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Homological duality and exact sequences of Hopf algebras

Julian Le Clainche

TL;DR

The paper develops a Hopf-algebraic framework for homological duality, establishing that duality properties are stable under exact sequences of Hopf algebras when the middle algebra is faithfully flat over the subalgebra and the outer algebras are smooth. It introduces Hopf-algebra Lyndon–Hochschild–Serre spectral sequences to connect Ext and Tor groups across the exact sequence, then proves that $\mathrm{cd}(A)=\mathrm{cd}(B)+\mathrm{cd}(H)$ and that duality transfers from $B$ and $H$ to $A$ (and conversely). It further shows that twisted Calabi–Yau properties are preserved in the sum of cohomological dimensions and provides concrete examples, including universal pointed Hopf algebras and quantum groups of $GL(2)$, where these dualities yield new instances and explicit structures.

Abstract

We study the stability of homological duality properties of Hopf algebras under extensions.

Homological duality and exact sequences of Hopf algebras

TL;DR

The paper develops a Hopf-algebraic framework for homological duality, establishing that duality properties are stable under exact sequences of Hopf algebras when the middle algebra is faithfully flat over the subalgebra and the outer algebras are smooth. It introduces Hopf-algebra Lyndon–Hochschild–Serre spectral sequences to connect Ext and Tor groups across the exact sequence, then proves that and that duality transfers from and to (and conversely). It further shows that twisted Calabi–Yau properties are preserved in the sum of cohomological dimensions and provides concrete examples, including universal pointed Hopf algebras and quantum groups of , where these dualities yield new instances and explicit structures.

Abstract

We study the stability of homological duality properties of Hopf algebras under extensions.
Paper Structure (14 sections, 38 theorems, 97 equations)

This paper contains 14 sections, 38 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Hochschild cohomology and homological duality
  3. Finiteness conditions on modules
  4. Poincaré duality modules
  5. Hochschild (co)homology and homological duality
  6. Cohomology of Hopf algebras
  7. Exact sequences of Hopf algebras
  8. Definitions
  9. Spectral sequences for exact sequences of Hopf algebras
  10. Homological smoothness and exact sequences of Hopf algebras
  11. Homological duality for exact sequences of Hopf algebras
  12. Examples
  13. The universal pointed Hopf algebra coacting on $k[x,y]$
  14. Quantum groups of $GL(2)$ representation type

Key Result

Theorem 1.1

Let $k \longrightarrow B \overset{i}{\longrightarrow} A \overset{p}{\longrightarrow} H \longrightarrow k$ be an exact sequence of Hopf algebras with bijective antipodes. Assume that $A$ is faithfully flat as a left and right $B$-module, and that $H$ and $B$ are smooth algebras. Then the following as If these conditions hold, we have $\mathrm{cd} (A) = \mathrm{cd}(B) + \mathrm{cd}(H).$

Theorems & Definitions (85)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • ...and 75 more