Duality and Serre functor in homotopy categories

TL;DR

This paper establishes a duality between certain homotopy categories of modules over coherent rings and artin algebras, leading to the existence of Serre functors and Auslander-Reiten triangles in these categories.

Contribution

It introduces a duality between homotopy categories of modules over coherent rings and artin algebras, and shows the existence of Serre functors in these categories.

Findings

Duality between homotopy categories for coherent rings and artin algebras.

Existence of Serre functor in the category of acyclic complexes.

Presence of Auslander-Reiten triangles in these categories.

Abstract

For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A^{{\rm op}})$ and ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A)$. If $A=\Lambda$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, ${\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda^{\rm op})$ and ${\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda).$ As a result, it will be shown that, in this case, ${\mathbb{K}}_{\rm ac}^{{\rm{b}}}({\rm mod}{\mbox{-}}\Lambda)$ admits a Serre functor and hence has Auslander-Reiten triangles.

Equations

$$\mu :\mathbb{D}(\mathrm{Mod}\text{mbox}-(\mathrm{mod}\text{mbox}-A^{\mathrm{op}}))⟶\mathbb{D}(\mathrm{Mod}\text{mbox}-(\mathrm{mod}\text{mbox}-A{)}^{\mathrm{op}})$$

$$\mathfrak{D}:\mathrm{mod}\text{mbox}-(\mathrm{mod}\text{mbox}-A^{\mathrm{op}}{)}^{\mathrm{op}}⟶\mathrm{mod}\text{mbox}-(\mathrm{mod}\text{mbox}-A{)}^{\mathrm{op}}.$$

$$\frac{{\mathbb{K}}^{\mathrm{b}}(\mathrm{mod}\text{mbox}-\Lambda )}{{\mathbb{K}}^{\mathrm{b}}(\mathrm{prj}\text{mbox}-\Lambda )}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(-,X)⟶(-,Y)⟶F⟶0,$$

$$resp.(X,-)⟶(Y,-)⟶F⟶0,$$

Full text

1. Introduction

A contravariant functor between two categories that is an equivalence is called a duality. The role and importance of dualities is known in representation theory of algebras. Let $A$ be a right and left coherent ring. In this paper, we introduce and study a duality between the bounded homotopy categories of finitely generated right and finitely generated left $A$-modules, denoted by $\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}A)$ and $\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}A^{{\rm{op}}})$, respectively. We gain this duality starting from an equivalence

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of derived categories of functor categories.

The relationship between $\mu$ and some known dualities will be discussed. In particular, it is shown that, Proposition LABEL:AusGJ-Dualtity below, there is a close relationship between $\mu$ and the Auslander-Gruson-Jensen duality

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Let $\Lambda$ be an artin algebra of finite global dimension over a commutative artinian ring $R$. We show that in this case, the above duality between $\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}\Lambda^{{\rm{op}}})$ and $\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}\Lambda)$ restricts to a duality between $\mathbb{K}^{{\rm{b}}}_{{\rm{ac}}}({\rm{{mod\mbox{-}}}}\Lambda^{{\rm{op}}})$ and $\mathbb{K}^{{\rm{b}}}_{{\rm{ac}}}({\rm{{mod\mbox{-}}}}\Lambda),$ where for an abelian category $\mathcal{A}$, $\mathbb{K}_{{\rm{ac}}}^{{\rm{b}}}(\mathcal{A})$ is the full subcategory of $\mathbb{K}^{{\rm{b}}}(\mathcal{A})$ consisting of all acyclic complexes. This, in turn, implies that there is an equivalence of triangulated categories

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Note that under certain conditions, the quotient $\frac{\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}\Lambda)}{\mathbb{K}^{{\rm{b}}}({\rm{prj}\mbox{-}}\Lambda)}$ is equivalent to the relative singularity category introduced and studied recently in [KY], see Remark LABEL:KY.

Finally, we show that $\mathbb{K}_{{\rm{ac}}}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}\Lambda)$ admits a Serre functor $\mathbb{S}$ in the sense of [BK]. By a well-known result of Reiten and Van den Bergh [RV, Theorem I.2.4], the existence of a Serre functor is equivalent to the existence of Auslander-Reiten triangles in a category and so we deduce that $\mathbb{K}_{{\rm{ac}}}^{{\rm{b}}}({\rm{{mod\mbox{-}}}}\Lambda)$ admits Auslander-Reiten triangles.

2. Preliminaries

Throughout the paper, $A$ denotes a right and left coherent ring. $A$-module means right $A$-module. ${\rm{Mod\mbox{-}}}A$, resp. ${\rm{{mod\mbox{-}}}}A$, denotes the category of $A$-modules, resp. finitely presented $A$-modules. ${\rm{Prj}\mbox{-}}A$, resp. ${\rm{prj}\mbox{-}}A$, denotes the full subcategory of ${\rm{Mod\mbox{-}}}A$, resp. ${\rm{{mod\mbox{-}}}}A$, consisting of projective $A$-modules. ${\rm{Inj}\mbox{-}}A$ and ${\rm{inj}\mbox{-}}A$ represent injectives and finitely presented injectives, resp. For an additive category $\mathcal{A}$, $\mathbb{D}(\mathcal{A})$, resp. $\mathbb{K}(\mathcal{A})$, denotes the derived category, resp. homotopy category, of $\mathcal{A}$. As usual, the bounded derived, resp. homotopy, category of $\mathcal{A}$, will be denoted by $\mathbb{D}^{{\rm{b}}}(\mathcal{A})$, resp. $\mathbb{K}^{{\rm{b}}}(\mathcal{A})$.

2.1**.**

Following Auslander we let ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)$, resp. ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)^{{\rm{op}}}$, denote the category of all contravariant, resp. covariant, additive functors from ${\rm{{mod\mbox{-}}}}A$ to $\mathcal{A}b$, the category of abelian groups. Throughout we shall use parenthesis to denote the Hom sets. An object $F$ of ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)$, resp. ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)^{{\rm{op}}}$, is called coherent if there exists a short exact sequence

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of functors, where $X$ and $Y$ belong to ${\rm{{mod\mbox{-}}}}A$. We let ${\rm{{mod\mbox{-}}}}({\rm{{mod\mbox{-}}}}A)$, resp. ${\rm{{mod\mbox{-}}}}({\rm{{mod\mbox{-}}}}A)^{{\rm{op}}}$, denote the full subcategory of ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)$, resp. ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)^{{\rm{op}}}$, consisting of all coherent functors. It is known [As1] that both ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)$ and ${\rm{{mod\mbox{-}}}}({\rm{{mod\mbox{-}}}}A)$ and also their counterparts of covariant functors are abelian categories with enough projective objects.

Special objects of such categories have been studied by several authors. In particular, an object $F$ of ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)$ is flat if and only if there exists an $A$-module $M$ such that $F\cong{\rm{Hom}}_{A}(-,M)$, see [JL, Theorem B.10]. We let $\mathcal{F}({\rm{{mod\mbox{-}}}}A)$ denote the full subcategory of ${\rm{Mod\mbox{-}}}({\rm{{mod\mbox{-}}}}A)$ consisting of all flat functors.