On Dual Algebras of Hopf Algebroids

1. Problem Statement

The paper addresses the algebraic and categorical relationship between Hopf algebroids and their dual algebras.

• Context: Hopf algebroids $(A, \Gamma)$ are fundamental in algebraic topology (e.g., in the computation of Topological Cyclic Homology, $TC$) and algebraic geometry, where they encode the descent data for quasi-coherent sheaves on stacks. Specifically, a quasi-coherent sheaf on a stack $X$ associated with $(A, \Gamma)$ is equivalent to a right $\Gamma$-comodule.
• The Gap: While the category of $\Gamma$-comodules ($RCoMod_\Gamma$) is well-understood, the paper seeks to reinterpret these comodules as modules over a dual algebra $\Gamma^\vee$.
• The Challenge: The dual algebra $\Gamma^\vee = \text{Hom}_{A, \eta_L}(\Gamma, A)$ is generally non-commutative and carries a natural topology (compact-open). Standard module theory over discrete rings is insufficient to capture the full structure of comodules, especially when $\Gamma$ is infinite or topological. The paper aims to establish an equivalence of categories between comodules and a specific class of topological modules, and to understand how the symmetric monoidal structure and cohomology translate in this dual setting.

2. Methodology

The author employs a blend of homological algebra, topological ring theory, and category theory.

• Definition of the Dual Algebra:
  The dual algebra is defined as $\Gamma^\vee := \text{Hom}_{A, \eta_L}(\Gamma, A)$, where $\Gamma$ is viewed as a free $A$-module via the left unit $\eta_L$. The multiplication in $\Gamma^\vee$ is induced by the comultiplication $\Delta$ of the Hopf algebroid, and the unit is induced by the counit $\epsilon$.
• Topological Framework:
  Since $\Gamma$ is often infinite, $\Gamma^\vee$ is equipped with the compact-open topology. The paper treats $\Gamma^\vee$ as a topological ring.
  • Discrete Modules: The category of interest is $\text{Mod}^d_{\Gamma^\vee}$, the category of discrete left $\Gamma^\vee$-modules with continuous actions.
• Categorical Equivalence:
  The core methodology involves constructing a functor between the category of right $\Gamma$-comodules ($RCoMod_\Gamma$) and $\text{Mod}^d_{\Gamma^\vee}$. This relies on the canonical isomorphism $M \otimes_{A, \eta_L} \Gamma \cong \text{Hom}_A(\Gamma^\vee, M)$ for free modules, extended to comodules via adjunction.
• Functoriality and Factorization:
  To handle morphisms between Hopf algebroids (which do not naturally induce morphisms between dual algebras in the reverse direction), the author utilizes a factorization technique. A morphism $f: (A, \Gamma) \to (B, \Sigma)$ is factored through an intermediate Hopf algebroid $(B, \Gamma_f)$, allowing the construction of a morphism between dual algebras $\Sigma^\vee \to \Gamma_f^\vee$ that induces the correct functor on module categories.
• Profinite and Solid Structures:
  In cases where $A$ is a finite ring, $\Gamma^\vee$ becomes a profinite ring. The paper leverages Pontryagin duality and the theory of solid abelian groups (referencing Scholze's work) to handle cohomological computations and tensor products, suggesting that a full theory requires the "solid" framework to unify discrete and profinite modules.

3. Key Contributions and Results

A. Equivalence of Categories (The Main Theorem)

Theorem 2.22: Under the assumption that $\Gamma$ is a free $A$-module with a countable basis, there is an equivalence of categories:
$$RCoMod_\Gamma \xrightarrow{\sim} \text{Mod}^d_{\Gamma^\vee}$$
where $\text{Mod}^d_{\Gamma^\vee}$ is the category of discrete $\Gamma^\vee$-modules with continuous actions.

• Significance: This establishes that studying comodules over a Hopf algebroid is equivalent to studying modules over a specific topological algebra. This allows the use of standard module-theoretic tools (like projective resolutions) in the context of Hopf algebroids.

B. Structure of the Dual Algebra

• Non-commutativity: The paper clarifies that $\Gamma^\vee$ is generally non-commutative, even if $A$ and $\Gamma$ are commutative. The left and right $A$-module structures on $\Gamma^\vee$ differ unless $\eta_L = \eta_R$.
• Generators and Relations: In the specific example of the Hopf algebroid arising from prismatic envelopes (related to $TC$ computations), the dual algebra is generated by a differential operator $\nabla_q$ satisfying specific commutation relations (e.g., $\nabla_q \circ f(q) = f(q^{1+p}) \circ \nabla_q + \dots$).

C. Functoriality of Dual Algebras

The paper resolves the lack of direct functoriality for dual algebras.

• Construction 3.4 & 3.6: It shows that while a morphism of Hopf algebroids $f: (A, \Gamma) \to (B, \Sigma)$ does not directly yield $\Sigma^\vee \to \Gamma^\vee$, it factors through an intermediate object. This allows the construction of a continuous homomorphism $\beta^\vee: \Sigma^\vee \to \Gamma_f^\vee$ which induces the correct pullback functor on module categories, matching the pullback of comodules.

D. Symmetric Monoidal Structure

The paper investigates how the tensor product of comodules translates to the dual side.

• Challenge: The tensor product $M \otimes_A N$ of comodules corresponds to a "co-multiplication" on $\Gamma^\vee$. However, defining a tensor product for topological modules is non-trivial.
• Proposal: The author suggests that the solid tensor product of solid abelian groups is the correct framework to define the symmetric monoidal structure on $\text{Mod}^d_{\Gamma^\vee}$, reflecting the monoidal structure of $RCoMod_\Gamma$.

E. Cohomology and Profinite Rings

• Profinite Case: When $A$ is finite, $\Gamma^\vee$ is a profinite ring. The paper reviews the cohomological algebra for profinite rings, distinguishing between discrete modules and profinite modules.
• Ext Groups: It establishes that $\text{Ext}_{\Gamma^\vee}(M, N)$ can be computed via projective resolutions of profinite modules or injective resolutions of discrete modules, provided one works within the appropriate category (or the unified category of solid modules).
• Dualizing Object: The paper introduces the dualizing object $\omega = R\text{Hom}_{\Gamma^\vee}(A, \Gamma^\vee)$. In the prismatic example, this object is computed to be related to the module of differentials, suggesting a link to Poincaré duality in this context.

4. Significance and Applications

1. Computational Tool for Topological Cyclic Homology ($TC$):
   The primary motivation is the computation of $TC$ for rings of integers in local fields. These computations often reduce to cohomology of comodules over Hopf algebroids derived from prismatic envelopes. By converting these to module problems over dual algebras, the paper provides a pathway to use projective resolutions (which are often easier to construct than injective comodules) to compute these cohomology groups.

2. Geometrization of $p$-adic Cohomology:
   The work bridges the gap between the geometric language of stacks (where comodules live) and the algebraic language of rings and modules. It formalizes the idea that the "dual algebra" encodes the monadicity of the stack, complementing the comonadicity of the cover.

3. Foundations for Solid Mathematics:
   The paper explicitly advocates for the use of solid abelian groups (as developed by Clausen and Scholze) to handle the topological nuances of dual algebras. It suggests that the theory of Hopf algebroids and their duals is a natural testing ground for solid mathematics, particularly regarding tensor products and duality in non-discrete settings.

4. Generalization of Classical Duality:
   It generalizes classical results (like the duality between Hopf algebras and their duals) to the more complex setting of Hopf algebroids, handling the non-commutativity and topological constraints inherent in modern arithmetic geometry.

Conclusion

Jingbang Guo's paper provides a rigorous framework for treating comodules over Hopf algebroids as modules over a topological dual algebra. By establishing an equivalence of categories and addressing the topological and functorial subtleties, the paper offers a powerful new perspective for computing cohomology in arithmetic topology and algebraic geometry, particularly in the context of $p$-adic cohomology theories and Topological Cyclic Homology. It strongly points toward the necessity of solid mathematics for a complete theoretical treatment of these dual structures.