Gorenstein homological invariants and monoidal model categories of Hopf algebras

TL;DR

This work studies Gorenstein homological invariants in Hopf algebras, establishing that the left and right Gorenstein global dimensions of a Hopf algebra $H$ (with bijective antipode) align with the Gorenstein projective dimension of the trivial module $_{\varepsilon}k$, and that finiteness of this quantity characterizes finite dimensionality. It proves that monoidal Morita–Takeuchi equivalence preserves both the Gorenstein global dimension and the AS Gorenstein property, and provides a precise equality linking several Gorenstein invariants across monoidally equivalent Hopf algebras. Furthermore, when the Gorenstein projective dimension is finite, the paper constructs monoidal Gorenstein projective model structures on module categories $_H\mathcal{M}$ and $_B\mathcal{M}_B^H$, showing their homotopy (and stable) categories are tensor triangulated, thereby connecting Gorenstein homological algebra with monoidal Hopf theory and providing tools for Brown–Goodearl type questions about AS Gorenstein Hopf algebras.

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.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Proposition 3
• Theorem 4
• Lemma 1.1
• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof