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Monoidal invariance of the cohomological dimension of Hopf algebras: the finite case

Julien Bichon

TL;DR

The paper addresses the question of whether the global dimension of Hopf algebras is preserved under monoidal equivalence of their comodule categories. It introduces a direct, Gorenstein-free proof that finite global dimensions are invariant under such equivalences and shows the bijectivity of the antipode is not essential. A general transfer principle for projective dimensions under exact adjoint functors is established and then applied to Hopf-Galois objects, linking global dimension to Hochschild cohomological dimension via Miyashita–Ulbrich actions. Consequently, if $A$ and $B$ have monoidally equivalent comodule categories and finite global dimensions, then $\mathrm{gldim}(A)=\mathrm{gldim}(B)$, realized through a bi-Galois object $R$ with $\mathrm{gldim}(A)=\mathrm{cd}(R)=\mathrm{gldim}(B)$.

Abstract

A consequence of the recent work of Ren and Zhu on Gorenstein projective dimensions of modules over Hopf algebras is that if $A$ and $B$ are Hopf algebras with bijective antipodes having equivalent linear tensor categories of comodules and both having finite global dimensions, then their global dimensions coincide. In this note we provide a direct proof of this result, without using Gorenstein projective dimensions, and we notice that the assumption on the bijectivity of the antipodes can be removed.

Monoidal invariance of the cohomological dimension of Hopf algebras: the finite case

TL;DR

The paper addresses the question of whether the global dimension of Hopf algebras is preserved under monoidal equivalence of their comodule categories. It introduces a direct, Gorenstein-free proof that finite global dimensions are invariant under such equivalences and shows the bijectivity of the antipode is not essential. A general transfer principle for projective dimensions under exact adjoint functors is established and then applied to Hopf-Galois objects, linking global dimension to Hochschild cohomological dimension via Miyashita–Ulbrich actions. Consequently, if and have monoidally equivalent comodule categories and finite global dimensions, then , realized through a bi-Galois object with .

Abstract

A consequence of the recent work of Ren and Zhu on Gorenstein projective dimensions of modules over Hopf algebras is that if and are Hopf algebras with bijective antipodes having equivalent linear tensor categories of comodules and both having finite global dimensions, then their global dimensions coincide. In this note we provide a direct proof of this result, without using Gorenstein projective dimensions, and we notice that the assumption on the bijectivity of the antipodes can be removed.
Paper Structure (3 sections, 5 theorems, 13 equations)

This paper contains 3 sections, 5 theorems, 13 equations.

Table of Contents

  1. introduction
  2. The key result
  3. Cohomological dimension of Hopf-Galois objects

Key Result

Theorem 1.2

Let $A$, $B$ be Hopf algebras with bijective antipodes such that $\mathcal{M}^A \simeq^\otimes\mathcal{M}^B$. If $\mathop{\mathrm{gldim}}\nolimits(A)$ and $\mathop{\mathrm{gldim}}\nolimits(B)$ are finite, then $\mathop{\mathrm{gldim}}\nolimits(A)=\mathop{\mathrm{gldim}}\nolimits(B)$.

Theorems & Definitions (12)

  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 2 more