On Serre duality for compact homologically smooth DG algebras

TL;DR

This paper establishes a Serre functor for perfect modules over compact, smooth DG algebras and demonstrates a non-degenerate pairing on their Hochschild homology, paralleling classical Hodge theory.

Contribution

It introduces a Serre functor for arbitrary compact, smooth DG algebras and proves the existence of a duality pairing on Hochschild homology.

Findings

Defined a Serre functor for perfect modules over DG algebras.

Proved the existence of a non-degenerate pairing on Hochschild homology.

Connected algebraic structures to classical Hodge theory.

Abstract

The bounded derived category of coherent sheaves on a smooth projective variety is known to be equivalent to the triangulated category of perfect modules over a DG algebra. DG algebras, arising in this way, have to satisfy some compactness and smoothness conditions. In this paper, we describe a Serre functor on the category of perfect modules over an arbitrary compact and smooth DG algebra and use it to prove the existence of a non-degenerate pairing on the Hochschild homology of the DG algebra. This pairing is an algebraic analog of a well-known pairing on the Hodge cohomology of a smooth projective variety.