Dualities of artinian coalgebras with applications to noetherian complete algebras

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 ${}_1{C}_{{\sigma }^{\ast }}$ with $\sigma$ a coalgebra automorphism. This yields that the balanced dualinzing complex of $A$ is a shift of the twisted bimodule ${}_{{\sigma }^{\ast }}{A}_1$. If $\sigma$ is an inner automorphism, then $A$ is Calabi-Yau.

References in corpus (2)

• Calabi-Yau algebras
• Calabi-Yau coalgebras