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On Dual Algebras of Hopf Algebroids

Jingbang Guo

TL;DR

This work develops a dual-algebra perspective on Hopf algebroids by introducing $\Gamma^\vee=\mathrm{Hom}_{A,\eta_L}(\Gamma,A)$ as a topological algebra that encodes the comodule structure of $\Gamma$-comodules. Under a freeness hypothesis, comodules correspond to discrete $\Gamma^\vee$-modules, enabling a two-term resolution $0\to\Gamma^\vee\xrightarrow{-\cdot\nabla_q}\Gamma^\vee\xrightarrow{\eta_L^\vee}A\to 0$ and linking comodule cohomology to Ext groups in the profinite/discrete setting; this also connects to prismatic vector bundles via a $q$-connection $\nabla_q$ with a specific relation. The paper analyzes functoriality, monoidal structures, and base-change for dual algebras, showing how pullbacks of comodules correspond to dual maps after suitable factorizations, and discusses how to recover cohomological information through $\Gamma^\vee$-modules. It further surveys profinite-ring theory and Pontryagin duality to situate Ext computations for dual algebras within a robust cohomological framework, offering a pathway to compute comodule cohomology via derived functors in the profinite/discrete setting and hinting at applications to prismatic cohomology and topological cyclic homology. The authors propose extending the program to graded/filtered contexts and solid-abelian-group formalism to solidify the topology-sensitive aspects of the duality, aiming for intrinsic, stack-level invariants $\omega=\mathrm{RHom}_{\Gamma^\vee}(A,\Gamma^\vee)$ and presentation-independent cohomology.

Abstract

We study the dual algebras of (discrete) Hopf algebroids. In particular, we understand comodules over a Hopf algebroid as (discrete) modules over its dual algebra.

On Dual Algebras of Hopf Algebroids

TL;DR

This work develops a dual-algebra perspective on Hopf algebroids by introducing as a topological algebra that encodes the comodule structure of -comodules. Under a freeness hypothesis, comodules correspond to discrete -modules, enabling a two-term resolution and linking comodule cohomology to Ext groups in the profinite/discrete setting; this also connects to prismatic vector bundles via a -connection with a specific relation. The paper analyzes functoriality, monoidal structures, and base-change for dual algebras, showing how pullbacks of comodules correspond to dual maps after suitable factorizations, and discusses how to recover cohomological information through -modules. It further surveys profinite-ring theory and Pontryagin duality to situate Ext computations for dual algebras within a robust cohomological framework, offering a pathway to compute comodule cohomology via derived functors in the profinite/discrete setting and hinting at applications to prismatic cohomology and topological cyclic homology. The authors propose extending the program to graded/filtered contexts and solid-abelian-group formalism to solidify the topology-sensitive aspects of the duality, aiming for intrinsic, stack-level invariants and presentation-independent cohomology.

Abstract

We study the dual algebras of (discrete) Hopf algebroids. In particular, we understand comodules over a Hopf algebroid as (discrete) modules over its dual algebra.
Paper Structure (6 sections, 16 theorems, 106 equations)

This paper contains 6 sections, 16 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. The Dual Algebras of Hopf algebroids
  3. The Functoriality
  4. The Symmetric Monoidal Structure
  5. Profinite Rings and Dual Algebras
  6. Further Development

Key Result

Theorem 1.18

Assume that, through the left unit map $\eta_L:A\rightarrow \Gamma$, $\Gamma$ is a free $A$-module with a countable basis. Then, by regarding comodules as modules, there is an equivalence of categories where $\mathrm{RCoMod}_\Gamma$ is the category of right $\Gamma$-comodules and $\mathrm{Mod}^d_{\Gamma^\vee}$ is the category of discrete $\Gamma^\vee$-modules (with continuous $\Gamma^\vee$-action

Theorems & Definitions (78)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Example 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 68 more