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Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $$\mathsf V$$, as well as for small $$\mathsf V$$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $$C_n$$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $$K$$-theory to twisted topological Hochschild homology.
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Hochschild homology of mod-𝑝 motivic cohomology over algebraically closed fields
We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod-p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields.
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- Award ID(s):
- 2204365
- PAR ID:
- 10633991
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Communications of the American Mathematical Society
- Volume:
- 4
- Issue:
- 13
- ISSN:
- 2692-3688
- Page Range / eLocation ID:
- 578-606
- Subject(s) / Keyword(s):
- Motivic homotopy theory motivic cohomology Hochschild homology
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/cams/36
