Core Concepts

This article introduces a noncommutative analogue of the relative de Rham–Witt complex, called relative Hochschild–Witt homology, and proves an HKR-type theorem relating it to the relative de Rham–Witt complex in the commutative case.

Abstract

**Bibliographic Information:**Mao, Zhouhang. "Noncommutative relative de Rham–Witt complex via the norm." (Date information not provided)**Research Objective:**The main objective of this paper is to construct a noncommutative analogue of the relative de Rham–Witt complex, extending previous work by Illusie, Langer–Zink, and Kaledin.**Methodology:**The author utilizes tools from algebraic topology, specifically the theory of p-cyclotomic spectra, Tambara functors, and the Hill–Hopkins–Ravenel norm. They define relative Hochschild–Witt homology as topological Hochschild homology relative to the Tambara functor W(Fp).**Key Findings:**- The author establishes a connection between Kaledin's polynomial Witt vectors and the Hill–Hopkins–Ravenel norm, providing an intrinsic description of the former.
- They prove an HKR-type theorem, demonstrating that for smooth commutative algebras, the relative Hochschild–Witt homology groups are isomorphic to the relative de Rham–Witt forms.
- The paper also establishes an equivalence between topological restriction homology and non-truncated Hochschild–Witt homology for associative Fp-algebras.

**Main Conclusions:**This work successfully constructs a noncommutative generalization of the relative de Rham–Witt complex, bridging a gap in the existing literature. The HKR-type theorem provides a concrete link to the commutative setting, while the connection to topological restriction homology opens up further avenues for exploration.**Significance:**This research significantly contributes to the field of algebraic topology, particularly in the study of cyclotomic spectra and their applications to algebraic geometry. The results have implications for understanding noncommutative geometry and deformation theory.**Limitations and Future Research:**The author acknowledges that a relative, mixed-characteristic version of Proposition 3.2 could be explored in future work. Additionally, further investigation into the applications of relative Hochschild–Witt homology in noncommutative geometry and related areas is warranted.

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by Zhouhang Mao at **arxiv.org** 10-10-2024

Deeper Inquiries

Extending the concept of relative Hochschild–Witt homology to algebraic structures beyond associative algebras is a fruitful avenue for future research. Here are some potential approaches:
Generalizing Hochschild Homology: The foundation of Hochschild–Witt homology lies in Hochschild homology, which itself can be defined for a wider class of algebraic structures. For instance:
Operads: One can define Hochschild homology for algebras over operads, encompassing structures like commutative algebras and Lie algebras. This suggests exploring a "Hochschild–Witt homology for operads" by incorporating Witt vectors into the picture.
Categorical Constructions: Hochschild homology has categorical interpretations, such as in terms of derived tensor products and bimodules. These categorical frameworks might offer a path to define Hochschild–Witt homology for categories with suitable extra structure.
Exploiting Trace-Like Structures: Kaledin's construction relies heavily on the notion of "trace theory." Investigating whether analogous trace-like structures exist for other algebraic objects could pave the way for defining their Hochschild–Witt homology. For example:
Cyclic Objects: The presence of cyclic structures often hints at underlying trace-like behavior. Exploring connections between cyclic objects and Witt vectors might be promising.
Geometric Approaches: Drawing inspiration from the geometric interpretations of Hochschild and cyclic homology:
Noncommutative Geometry: Investigate how relative Hochschild–Witt homology might capture geometric invariants of "noncommutative spaces" associated with more general algebraic structures.
Derived Algebraic Geometry: Explore interpretations within the framework of derived algebraic geometry, where objects like derived stacks might provide a natural setting.

While the Hill–Hopkins–Ravenel norm provides an elegant framework for constructing the relative Hochschild–Witt complex, exploring alternative approaches is a worthwhile endeavor. Here are some possibilities:
Direct Construction of F-V Procomplexes: One could attempt to directly construct F-V procomplexes mimicking the structure of the relative de Rham–Witt complex, but in a noncommutative setting. This would likely involve developing new techniques for handling the complexities of noncommutative differential forms and their interactions with Witt vectors.
Operadic Methods: Operads provide a powerful tool for encoding algebraic structures. It might be possible to construct an operad whose algebras naturally give rise to noncommutative relative de Rham–Witt complexes. This approach could leverage existing operadic techniques for handling differential forms and cyclic homology.
Deformation Theory: Deformation theory studies how algebraic structures vary under infinitesimal perturbations. One could try to deform the classical relative de Rham–Witt complex to obtain a noncommutative version. This approach might involve working with differential graded Lie algebras or other homotopical algebraic structures.
Motivic Homotopy Theory: Motivic homotopy theory provides a framework for studying algebraic varieties and their invariants using tools from homotopy theory. It might be possible to develop a motivic version of the relative de Rham–Witt complex that naturally incorporates noncommutative spaces.

The development of relative Hochschild–Witt homology holds intriguing implications for the study of motives and motivic cohomology in the realm of noncommutative geometry:
Noncommutative Motives: Motives are supposed to capture the "universal" cohomological information of algebraic varieties. Relative Hochschild–Witt homology, as a refined invariant sensitive to noncommutative structures, could provide insights into the elusive notion of "noncommutative motives." It might lead to new ways of constructing and studying motivic categories for noncommutative spaces.
Motivic Cohomology and Witt Vectors: The appearance of Witt vectors in relative Hochschild–Witt homology suggests a deeper connection between motivic cohomology and Witt-theoretic constructions. This could lead to:
New Motivic Invariants: The development of new motivic invariants built using Witt vectors, potentially revealing finer geometric information.
Connections to Arithmetic Geometry: Witt vectors play a fundamental role in arithmetic geometry. This connection could illuminate arithmetic aspects of noncommutative spaces and their motivic cohomology.
Noncommutative Hodge Theory: Hodge theory studies the interplay between the geometry and topology of complex manifolds. Relative Hochschild–Witt homology, with its connections to de Rham–Witt complexes, might provide tools for developing a "noncommutative Hodge theory" that applies to noncommutative spaces. This could lead to new insights into the structure of noncommutative spaces and their invariants.
Overall, the work on relative Hochschild–Witt homology opens up exciting new avenues for research at the intersection of noncommutative geometry, motivic homotopy theory, and arithmetic geometry.

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