Core Concepts

This article proves that the ring of differential operators on a cuspidal curve defined by a numerical semigroup is a Hopf algebroid, utilizing a novel descent theory approach for these structures.

Abstract

Krähmer, U., & Mahaman, M. (2024). THE RING OF DIFFERENTIAL OPERATORS ON A MONOMIAL CURVE IS A HOPF ALGEBROID. *arXiv preprint arXiv:2405.08490v2*.

This paper investigates whether the ring of differential operators (DA) on a k-algebra A possesses a Hopf algebroid structure, particularly focusing on the case where A represents the coordinate ring of a cuspidal curve defined by a numerical semigroup.

The authors employ a descent theory approach to determine the conditions under which a Hopf algebroid structure on a bialgebroid C over a commutative k-algebra K can descend to a subalgebra, specifically focusing on the ring of differential operators. They leverage the concept of local projectivity relative to a subset of the dual module to establish criteria for this descent.

- The authors prove a general theorem (Theorem 1.3) stating that if a certain local projectivity condition is satisfied, a Hopf algebroid structure on C can descend to a subalgebra CpB, Aq.
- Applying this theorem to the ring of differential operators, they demonstrate that for A being the coordinate ring of a cuspidal curve defined by a numerical semigroup, DA is a cocommutative and conilpotent left Hopf algebroid over A.
- Furthermore, they show that if the numerical semigroup is symmetric (and thus A is Gorenstein), DA becomes a full Hopf algebroid, admitting an antipode.
- The authors provide an explicit computation of the Hopf algebroid structure for the basic example of A = k[t², t³].

The research establishes a novel connection between the theory of Hopf algebroids and the study of differential operators on singular curves. The results have implications for understanding the homological properties of DA and provide a new perspective on the classification of Hopf algebroids beyond the projective case.

This work contributes significantly to the field of algebraic geometry by providing a new class of examples of Hopf algebroids and highlighting the utility of descent theory in their study. It opens up avenues for further research into the classification of Hopf algebroids and their applications in understanding the algebraic structures associated with singular varieties.

While the paper focuses on cuspidal curves defined by numerical semigroups, the descent theory approach developed here has the potential to be applied to a broader class of algebras and geometric objects. Future research could explore extending these results to higher-dimensional singular varieties and investigating the implications of the Hopf algebroid structure on DA for the representation theory of these varieties.

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Yes, the descent theory approach developed in the paper has the potential to be applied to a broader class of singular varieties beyond just monomial curves. Here's why:
Generality of the Descent Theorem: The core of the paper is Theorem 1.3, a general result about the descent of Hopf algebroid structures. This theorem doesn't rely on the specific geometry of monomial curves. It only requires certain properties of the algebras involved, particularly the local projectivity condition.
Étale Morphisms and Localizations: The proof leverages the properties of étale morphisms and localizations. These concepts are fundamental in algebraic geometry and appear in the study of various singular varieties. If one can find a suitable étale morphism relating the ring of differential operators on a singular variety to a Hopf algebroid over a "nicer" variety (like the normalization in the monomial curve case), the descent theorem could be applied.
Characteristic p and Beyond: The authors even hint at the possibility of extending the results to characteristic p, where differential operators behave differently. This suggests a broader applicability of the descent approach.
However, applying this method to other singular varieties comes with challenges:
Finding Suitable Étale Extensions: Identifying an appropriate étale extension that allows for the descent machinery to work might be difficult and depend on the specific type of singularity.
Verifying Local Projectivity: Checking the local projectivity condition for the descended object might be non-trivial and require specific techniques depending on the chosen variety.
Therefore, while the descent theory approach offers a promising avenue for investigating Hopf algebroid structures on rings of differential operators of more general singular varieties, further research is needed to overcome the challenges and determine its full scope.

Extending the results to prime characteristic is a natural question, but it poses significant challenges and requires substantial modifications. Here's a breakdown:
Challenges in Prime Characteristic:
Differential Operators in Characteristic p: The theory of differential operators behaves quite differently in prime characteristic. The usual definition of $D_A$ leads to a ring that's too large and not well-behaved. Instead, one often works with the ring of pd-differential operators, which incorporates the Frobenius map.
Failure of the Poincaré-Birkhoff-Witt Theorem: The classical Poincaré-Birkhoff-Witt (PBW) theorem, crucial for relating universal enveloping algebras and symmetric algebras, doesn't hold in its usual form in characteristic p. This makes it harder to establish isomorphisms between graded objects associated with differential operators.
Gorenstein Property and Antipodes: The existence of an antipode in the Hopf algebroid structure is linked to the Gorenstein property of the coordinate ring. The characterization of Gorenstein rings in characteristic p can be more subtle.
Possible Modifications and Approaches:
pd-Differential Operators: One needs to replace the usual ring of differential operators with the ring of pd-differential operators. This ring is better suited for characteristic p and has connections to the Frobenius map.
Restricted Lie Algebras: Instead of ordinary Lie algebras, one might need to work with restricted Lie algebras (also known as p-Lie algebras), which are better adapted to characteristic p.
Modified PBW-Type Results: While the classical PBW theorem fails, there might be modified versions or analogous results for pd-differential operators and restricted Lie algebras that could be used.
Alternative Characterizations of Hopf Algebroids: It might be necessary to explore alternative characterizations of Hopf algebroids in characteristic p that are less reliant on the classical PBW theorem.
In summary, extending the results to prime characteristic is a non-trivial task. It demands a deep understanding of pd-differential operators, restricted Lie algebras, and potential modifications to the theory of Hopf algebroids in this setting.

The Hopf algebroid structure on the rings of differential operators of monomial curves has several interesting implications for their Hochschild cohomology:
Gerstenhaber Algebra Structure: The Hochschild cohomology $HH^*(D_A, D_A)$ of an algebra $D_A$ is naturally a Gerstenhaber algebra, carrying both a graded commutative cup product and a Lie bracket (of degree -1) that are compatible. The Hopf algebroid structure on $D_A$ can provide tools for computing and understanding this Gerstenhaber algebra structure.
Additional Structures on $HH^*(D_A, M)$: For a $D_A$-bimodule $M$, the Hochschild cohomology $HH^(D_A, M)$ is a module over the Gerstenhaber algebra $HH^(D_A, D_A)$. The Hopf algebroid structure can induce additional structures on $HH^*(D_A, M)$, such as comodule structures or actions of related Hopf algebroids.
Deformation Theory: Hochschild cohomology plays a crucial role in deformation theory. The Hopf algebroid structure might provide insights into deformations of $D_A$ as an algebra or as a Hopf algebroid. This could be related to deformations of the corresponding monomial curve or its singularity.
Connections to Calabi-Yau Structures: Hopf algebras and Hopf algebroids are intimately connected to the notion of Calabi-Yau algebras. The existence of a Hopf algebroid structure on $D_A$ might suggest the presence of a Calabi-Yau structure on a suitable category of $D_A$-modules. This could have implications for the homological properties of $D_A$.
Comparison with Smooth Case: The Hochschild cohomology of rings of differential operators on smooth varieties is relatively well-understood. The Hopf algebroid structure allows for a comparison between the Hochschild cohomology of $D_A$ in the singular case and the corresponding smooth case (e.g., the ring of differential operators on the normalization of the monomial curve).
In conclusion, the Hopf algebroid structure on $D_A$ provides a rich algebraic framework for studying its Hochschild cohomology. It offers tools for computations, reveals connections to other areas like deformation theory and Calabi-Yau algebras, and facilitates comparisons with the smooth case. Further research is needed to explore these implications fully.

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