insight - Scientific Computing - # Hochschild-Kostant-Rosenberg Theorem

The Space of Functorial Hochschild-Kostant-Rosenberg Isomorphisms

Q: How does the characterization of HKR isomorphisms in this paper relate to other approaches to the HKR theorem, such as those using deformation quantization or operads?

This paper focuses on a specific flavor of HKR isomorphisms: those compatible with the rich geometric structure encoded by the filtered circle. This approach, rooted in derived algebraic geometry, provides a unique perspective compared to deformation quantization or operadic methods. Deformation Quantization: This approach views the HKR theorem through the lens of noncommutative geometry. It establishes a correspondence between the Poisson structure on a variety and a deformation of its structure sheaf into a noncommutative algebra. While powerful, this perspective doesn't directly address the functoriality and compatibility with the circle action emphasized in the paper. Operads: Operadic methods provide an abstract framework to study algebraic structures, including those present in Hochschild homology and differential forms. They offer a way to understand the HKR isomorphism as arising from a natural comparison map between operads. However, this approach often focuses on the algebraic structure rather than the geometric aspects highlighted by the filtered circle. In contrast, the paper leverages the geometric interpretation of Hochschild homology as functions on the derived loop space. By studying the HKR filtration through the lens of the filtered circle, it reveals a deep connection between the choice of HKR isomorphisms and formal exponentials. This geometric perspective provides a novel understanding of the HKR theorem, complementing the insights gained from deformation quantization and operadic approaches.

Q: Could there be a weaker notion of "functoriality" for HKR isomorphisms that leads to a larger space of possibilities, even in positive characteristic?

The paper focuses on a strong notion of functoriality, requiring the HKR isomorphisms to be compatible with the group structure of the filtered circle. This rigidity restricts the space of possible isomorphisms, especially in positive characteristic where it becomes empty. Exploring weaker notions of functoriality could indeed lead to a richer landscape of HKR isomorphisms. Some potential avenues include: Relaxing the group structure: Instead of demanding compatibility with the full group structure of the filtered circle, one could consider isomorphisms respecting only a weaker structure, such as a monoid or a pointed space. Restricting the category of schemes: Instead of requiring functoriality over all derived schemes, one could focus on specific subcategories, such as smooth schemes or schemes with additional geometric structures. This could potentially allow for HKR isomorphisms that depend on these extra structures. Modifying the target: Instead of aiming for isomorphisms with the split filtered circle, one could consider maps to other geometric objects encoding the HKR filtration, potentially allowing for more flexibility in positive characteristic. Investigating these weaker notions of functoriality could unveil new classes of HKR isomorphisms with interesting arithmetic and geometric properties, particularly in the challenging realm of positive characteristic.

Q: What insights can we gain about the geometry of the "filtered circle" by studying its space of splittings, and how does this relate to broader questions in homotopy theory and algebraic topology?

The space of splittings of the filtered circle, as explored in the paper, provides a window into its intricate geometry. The key observation is that this space is surprisingly rigid, being controlled by formal exponentials. This rigidity has profound implications: Relationship to K-theory and the Todd class: The paper highlights a connection between splittings of the filtered circle and the existence of Chern characters in K-theory. The obstruction to finding cogroup splittings is linked to the Todd class, hinting at a deep relationship between the geometry of the filtered circle and fundamental constructions in algebraic topology. Hodge degeneration and formality: The filtered circle plays a crucial role in understanding the Hodge degeneration theorem, which relates the de Rham cohomology of a smooth projective variety to its singular cohomology. The rigidity of its splittings suggests a connection between formality properties of spaces and the structure of the filtered circle. Connections to homotopy theory: The filtered circle can be viewed as an object in a suitable category of spectra, bridging the gap between algebraic geometry and homotopy theory. Studying its space of splittings in this context could shed light on the relationship between algebraic and topological approaches to K-theory and other cohomology theories. In summary, the space of splittings of the filtered circle acts as a bridge between seemingly disparate areas of mathematics. Its rigidity reveals deep connections between algebraic geometry, homotopy theory, and index theory, offering a fertile ground for further exploration.

Core Concepts

The set of all possible functorial Hochschild-Kostant-Rosenberg (HKR) isomorphisms, which respect key geometric structures, is surprisingly small and can be precisely characterized.

Abstract

This research paper delves into the mathematical intricacies of the Hochschild-Kostant-Rosenberg (HKR) theorem within the realm of derived algebraic geometry.

Bibliographic Information: Robalo, M. (2024). Choices of HKR isomorphisms. arXiv:2310.05859v2 [math.AG]

Research Objective: The paper aims to classify all possible HKR isomorphisms that are functorial and compatible with essential geometric structures, such as the HKR filtration, circle action on loop spaces, and the de Rham differential.

Methodology: The author employs advanced mathematical tools from derived algebraic geometry, including the theory of filtered stacks, Cartier duality, and formal group schemes. They leverage previous results on the structure of the filtered circle and its relation to the HKR filtration.

Key Findings: The main result demonstrates that the set of desired HKR isomorphisms is in bijection with the set of formal exponentials between specific formal groups. Notably, over a field of characteristic zero, this set is simply the multiplicative group of the field, while it is empty in positive characteristic.

Main Conclusions: The paper provides a complete and elegant characterization of functorial HKR isomorphisms, revealing a deep connection between seemingly disparate mathematical objects. This result has significant implications for understanding the interplay between algebraic and geometric structures in derived algebraic geometry.

Significance: This research contributes significantly to the field of derived algebraic geometry by providing a precise understanding of the HKR theorem's flexibility and limitations. It sheds light on the relationship between algebraic cycles, differential forms, and K-theory, with potential applications to areas like deformation theory and mirror symmetry.

Limitations and Future Research: The paper primarily focuses on the case of derived schemes over a field. Exploring similar questions for more general derived geometric objects, such as stacks or derived Artin stacks, could be a fruitful avenue for future research.

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Choices of HKR isomorphisms

by Marco Robalo at arxiv.org 11-12-2024

https://arxiv.org/pdf/2310.05859.pdf

Deeper Inquiries

How does the characterization of HKR isomorphisms in this paper relate to other approaches to the HKR theorem, such as those using deformation quantization or operads?

This paper focuses on a specific flavor of HKR isomorphisms: those compatible with the rich geometric structure encoded by the filtered circle. This approach, rooted in derived algebraic geometry, provides a unique perspective compared to deformation quantization or operadic methods.

Deformation Quantization: This approach views the HKR theorem through the lens of noncommutative geometry. It establishes a correspondence between the Poisson structure on a variety and a deformation of its structure sheaf into a noncommutative algebra. While powerful, this perspective doesn't directly address the functoriality and compatibility with the circle action emphasized in the paper.

Operads: Operadic methods provide an abstract framework to study algebraic structures, including those present in Hochschild homology and differential forms. They offer a way to understand the HKR isomorphism as arising from a natural comparison map between operads. However, this approach often focuses on the algebraic structure rather than the geometric aspects highlighted by the filtered circle.
In contrast, the paper leverages the geometric interpretation of Hochschild homology as functions on the derived loop space. By studying the HKR filtration through the lens of the filtered circle, it reveals a deep connection between the choice of HKR isomorphisms and formal exponentials. This geometric perspective provides a novel understanding of the HKR theorem, complementing the insights gained from deformation quantization and operadic approaches.

Could there be a weaker notion of "functoriality" for HKR isomorphisms that leads to a larger space of possibilities, even in positive characteristic?

The paper focuses on a strong notion of functoriality, requiring the HKR isomorphisms to be compatible with the group structure of the filtered circle. This rigidity restricts the space of possible isomorphisms, especially in positive characteristic where it becomes empty.
Exploring weaker notions of functoriality could indeed lead to a richer landscape of HKR isomorphisms. Some potential avenues include:

Relaxing the group structure: Instead of demanding compatibility with the full group structure of the filtered circle, one could consider isomorphisms respecting only a weaker structure, such as a monoid or a pointed space.

Restricting the category of schemes:  Instead of requiring functoriality over all derived schemes, one could focus on specific subcategories, such as smooth schemes or schemes with additional geometric structures. This could potentially allow for HKR isomorphisms that depend on these extra structures.

Modifying the target: Instead of aiming for isomorphisms with the split filtered circle, one could consider maps to other geometric objects encoding the HKR filtration, potentially allowing for more flexibility in positive characteristic.
Investigating these weaker notions of functoriality could unveil new classes of HKR isomorphisms with interesting arithmetic and geometric properties, particularly in the challenging realm of positive characteristic.

What insights can we gain about the geometry of the "filtered circle" by studying its space of splittings, and how does this relate to broader questions in homotopy theory and algebraic topology?

The space of splittings of the filtered circle, as explored in the paper, provides a window into its intricate geometry. The key observation is that this space is surprisingly rigid, being controlled by formal exponentials. This rigidity has profound implications:

Relationship to K-theory and the Todd class: The paper highlights a connection between splittings of the filtered circle and the existence of Chern characters in K-theory. The obstruction to finding cogroup splittings is linked to the Todd class, hinting at a deep relationship between the geometry of the filtered circle and fundamental constructions in algebraic topology.

Hodge degeneration and formality: The filtered circle plays a crucial role in understanding the Hodge degeneration theorem, which relates the de Rham cohomology of a smooth projective variety to its singular cohomology. The rigidity of its splittings suggests a connection between formality properties of spaces and the structure of the filtered circle.

Connections to homotopy theory: The filtered circle can be viewed as an object in a suitable category of spectra, bridging the gap between algebraic geometry and homotopy theory. Studying its space of splittings in this context could shed light on the relationship between algebraic and topological approaches to K-theory and other cohomology theories.
In summary, the space of splittings of the filtered circle acts as a bridge between seemingly disparate areas of mathematics. Its rigidity reveals deep connections between algebraic geometry, homotopy theory, and index theory, offering a fertile ground for further exploration.