Gorenstein homological invariants and monoidal model categories of Hopf algebras

TL;DR

This paper explores Gorenstein homological invariants in Hopf algebras, showing their invariance under certain equivalences and establishing monoidal model category structures related to Gorenstein projective modules.

Contribution

It proves the equivalence of Gorenstein global and projective dimensions for Hopf algebras, and demonstrates invariance of Gorenstein properties under monoidal Morita-Takeuchi equivalence.

Findings

Gorenstein global dimension equals Gorenstein projective dimension of the trivial module

Finite dimensionality of Hopf algebra characterized by trivial Gorenstein projective dimension of k

Categories of modules form monoidal model categories with tensor triangulated stable categories

Abstract

Let $H$ be a Hopf algebra over a field $k$ with a bijective antipode. It is proved that the Gorenstein global dimension of $H$ coincides with the Gorenstein projective dimension of the trivial left (or right) $H$-module $k$. Then, $H$ is finite dimensional if and only if the Gorenstein projective dimension of $k$ is trivial. Although monoidal Morita-Takeuchi equivalence of Hopf algebras does not preserve the global dimension, we demonstrate that it does preserve the Gorenstein global dimension and the Artin-Schelter Gorenstein property; this supports Brown-Goodearl's question of whether every noetherian (affine) Hopf algebra is AS Gorenstein. Finally, for $H$ and an $H$-Galois object $B$, we show the categories of modules $_H\mathcal{M}$ and $_B\mathcal{M}_B^H$ are monoidal model categories regarding Gorenstein projective model structure, provided that the Gorenstein global dimension of $H$ is finite. The corresponding stable categories are tensor triangulated categories.