Dualities of artinian coalgebras with applications to noetherian complete algebras

TL;DR

This paper establishes a duality theorem for derived categories of comodules over artinian coalgebras and explores implications for noetherian complete algebras, including conditions for Calabi-Yau properties.

Contribution

It introduces a duality theorem linking derived categories of comodules with applications to noetherian complete algebras, revealing conditions for Calabi-Yau structures.

Findings

Duality theorem for derived categories of comodules over artinian coalgebras

Isomorphism between local cohomology and twisted bimodules in certain algebras

Characterization of Calabi-Yau algebras via inner automorphisms

Abstract

A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual coalgebra. If $A$ is Artin-Schelter regular, then the local cohomology of $A$ is isomorphic to a shift of twisted bimodule ${}_1C_{\sigma^*}$ with $\sigma$ a coalgebra automorphism. This yields that the balanced dualinzing complex of $A$ is a shift of the twisted bimodule ${}_{\sigma^*}A_1$. If $\sigma$ is an inner automorphism, then $A$ is Calabi-Yau.