is Hom-finite if each morphism set is a $k$-module of finite length. Suppose now that ${\mathcal{A}}$ is a $k$-linear abelian category. Then the extension groups are naturally modules over $k$, and ${\mathcal{A}}$ is called Ext-finite if $\operatorname{Ext}^{n}_{\mathcal{A}}(A,B)$ is of finite length over $k$ for all $A,B$ in ${\mathcal{A}}$ and $n\geq 0$.

Fix a field $k$ and a Hom-finite $k$-linear abelian category ${\mathcal{A}}$. The category ${\mathcal{A}}$ is said to satisfy Serre duality222This is the appropriate notion of Serre duality for hereditary abelian categories. Higher dimensional analogues involving $D\operatorname{Ext}^{n}_{\mathcal{A}}(A,-)$ appear in algebraic geometry; see also §3.4. if there exists an equivalence $\tau\colon{\mathcal{A}}\xrightarrow{\sim}{\mathcal{A}}$ with functorial $k$-linear isomorphisms

[TABLE]

for all $A,B$ in ${\mathcal{A}}$, where $D=\operatorname{Hom}_{k}(-,k)$ denotes the standard $k$-duality. The functor $\tau$ is called Serre functor or Auslander-Reiten translation. Note that a Serre functor is $k$-linear and essentially unique provided it exists; this follows from Yoneda’s lemma.

Recall that ${\mathcal{A}}_{0}$ denotes the full subcategory consisting of all finite length objects in ${\mathcal{A}}$. Denote by ${\mathcal{A}}_{+}$ the full subcategory consisting of all objects $A$ in ${\mathcal{A}}$ satisfying $\operatorname{Hom}_{\mathcal{A}}(A_{0},A)=0$ for all $A_{0}$ in ${\mathcal{A}}_{0}$.

Proposition 1.8.1**.**

Let ${\mathcal{A}}$ be a Hom-finite $k$-linear abelian category and suppose ${\mathcal{A}}$ admits a Serre functor $\tau$. Then the following holds:

1. (1)

The category ${\mathcal{A}}$ is hereditary. 2. (2)

The category ${\mathcal{A}}$ has no non-zero projective or injective objects. 3. (3)

A noetherian object $A$ has a unique maximal subobject $A_{0}$ of finite length. Moreover, $A_{0}$ is a direct summand of $A$ and $A/A_{0}$ belongs to ${\mathcal{A}}_{+}$. 4. (4)

For each indecomposable object $A$ in ${\mathcal{A}}$, there is an almost split sequence333For the notion of an almost split sequence, we refer to [1].* $0\rightarrow\tau A\rightarrow E\rightarrow A\rightarrow 0$.* 5. (5)

For each object $A$ in ${\mathcal{A}}$, we have $A^{\perp}={{}^{\perp}\tau A}$.

Proof.

(1) For each object $A$, the functor $\operatorname{Ext}^{1}_{\mathcal{A}}(A,-)$ is right exact. Thus the category ${\mathcal{A}}$ is hereditary by Lemma 1.5.1.

(2) Let $A$ be a projective object. Then $\operatorname{Hom}_{\mathcal{A}}(-,\tau A)\cong D\operatorname{Ext}^{1}_{\mathcal{A}}(A,-)=0$. Thus $\tau A=0$ and therefore $A=0$. The dual argument works for injective objects.

(3) Choose a maximal subobject $A_{0}$ of finite length. Then $A/A_{0}$ belongs to ${\mathcal{A}}_{+}$, and every finite length subobject of $A$ is contained in $A_{0}$. In particular, $A_{0}$ is unique. We have $\operatorname{Ext}^{1}_{\mathcal{A}}(A/A_{0},A_{0})=0$ by Serre duality, and therefore $A_{0}$ is a direct summand of $A$.

(4) Let $A$ be an indecomposable object. The endomorphism ring $\operatorname{End}_{\mathcal{A}}(A)$ is local and we denote by $\mathfrak{m}$ its maximal ideal. Choose any non-zero $k$-linear map $\omega\colon\operatorname{End}_{\mathcal{A}}(A)\rightarrow k$ such that $\omega$ vanishes on $\mathfrak{m}$. The map $\omega$ corresponds via Serre duality to a non-split short exact sequence $\xi\colon 0\rightarrow\tau A\xrightarrow{}E\xrightarrow{}A\rightarrow 0$. We claim that $\xi$ is an almost split sequence. For this one needs to show that each morphism $\alpha\colon A^{\prime}\to A$ factors through the morphism $E\to A$, provided that $\alpha$ is not a split epimorphism. Thus one needs to show that $\xi\alpha=0$. The element $\xi\alpha$ corresponds via Serre duality to $\omega\alpha$ which sends $\phi\in\operatorname{Hom}_{\mathcal{A}}(A,A^{\prime})$ to $\omega(\alpha\phi)$. Thus $\xi\alpha=0$, since $\alpha\phi$ belongs to $\mathfrak{m}$.

(5) This is clear from the definitions. ∎

Next observe that a Serre functor $\tau$ on ${\mathcal{A}}$ restricts to a Serre functor on the subcategory ${\mathcal{A}}_{0}$ of finite length objects. The following result describes the structure of a length category with Serre duality. Let us recall the shape of the relevant diagrams.

[TABLE]

Proposition 1.8.2**.**

Let ${\mathcal{A}}$ be a Hom-finite $k$-linear length category and suppose ${\mathcal{A}}$ admits a Serre functor $\tau$. Then ${\mathcal{A}}$ is uniserial. The category ${\mathcal{A}}$ admits a unique decomposition ${\mathcal{A}}=\coprod_{i\in I}{\mathcal{A}}_{i}$ into connected uniserial categories with Serre duality, where the index set equals the set of $\tau$-orbits of simple objects in ${\mathcal{A}}$. The Ext-quiver of each ${\mathcal{A}}_{i}$ is either of type $\mathbb{A}_{\infty}^{\infty}$ (with linear orientation) or of type $\tilde{\mathbb{A}}_{n}$ (with cyclic orientation).

Proof.

We apply the criterion of Theorem 1.7.1 to show that ${\mathcal{A}}$ is uniserial. To this end fix a simple object $S$. Then $\operatorname{Ext}_{\mathcal{A}}^{1}(S,S^{\prime})\cong D\operatorname{Hom}_{\mathcal{A}}(S^{\prime},\tau S)\neq 0$ for some $S^{\prime}\in\Sigma_{0}$ if and only if $S^{\prime}\cong\tau S$. Moreover, $\operatorname{dim}_{\Delta(S)}\operatorname{Ext}_{\mathcal{A}}^{1}(S,\tau S)=1$. Thus the category ${\mathcal{A}}$ is uniserial.

The structure of the Ext-quiver of ${\mathcal{A}}$ follows from the condition (1.7.1). The Serre functor acts on $\Sigma_{0}$ and the set of $\tau$-orbits $I=\Sigma_{0}/\tau$ is the index set of the decomposition ${\mathcal{A}}=\coprod_{i\in I}{\mathcal{A}}_{i}$ into connected components; see Lemma 1.6.1. The Ext-quiver of ${\mathcal{A}}_{i}$ is of type $\mathbb{A}_{\infty}^{\infty}$ if the corresponding $\tau$-orbit is infinite. Otherwise, the Ext-quiver of ${\mathcal{A}}_{i}$ is of type $\tilde{\mathbb{A}}_{n}$ where $n+1$ equals the cardinality of the $\tau$-orbit. ∎

Let ${\mathcal{A}}$ be a Hom-finite $k$-linear length category and suppose ${\mathcal{A}}$ admits a Serre functor. Then a complete set of representatives of the isomorphism classes of indecomposable objects of ${\mathcal{A}}$ is given by $\{S^{[n]}\mid S\in\Sigma_{0},\,n\geq 1\}$ and also by $\{S_{[n]}\mid S\in\Sigma_{0},\,n\geq 1\}$; see Remark 1.7.5.

Example 1.8.3**.**

Let $k$ be a field and $\Gamma$ a quiver of extended Dynkin type $\tilde{\mathbb{A}}_{n}$ with cyclic orientation. Denote by ${\mathcal{A}}=\operatorname{rep}_{0}(\Gamma,k)$ the full subcategory of $\operatorname{rep}(\Gamma,k)$ consisting of all nilpotent representations. Then ${\mathcal{A}}$ satisfies Serre duality and the Ext-quiver of ${\mathcal{A}}$ is isomorphic to $\Gamma$, with valuation $(1,1)$ for each arrow. Note that the Serre functor on ${\mathcal{A}}$ has order $n+1$ and every simple object has endomorphism algebra $k$. In fact, a connected Hom-finite $k$-linear length category with Serre duality satisfying these properties is equivalent to $\operatorname{rep}_{0}(\Gamma,k)$.

2. Derived categories

To each abelian category is associated its derived category. This section provides a brief introduction. We present the definition and discuss two cases where one has a convenient description: If the abelian category is hereditary, then each complex is isomorphic to its cohomology. On the other hand, if there are enough projective objects, then one can compute morphisms in the derived category by passing to the homotopy category of projective objects.

2.1. Categories of complexes

Let ${\mathcal{A}}$ be an additive category. A cochain complex in ${\mathcal{A}}$ is a sequence of morphisms

[TABLE]

such that $d^{n}d^{n-1}=0$ for all $n\in\mathbb{Z}$. We denote by $\mathbf{C}({\mathcal{A}})$ the category of cochain complexes, where a morphism $\phi\colon X\to Y$ between cochain complexes consists of morphisms $\phi^{n}\colon X^{n}\to Y^{n}$ with $d_{Y}^{n}\phi^{n}=\phi^{n+1}d_{X}^{n}$ for all $n\in\mathbb{Z}$.

A chain complex in ${\mathcal{A}}$ is a sequence of morphisms

[TABLE]

such that $d_{n}d_{n+1}=0$ for all $n\in\mathbb{Z}$. Any chain complex may be viewed as a cochain complex by changing its indices, and vice versa. Thus we often confuse both concepts and simply use the term complex.

A morphism $\phi\colon X\to Y$ between complexes is null-homotopic if there are morphisms $\rho^{n}\colon X^{n}\to Y^{n-1}$ such that $\phi^{n}=d_{Y}^{n-1}\rho^{n}+\rho^{n+1}d_{X}^{n}$ for all $n\in\mathbb{Z}$. The null-homotopic morphisms form an ideal ${\mathcal{N}}$ in $\mathbf{C}({\mathcal{A}})$, that is, for each pair $X,Y$ of complexes a subgroup

[TABLE]

such that any composition $\psi\phi$ of morphisms in $\mathbf{C}({\mathcal{A}})$ belongs to ${\mathcal{N}}$ if $\phi$ or $\psi$ belongs to ${\mathcal{N}}$. The homotopy category $\mathbf{K}({\mathcal{A}})$ is the quotient of $\mathbf{C}({\mathcal{A}})$ with respect to this ideal. Thus

[TABLE]

for every pair of complexes $X,Y$.

Now let ${\mathcal{A}}$ be an abelian category. The cohomology of a complex $X$ in degree $n$ is by definition $H^{n}X=\operatorname{Ker}d^{n}/\operatorname{Im}d^{n-1}$, and each morphism $\phi\colon X\to Y$ of complexes induces a morphism $H^{n}\phi\colon H^{n}X\to H^{n}Y$. The morphism $\phi$ is a quasi-isomorphism if $H^{n}\phi$ is an isomorphism for all $n\in\mathbb{Z}$. Note that two morphisms $\phi,\psi\colon X\to Y$ induce the same morphism $H^{n}\phi=H^{n}\psi$, if $\phi-\psi$ is null-homotopic.

The derived category $\mathbf{D}({\mathcal{A}})$ of ${\mathcal{A}}$ is obtained from $\mathbf{K}({\mathcal{A}})$ by formally inverting all quasi-isomorphisms. To be precise, one defines

[TABLE]

as the localization of $\mathbf{K}({\mathcal{A}})$ with respect to the class of all quasi-isomorphisms. The full subcategory consisting of objects that are isomorphic to a complex $X$ such that $X^{n}=0$ for almost all $n\in\mathbb{Z}$ is denoted by $\mathbf{D}^{b}({\mathcal{A}})$.

An object $A$ in ${\mathcal{A}}$ is identified with the complex

[TABLE]

concentrated in degree zero, and this complex is also denoted by $A$. Given any complex $X$ in ${\mathcal{A}}$ and $p\in\mathbb{Z}$, we denote by $X[p]$ the shifted complex with

[TABLE]

This operation induces an isomorphism $\mathbf{D}({\mathcal{A}})\xrightarrow{\sim}\mathbf{D}({\mathcal{A}})$ and is called shift.

The derived category $\mathbf{D}({\mathcal{A}})$ is an additive category with some additional structure: it is a triangulated category in the sense of Verdier [25]. For instance, any exact sequence $0\to A\to B\to C\to 0$ in ${\mathcal{A}}$ induces an exact triangle $A\to B\to C\to A[1]$ in $\mathbf{D}({\mathcal{A}})$.

Given two abelian categories ${\mathcal{A}}$ and ${\mathcal{A}}^{\prime}$, a functor $F\colon\mathbf{D}({\mathcal{A}})\to\mathbf{D}({\mathcal{A}}^{\prime})$ is by definition a derived equivalence if it is an equivalence of triangulated categories, that is, $F$ is an equivalence, there is a functorial isomorphism $(FX)[1]\cong F(X[1])$ for each $X$ in $\mathbf{D}({\mathcal{A}})$, and $F$ preserves exact triangles.

The following statement justifies the study of derived categories.

Proposition 2.1.1**.**

Let $A,B$ be objects in ${\mathcal{A}}$. Then

[TABLE]

Proof.

For the case that ${\mathcal{A}}$ has enough injectives or enough projectives, see [26, Corollary 10.7.5]. For the general case, see [25, III.3]. ∎

2.2. Hereditary abelian categories

Let ${\mathcal{A}}$ be a hereditary abelian category, that is, $\operatorname{Ext}_{\mathcal{A}}^{2}(-,-)$ vanishes. In this case, there is an explicit description of all objects and morphisms in $\mathbf{D}^{b}({\mathcal{A}})$ via the ones in ${\mathcal{A}}$. Every complex $X$ is completely determined by its cohomology because there is an isomorphism between $X$ and the following complex with trivial differential.

[TABLE]

To construct this isomorphism, note that the vanishing of $\operatorname{Ext}^{2}_{\mathcal{A}}(H^{n}X,-)$ implies the existence of a commutative diagram

[TABLE]

with exact rows. We obtain the following commutative diagram.

[TABLE]

The vertical morphisms yield two morphisms in $\mathbf{D}^{b}({\mathcal{A}})$. The upper one is a quasi-isomorphism, and the lower one induces a cohomology isomorphism in degree $n$. This yields for each $n\in\mathbb{Z}$ a morphism $(H^{n}X)[-n]\to X$ in $\mathbf{D}^{b}({\mathcal{A}})$ and therefore the following description of $X$.

Lemma 2.2.1**.**

Let ${\mathcal{A}}$ be a hereditary abelian category and $X$ a complex in ${\mathcal{A}}$. In $\mathbf{D}^{b}({\mathcal{A}})$ there is a (non-canonical) isomorphism

[TABLE]

Proof.

The morphism is a quasi-isomorphism by construction. ∎

For an abelian category ${\mathcal{A}}$ one defines its repetitive category $\bigsqcup_{n\in\mathbb{Z}}{\mathcal{A}}[n]$ as the additive closure of the union of disjoint copies ${\mathcal{A}}[n]$ of ${\mathcal{A}}$ with morphisms

[TABLE]

and composition given by the Yoneda product of extensions. It follows from Proposition 2.1.1 that the family of functors ${\mathcal{A}}[n]\to\mathbf{D}^{b}({\mathcal{A}})$ ($n\in\mathbb{Z}$) sending an object $A$ to $A[n]$ induces a fully faithful functor

[TABLE]

Corollary 2.2.2**.**

The canonical functor $\bigsqcup_{n\in\mathbb{Z}}{\mathcal{A}}[n]\to\mathbf{D}^{b}({\mathcal{A}})$ is an equivalence for any hereditary abelian category ${\mathcal{A}}$. ∎

2.3. Abelian categories with enough projectives

We describe the derived category of an abelian category ${\mathcal{A}}$ in terms of its projective objects. The crucial observation is the following.

Lemma 2.3.1**.**

Let $X,Y$ be a pair of complexes in ${\mathcal{A}}$. Suppose that each $X^{n}$ is projective and $X^{n}=0$ for $n\gg 0$. Then the map $\operatorname{Hom}_{\mathbf{K}({\mathcal{A}})}(X,Y)\to\operatorname{Hom}_{\mathbf{D}({\mathcal{A}})}(X,Y)$ is bijective.

Proof.

See for example [26, Corollary 10.4.7]. ∎

Let $\operatorname{Proj}{\mathcal{A}}$ denote the full subcategory of ${\mathcal{A}}$ consisting of all objects that are projective. Denote by $\mathbf{K}^{-,b}(\operatorname{Proj}{\mathcal{A}})$ the full subcategory of complexes $X$ in $\mathbf{K}(\operatorname{Proj}{\mathcal{A}})$ such that $X^{n}=0$ for $n\gg 0$ and $H^{n}X=0$ for almost all $n\in\mathbb{Z}$. One says that ${\mathcal{A}}$ has enough projectives if each object in ${\mathcal{A}}$ is the quotient of some projective object.

Proposition 2.3.2**.**

Let ${\mathcal{A}}$ be an abelian category having enough projectives. Then the canonical functor $\mathbf{K}({\mathcal{A}})\to\mathbf{D}({\mathcal{A}})$ induces an equivalence

[TABLE]

Proof.

The functor $F\colon\mathbf{K}^{-,b}(\operatorname{Proj}{\mathcal{A}})\to\mathbf{D}({\mathcal{A}})$ is by definition the identity on objects, and $F$ is fully faithful by Lemma 2.3.1. It is clear that each object in the image of $F$ is isomorphic to one in $\mathbf{D}^{b}({\mathcal{A}})$. To show that each complex $X$ in $\mathbf{D}^{b}({\mathcal{A}})$ is isomorphic to one in the image of $F$, we use induction on

[TABLE]

Note that each bounded complex $X\neq 0$ fits into an exact triangle $X^{\prime}\to X^{\prime\prime}\to X\to X^{\prime}[1]$ such that $\ell(X^{\prime})=1$ and $\ell(X^{\prime\prime})=\ell(X)-1$. If $\ell(X)=1$, say $X^{n}\neq 0$, then $X\cong F(P[-n])$ where $P$ denotes a projective resolution of $X^{n}$. Such a resolution exists since ${\mathcal{A}}$ has enough projectives. If $\ell(X)>1$, then the induction hypothesis implies that the morphism $X^{\prime}\to X^{\prime\prime}$ is up to isomorphism of the form $F\phi$ for some morphism $\phi\colon P^{\prime}\to P^{\prime\prime}$ in $\mathbf{K}^{-,b}(\operatorname{Proj}{\mathcal{A}})$. Completing the morphism $\phi$ to an exact triangle $P^{\prime}\to P^{\prime\prime}\to P\to P^{\prime}[1]$ shows that $X$ belongs to $\operatorname{Im}F$ since $X\cong FP$. ∎

3. Tilting theory

Tilting provides a method to relate a category of coherent sheaves to a category of linear representations. For instance, a result of Beilinson [2] establishes for the category $\operatorname{coh}\mathbb{P}^{n}_{k}$ of coherent sheaves on the projective $n$-space over a field $k$ an equivalence of derived categories

[TABLE]

via a tilting object $T$ in $\operatorname{coh}\mathbb{P}^{n}_{k}$.444Except for $n=1$, the object $T={\mathcal{O}}(0)\oplus\dots\oplus{\mathcal{O}}(n)$ is not a tilting object in the strict sense of these notes; see Proposition 5.8.1.

In this section let $k$ be a field and ${\mathcal{A}}$ a $k$-linear abelian category that is Ext-finite. We show that each tilting object $T$ in ${\mathcal{A}}$ provides an equivalence of derived categories

[TABLE]

as in the example above. The principal reference for this result is [14], even though the proof given here is somewhat more direct, avoiding the formalism of torsion pairs and t-structures.

3.1. Tilting objects

Fix an object $T$ in ${\mathcal{A}}$. The object $T$ is called tilting object, provided that

1. (1)

$\operatorname{proj.dim}T\leq 1$, 2. (2)

$\operatorname{Ext}^{1}_{\mathcal{A}}(T,T)=0$, and 3. (3)

$\operatorname{Hom}_{\mathcal{A}}(T,A)=0=\operatorname{Ext}^{1}_{\mathcal{A}}(T,A)$ implies $A=0$ for each object $A$ in ${\mathcal{A}}$.

A morphism $T^{\prime}\to A$ in ${\mathcal{A}}$ is called right $T$-approximation of $A$ if it induces an epimorphism $\operatorname{Hom}_{\mathcal{A}}(T,T^{\prime})\to\operatorname{Hom}_{\mathcal{A}}(T,A)$ and $T^{\prime}$ belongs to $\operatorname{add}T$. An exact sequence $0\to A\to B\to T^{\prime}\to 0$ is called universal $T$-extension of $A$ if it induces an epimorphism $\operatorname{Hom}_{\mathcal{A}}(T,T^{\prime})\to\operatorname{Ext}^{1}_{\mathcal{A}}(T,A)$ and $T^{\prime}$ belongs to $\operatorname{add}T$. Such approximations and extensions exist for all $A$ in ${\mathcal{A}}$, since ${\mathcal{A}}$ is Ext-finite. Finally, set

[TABLE]

Lemma 3.1.1**.**

Let $T\in{\mathcal{A}}$ be a tilting object. Then the following holds:

1. (1)

Let $\pi\colon T^{\prime}\to A$ be a right $T$-approximation. Then $\operatorname{Ker}\pi$ is in ${\mathcal{T}}(T)$, $\operatorname{Hom}_{\mathcal{A}}(T,\operatorname{Coker}\pi)=0$, and $\operatorname{Ext}^{1}_{\mathcal{A}}(T,A)\xrightarrow{\sim}\operatorname{Ext}^{1}_{\mathcal{A}}(T,\operatorname{Coker}\pi)$. 2. (2)

Let $0\to A\to B\to T^{\prime}\to 0$ be a universal $T$-extension. Then $B\in{\mathcal{T}}(T)$. 3. (3)

The objects in ${\mathcal{T}}(T)$ are precisely the factor objects of objects in $\operatorname{add}T$.

Proof.

(1) Write the sequence $0\to A^{\prime}\to T^{\prime}\xrightarrow{\pi}A\to A^{\prime\prime}\to 0$ as composite of two exact sequences $0\to A^{\prime}\to T^{\prime}\to\bar{A}\to 0$ and $0\to\bar{A}\to A\to A^{\prime\prime}\to 0$. Then apply $\operatorname{Hom}_{\mathcal{A}}(T,-)$ to both sequences.

(2) Apply $\operatorname{Hom}_{\mathcal{A}}(T,-)$ to the sequence $0\to A\to B\to T^{\prime}\to 0$.

(3) Clearly, each factor of an object in $\operatorname{add}T$ belongs to ${\mathcal{T}}(T)$. For the other implication one uses (1). ∎

3.2. A derived equivalence

Let $T$ be an object in ${\mathcal{A}}$ and $\Lambda=\operatorname{End}_{\mathcal{A}}(T)$. We consider the functor

[TABLE]

This functor induces an equivalence $\operatorname{add}T\xrightarrow{\sim}\operatorname{proj}\Lambda$ and admits a left adjoint

[TABLE]

Given a $\Lambda$-module $M$ with projective presentation $P_{1}\to P_{0}\to M\to 0$, the cokernel of the corresponding morphism $T_{1}\to T_{0}$ in $\operatorname{add}T$ is by definition $M\otimes_{\Lambda}T$. For $i>0$, denote by

[TABLE]

the $i$-th left derived functor of $-\otimes_{\Lambda}T$ and set

[TABLE]

Lemma 3.2.1**.**

Let $T\in{\mathcal{A}}$ be a tilting object. Then $\operatorname{Tor}^{\Lambda}_{i}(-,T)=0$ for $i>1$.

Proof.

Let $M\in\operatorname{mod}\Lambda$ and choose a projective resolution

[TABLE]

Apply $-\otimes_{\Lambda}T$ and set $Z_{i}=\operatorname{Ker}(\delta_{i}\otimes_{\Lambda}T)$. The induced morphism $\bar{\delta}_{i+1}\colon P_{i+1}\otimes_{\Lambda}T\to Z_{i}$ is a right $T$-approximation for $i>0$, and therefore $Z_{i}$ belongs to ${\mathcal{T}}(T)$ for $i>1$ by Lemma 3.1.1. Thus $\bar{\delta}_{i+1}$ is an epimorphism for $i>1$, and this implies $\operatorname{Tor}^{\Lambda}_{i}(-,T)=0$. ∎

Lemma 3.2.2**.**

Let $T\in{\mathcal{A}}$ be a tilting object. Then $\operatorname{Hom}_{A}(T,-)$ and $-\otimes_{\Lambda}T$ restrict to equivalences between ${\mathcal{T}}(T)$ and ${\mathcal{Y}}(T)$.

Proof.

Fix objects $A\in{\mathcal{T}}(T)$ and $M\in{\mathcal{Y}}(T)$. We need to show that the adjunction morphisms

[TABLE]

are invertible.

Choose an exact sequence

[TABLE]

such that the induced morphism $T_{i}\to\operatorname{Im}\delta_{i}$ is a right $T$-approximation of $\operatorname{Im}\delta_{i}$ for each $i\geq 0$. Such a sequence exists by Lemma 3.1.1.

The functor $\operatorname{Hom}_{\mathcal{A}}(T,-)$ sends the sequence $\xi$ to a projective resolution of the $\Lambda$-module $\operatorname{Hom}_{\mathcal{A}}(T,A)$. Applying then $-\otimes_{\Lambda}T$ gives back $\xi$, that is, the adjunction morphism $\theta_{A}$ is invertible. Moreover, $\operatorname{Hom}_{\mathcal{A}}(T,A)$ belongs to ${\mathcal{Y}}(T)$.

Now choose an exact sequence $\pi\colon 0\to M^{\prime}\to P\to M\to 0$ such that $P$ is projective. Note that $M^{\prime}$ belongs to ${\mathcal{Y}}(T)$ by Lemma 3.2.1. The sequence $\pi\otimes_{\Lambda}T$ is exact since $M\in{\mathcal{Y}}(T)$, and the sequence $\operatorname{Hom}_{\mathcal{A}}(T,\pi\otimes_{\Lambda}T)$ is exact since $M^{\prime}\otimes_{\Lambda}T$ belongs to ${\mathcal{T}}(T)$. Thus there is the following commutative diagram with exact rows.

[TABLE]

The morphism $\eta_{P}$ is an isomorphism and it follows that $\eta_{M}$ is an epimorphism for all $M$ in ${\mathcal{Y}}(T)$. In particular, $\eta_{M^{\prime}}$ is an epimorphism. Using the snake lemma, it follows that $\eta_{M}$ is an isomorphism. ∎

Let $A$ be an object in ${\mathcal{A}}$. An $\operatorname{add}T$-resolution of $A$ is by definition a complex

[TABLE]

together with a quasi-isomorphism $Q\to A$ such that each $Q_{n}$ belongs to $\operatorname{add}T$.

Lemma 3.2.3**.**

Let $T\in{\mathcal{A}}$ be a tilting object and $Q\to A$ an $\operatorname{add}T$-resolution of an object $A\in{\mathcal{A}}$. Then

[TABLE]

for all $B\in{\mathcal{T}}(T)$ and $n\geq 0$.

Proof.

Use induction on $n$ and dimension shifting. ∎

Lemma 3.2.4**.**

Let $T\in{\mathcal{A}}$ be a tilting object. Then the functor $-\otimes_{\Lambda}T$ induces an isomorphism

[TABLE]

for all $M,N$ in ${\mathcal{Y}}(T)$ and $n\geq 0$.

Proof.

Choose a projective resolution $P\to M$ of $M$. Note that $N\cong\operatorname{Hom}_{\mathcal{A}}(T,N\otimes_{\Lambda}T)$ by Lemma 3.2.2, since $N$ belongs to ${\mathcal{Y}}(T)$. Then we obtain the following sequence of isomorphisms.

[TABLE]

The last isomorphism follows from Lemma 3.2.3, since $P\otimes_{\Lambda}T\to M\otimes_{\Lambda}T$ is an $\operatorname{add}T$-resolution. ∎

For a tilting object $T$ in ${\mathcal{A}}$, let us define the derived functor

[TABLE]

by taking a complex $P$ of projective $\Lambda$-modules with bounded cohomology to $P\otimes_{\Lambda}T$; see Proposition 2.3.2. The cohomology of $P\otimes_{\Lambda}T$ is bounded, since $\operatorname{Tor}_{i}^{\Lambda}(-,T)=0$ for $i>1$ by Lemma 3.2.1.

Theorem 3.2.5** (Happel-Reiten-Smalø).**

Let ${\mathcal{A}}$ be a $k$-linear abelian category that is Ext-finite. Let $T$ be a tilting object in ${\mathcal{A}}$ and $\Lambda=\operatorname{End}_{\mathcal{A}}(T)$. Then the functor

[TABLE]

is an equivalence of triangulated categories and its right adjoint $\operatorname{{\mathbf{R}}Hom}_{\mathcal{A}}(T,-)$ is a quasi-inverse.

We do not give the formal definition of the derived functor $\operatorname{{\mathbf{R}}Hom}_{\mathcal{A}}(T,-)$; all we use is the fact that it is a right adjoint of $-\otimes^{\mathbf{L}}_{\Lambda}T$.

Proof.

Set $F_{T}=-\otimes_{\Lambda}^{\mathbf{L}}T$. We identify objects in $\operatorname{mod}\Lambda$ and ${\mathcal{A}}$ with complexes that are concentrated in degree zero. For instance, $F_{T}M=M\otimes_{\Lambda}T$ for each $M$ in ${\mathcal{Y}}(T)$.

We need to show that for each pair of complexes $X,Y$ in $\operatorname{mod}\Lambda$, the induced map

[TABLE]

is bijective. Set

[TABLE]

and note that each bounded complex $X\neq 0$ fits into an exact triangle $X^{\prime}\to X\to X^{\prime\prime}\to X^{\prime}[1]$ such that $\ell(X^{\prime})=\ell(X)-1$ and $\ell(X^{\prime\prime})=1$.

Using the five lemma and induction on $\ell(X)+\ell(Y)$, one shows that $\phi_{X,Y}$ is bijective. The case $\ell(X)=\ell(Y)=1$ follows from Lemma 3.2.4. To be precise, one uses that each $\Lambda$-module $M$ fits into an exact sequence $0\to M^{\prime}\to P\to M\to 0$ with $M^{\prime},P$ in ${\mathcal{Y}}(T)$, which yields an exact triangle $M^{\prime}\to P\to M\to M^{\prime}[1]$ in $\mathbf{D}^{b}(\operatorname{mod}\Lambda)$.

Next we show that each object in $\mathbf{D}^{b}({\mathcal{A}})$ is isomorphic to one in the image of $F_{T}$. In fact, it suffices to show that each object in ${\mathcal{A}}$ belongs to the essential image $\operatorname{Im}F_{T}$, since $\operatorname{Im}F_{T}$ is a triangulated subcategory and $\mathbf{D}^{b}({\mathcal{A}})$ is generated (as a triangulated category) by the objects from ${\mathcal{A}}$.

It follows from Lemma 3.1.1 that each object $A$ in ${\mathcal{A}}$ fits into an exact triangle $A\to B\to C\to A[1]$ with $B,C$ in ${\mathcal{T}}(T)$. On the other hand, each $A$ in ${\mathcal{T}}(T)$ belongs to $\operatorname{Im}F_{T}$, since $A\cong F_{T}(\operatorname{Hom}_{\mathcal{A}}(T,A))$