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

Hiroyuki Minamoto

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 Căldăraru--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.

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

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 Căldăraru--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---bimodules, called a universal Auslander--Reiten triangle in the sense that the derived tensor product of this triangle with an indecomposable dg--module yields an Auslander--Reiten triangle starting from . 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.
Paper Structure (42 sections, 54 theorems, 110 equations)

This paper contains 42 sections, 54 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Serre duality and Mukai pairing
  3. Universal Auslander-Reinten triangle
  4. Related work and perspective
  5. Organization of the paper
  6. Notations and conventions
  7. Hochschild (co)homology, tensor formulas of dual dg-bimodules and string diagrams
  8. Preliminaries of dg-algebras and dg-bimodules
  9. Euler characteristic and (super) trace
  10. Hochschild (co)homology complex
  11. Representation of derived tensor products by string diagrams
  12. Dual of dg-bimodules and their derived tensor products
  13. Mukai pairing and Serre duality
  14. A canonical duality of Hochschild (co)homology group, Serre duality and Mukai pairing
  15. A canonical duality of Hochschild (co)homology group
  16. ...and 27 more sections

Key Result

Theorem 1.1

The three Mukai pairings coincide with each other: where $\langle -,+ \rangle_{\mathsf{Shk\textup{-}Mukai}}$ and $\langle -,+ \rangle_{\mathsf{CW\textup{-}Mukai}}$ are the Mukai pairings in the sense of Shklyarov and of Căldăraru–Willerton, respectively.

Theorems & Definitions (88)

  • Theorem 1.1: Theorem
  • Theorem 1.2: Universal Auslander--Reiten triangle (Corollary )
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7
  • proof : Proof of Lemma
  • ...and 78 more