🔢 mathematics

Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions

This paper investigates the Hochschild cohomology of restricted faithfully flat weak Hopf Galois extensions by establishing a spectral sequence that connects the cohomologies of the extension and base algebras, ultimately proving that the AS-Gorenstein property is preserved from the base algebra $B$ to the extension algebra $A$ under specific noetherian and PI conditions.

Original authors: Daowei Lu, Dingguo Wang

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daowei Lu, Dingguo Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to understanding the hidden symmetries and structural rules that govern complex systems. Imagine a universe built not of stars and planets, but of algebraic shapes and rules for how they can be combined, split, or transformed. Within this universe, mathematicians study objects called algebras, which are like sets of instructions for performing calculations. Some of these algebras are particularly well-behaved, possessing a kind of internal balance that makes them predictable and elegant. One such class of well-behaved algebras is known as Artin-Schelter Gorenstein algebras. These are special because they have a finite depth, a specific kind of symmetry in how they can be stretched or broken, and a unique way of measuring their complexity. Understanding when a new algebra inherits these beautiful properties from an older one is a central question, as it helps mathematicians map the boundaries between order and chaos in abstract structures.

To explore these boundaries, researchers often look at how algebras relate to one another through extensions. Think of an extension as building a larger structure on top of a smaller, existing foundation. In the world of algebra, this often involves a "Hopf algebra," which acts like a sophisticated toolkit containing rules for symmetry and transformation. When a new algebra is constructed using these rules, it is called a Hopf Galois extension. For decades, mathematicians have known that if the foundation is a well-behaved Artin-Schelter Gorenstein algebra, the new structure built upon it often inherits that same elegance. However, the toolkit used in these constructions has recently been expanded. A newer, more flexible version of the symmetry toolkit, called a "weak Hopf algebra," was introduced to handle more complex and varied situations. Unlike the traditional toolkit, this weaker version allows for rules that are slightly less rigid, opening the door to a wider range of mathematical possibilities. The question then became: does the beautiful property of being Artin-Schelter Gorenstein survive when we use this more flexible, weaker toolkit?

In a recent study, researchers Daowei Lu and Dingguo Wang set out to answer this question. They focused on a specific type of algebraic construction where a new algebra is built over a base algebra using the rules of a weak Hopf algebra. Their goal was to determine if the new, larger algebra would still possess the desirable Artin-Schelter Gorenstein property, provided the base algebra already had it. To do this, they had to navigate a complex web of relationships between the different parts of these structures. They began by developing a new mathematical lens, a tool known as a spectral sequence, which allows one to see how the cohomology, or the hidden topological features, of the base algebra relate to those of the new, extended algebra. This tool acts like a bridge, connecting the known properties of the foundation to the unknown properties of the structure built upon it.

The researchers first established a precise connection between the cohomology of the base algebra and the cohomology of the extended algebra. They showed that by using this bridge, one can translate information from the simpler, known world of the base algebra into the more complex world of the extension. They proved that under specific conditions—namely, when the extension is "faithfully flat," meaning the new structure is built without any gaps or distortions, and when the base algebra is a type of well-behaved algebra known as a Noetherian affine PI algebra—the relationship is strong enough to preserve the Artin-Schelter Gorenstein property. In simpler terms, they demonstrated that if you start with a perfectly balanced foundation and build upon it using these specific weak symmetry rules, the resulting structure will also be perfectly balanced.

The study confirms that the elegant property of being Artin-Schelter Gorenstein is robust enough to withstand the introduction of these weaker, more flexible symmetry rules. The authors proved that when the base algebra is Artin-Schelter Gorenstein, the extended algebra inherits this property, maintaining the same finite depth and symmetry. Furthermore, they were able to determine exactly how the injective dimension of the new algebra relates to the injective dimension of the base and the symmetry toolkit used, but only under an additional condition: if the symmetry toolkit (the weak Hopf algebra) is itself Artin-Schelter Gorenstein of a specific dimension, then the injective dimension of the new structure is simply the sum of the base's injective dimension and the toolkit's dimension. This finding extends previous results that were limited to the stricter, traditional symmetry rules, showing that the fundamental beauty of these algebraic structures persists even when the rules of their construction are relaxed. The work provides a definitive answer to a long-standing question in the field, confirming that the structural integrity of these algebras is preserved even in the more flexible landscape of weak Hopf algebras.

Technical Summary: Hochschild Cohomology and AS-Gorenstein Property of Weak Hopf Galois Extensions

Problem Statement
The paper addresses the homological properties of weak Hopf Galois extensions, specifically focusing on the preservation of the Artin-Schelter (AS) Gorenstein property. While the behavior of AS-Gorenstein algebras under classical Hopf Galois extensions was established by Zhu [25] using spectral sequences derived from Hochschild cohomology (following Stefan [21]), the analogous theory for weak Hopf algebras remained to be developed. Weak Hopf algebras, which generalize both classical Hopf algebras and groupoid algebras, relax the condition that the comultiplication must preserve the unit. The central problem is to determine whether a restricted faithfully flat weak Hopf Galois extension $A/B$ inherits the AS-Gorenstein property from the base algebra $B$ , and to construct the necessary cohomological machinery (spectral sequences) to prove this.

Methodology
The authors employ a combination of homological algebra and the specific structural theory of weak Hopf algebras. The methodology proceeds in three main stages:

Construction of Ext-Group Isomorphisms: The paper first establishes linear isomorphisms between Ext-groups involving weak Hopf modules. Specifically, it proves that $\text{Ext}^i_{Hop}(V \otimes W, X) \cong \text{Ext}^i_{Hop}(V, \text{Hom}^s_{Hop}(W, X))$ for right modules, and a similar result for left modules involving the tensor product $\hat{\otimes}$ . These isomorphisms rely on the bijectivity of the antipode and the specific module structures defined by the weak Hopf algebra's source and target counital subalgebras ( $H_s$ and $H_t$ ).
Definition of Restricted Extensions and Module Actions: The authors define a "restricted" weak Hopf Galois extension, a condition requiring the image of the translation map $\kappa$ applied to $H_s$ to lie within $B \otimes B B$ . They then construct a natural right $H$ -action on the Hochschild cohomology groups $H^q(B, M)$ for an $A^e$ -module $M$ . This action is derived using the translation map $\kappa: H \to A \otimes_B A$ .
Spectral Sequence Derivation: Utilizing the Grothendieck spectral sequence for the composition of functors, the authors relate the Hochschild cohomology of the extension $A$ to that of the base $B$ and the weak Hopf algebra $H$ . They define functors $F$ (Hochschild cohomology of $A$ ), $F_1$ (cohomology relative to $H_s$ ), and $F_2$ (Hochschild cohomology of $B$ ). By proving that injective $A^e$ -modules are acyclic for the composition, they establish a convergent spectral sequence:
$\text{Ext}^p_{Hop}(H_s, H^q(B, M)) \implies H^{p+q}(A, M)$
This sequence serves as the primary tool for transferring homological properties between $A$ and $B$ .

Key Contributions and Results

Spectral Sequence for Weak Hopf Extensions: The paper successfully extends Stefan's spectral sequence results from classical Hopf algebras to the weak Hopf setting. It establishes that for a restricted flat weak Hopf Galois extension $A/B$ , the Hochschild cohomology of $A$ is linked to the Hochschild cohomology of $B$ and the Ext-groups of $H_s$ over $H$ .
Separability Results: The authors prove that if $H$ is semisimple and $B$ is separable, then $A$ is separable. Conversely, under specific conditions (finite dimensional, semisimple, cosemisimple $H$ with a trace-like counit), if $A$ is separable, then $B$ is separable.
Dimension Bounds: The paper derives bounds for the global dimensions of the algebras involved, showing that $\text{rgl.dim}(B) \leq \text{rgl.dim}(H) + \text{rgl.dim}(A)$ and providing an upper bound for the projective dimension of $A$ as an $A^e$ -module.
Preservation of the AS-Gorenstein Property: The main theorem (Theorem 4.10) states that if $H$ $H$ is a weak Hopf algebra with a bijective antipode, and $A$ $A$ is a noetherian affine PI algebra that is a restricted faithfully flat weak $H$ $H$ -Galois extension of $B$ $B$ , then if $B$ $B$ is AS-Gorenstein, $A$ $A$ is also AS-Gorenstein.
- Furthermore, if $H$ is AS-Gorenstein of dimension $d_H$ , the injective dimension of $A$ satisfies $\text{inj.dim}(A_A) = \text{inj.dim}(B_B) + d_H$ .

Significance
The paper claims to extend the main results of Stefan [21] and Zhu [25] from the context of classical Hopf algebras to weak Hopf algebras. By constructing the appropriate spectral sequences and proving the preservation of the AS-Gorenstein property, the work provides a unified framework for studying the homological dimensions of algebras arising from weak Hopf Galois extensions. This is significant because weak Hopf algebras offer a natural setting for dynamical twists and representation categories of semisimple monoidal categories, and understanding their homological behavior is essential for classifying infinite-dimensional algebras with low Gelfand-Kirillov dimension. The results confirm that the homological integrity of the AS-Gorenstein property is robust even when the underlying symmetry is weakened from a Hopf algebra to a weak Hopf algebra.

More like this