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A filtered Hochschild-Kostant-Rosenberg theorem for real Hochschild homology
Algebraic Topology · Submitted 4 Mar 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex.
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The abstract, sentence by sentence
In this paper, we introduce a notion of derived involutive algebras in $ C_2 $-Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt--Mathew and Raksit. We show that the $ \infty $-category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. Using $ C_2 $-$ \infty $-categories of Barwick--Dotto--Glasman--Nardin--Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we introduce a notion of derived involutive algebras in $ C_2 $-Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt--Mathew and Raksit. | contribution 1.00 | 0.93 | 0.09 | asserted 1.00 |
| s1 | We show that the $ \infty $-category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. | contribution 0.52 | 0.94 | 0.17 | asserted 1.00 |
| s2 | We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. | result 0.70 | 0.95 | 0.52 | asserted 0.91 |
| s3 | Using $ C_2 $-$ \infty $-categories of Barwick--Dotto--Glasman--Nardin--Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit. | result 0.84 | 0.88 | 0.17 | asserted 1.00 |
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The paper
Original abstract
In this paper, we introduce a notion of derived involutive algebras in $ C_2 $-Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt--Mathew and Raksit. We show that the $ \infty $-category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. Using $ C_2 $-$ \infty $-categories of Barwick--Dotto--Glasman--Nardin--Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit.
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How the literature uses it
3 arXiv papers cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Cites it as support 1 paper
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By [ref] , there is an equivalence [equation] and thus an equivalence [equation] completing the proof.
On homological Real trace methods · Parametrized internal homs · 0.96
Cites it as background 2 papers
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The development of trace methods in the Real setting has been the subject of much research (see, for instance, [ref] ).
On homological Real trace methods · Introduction · 0.99
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Monoidal parametrized versions of module categories have been considered by other authors like Linskens–Nardin–Pol, Yang, and Pützstück [ref] .
Multiplicative Equivariant Thom Spectra & Structured Real Orientations · Relation to Other Work · 0.82
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