Serre duality, Mukai pairing and universal Auslander--Reiten triangle

TL;DR

This paper explores the connections between Serre duality, Mukai pairing, and Auslander--Reiten theory in the context of smooth proper dg-algebras, introducing new frameworks and functorial constructions for these concepts.

Contribution

It introduces an alternative definition of the Mukai pairing, unifies it with Serre duality, and constructs a universal Auslander--Reiten triangle functorially for dg-algebras.

Findings

Mukai pairing coincides with previous definitions.

Unified framework for Serre duality and Mukai pairing.

Functorial construction of Auslander--Reiten triangles.

Abstract

We study the relationship between Serre duality and the Mukai pairing for smooth and proper dg-algebras. We introduce an alternative definition of the Mukai pairing and prove that it coincides with the Mukai pairings defined by C\u{a}ld\u{a}raru--Willerton and by Shklyarov. Our construction places both the Mukai pairing and Serre duality within a unified framework based on an elementary pairing between Hochschild homology and Hochschild cohomology. As a consequence, the adjointness of the boundary--bulk and bulk--boundary maps follows naturally. As an application, we investigate Auslander--Reiten theory for the perfect derived category of a non-positive smooth and proper dg-algebra. We construct an exact triangle of dg-$A$-$A$-bimodules, called a universal Auslander--Reiten triangle in the sense that the derived tensor product of this triangle with an indecomposable dg-$A$-module $M$ yields an Auslander--Reiten triangle starting from $M$. In particular, this provides a functorial construction of Auslander--Reiten triangles. In the case of path algebras of quivers, our construction recovers the universal Auslander--Reiten triangle associated with quiver Heisenberg algebras.