arXiv subjects

Devarshi Mukherjee

Publications and source records attributed to Devarshi Mukherjee.

15 recordsLinked to original sources

arXiv2026

Bredon sheaf cohomology

For a finite group $G$, we compute the algebraic $K$-theory of the category of equivariant sheaves on a locally compact Hausdorff $G$-space, generalizing a result of Efimov, and determine the equivariant $E$-theory of the $C^*$-algebra of continuous functions. These invariants admit natural descriptions in terms of a new equivariant cohomology theory, which we call Bredon sheaf cohomology. This theory recovers classical Bredon cohomology for $G$-CW complexes and ordinary sheaf cohomology when $G$ is trivial. We establish its basic structural properties and prove a strong uniqueness theorem: any functor from the category of locally compact Hausdorff $G$-spaces to a dualizable stable category satisfying equivariant open descent and cofiltered compact codescent is equivalent to Bredon sheaf cohomology, generalizing a result of Clausen.

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arXiv2025

Operator algebras over the p-adic integers -- II

We continue the study of operator algebras over the $p$-adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of $p$-adic von Neumann algebras, and analyze those with trivial center, that we call ''factors''. In particular we show that ICC groups provide examples of factors. We then establish a characterization of $p$-simplicity for groupoid operator algebras, showing its relation to effectiveness and minimality. A central part of the paper is devoted to a $p$-adic analogue of the GNS construction, leading to a representation theorem for Banach $^*$-algebras over $\mathbb{Z}_p$. As applications, we exhibit large classes of $p$-adic operator algebras, including residually finite-rank algebras and affinoid algebras with the spectral norm. Finally, we investigate the $K$-theory of $p$-adic operator algebras, including the computation of homotopy analytic $K$-theory of continuous $\mathbb{Z}_p$-valued functions on a compact Hausdorff space and the analytic (non-homotopy invariant) $K$-theory of certain $p$-adically complete Banach algebras in terms of continuous $K$-theory. Together, these results extend the foundations of the emerging theory of $p$-adic operator algebras.

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arXiv2025

Localising invariants in derived bornological geometry

We study several categories of analytic stacks relative to the category of bornological modules over a Banach ring. When the underlying Banach ring is a non-Archimedean valued field, this category contains derived rigid analytic spaces as a full subcategory. When the underlying field is the complex numbers, it contains the category of derived complex analytic spaces. In the second part of the paper, we consider localising invariants of rigid categories associated to bornological algebras. The main results in this part include Nisnevich descent for derived analytic spaces and a version of the Grothendieck-Riemann-Roch Theorem for derived dagger analytic spaces over an arbitrary Banach ring.

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arXiv2024

Homology of Steinberg algebras

We study homological invariants of the Steinberg algebra $\mathcal{A}_k(\mathcal{G})$ of an ample groupoid $\mathcal{G}$ over a commutative ring $k$. For $\mathcal{G}$ principal or Hausdorff with ${\mathcal{G}}^{\rm{Iso}}\setminus{\mathcal{G}}^{(0)}$ discrete, we compute Hochschild and cyclic homology of $\mathcal{A}_k(\mathcal{G})$ in terms of groupoid homology. For any ample Hausdorff groupoid $\mathcal{G}$, we find that $H_*(\mathcal{G})$ is a direct summand of $HH_*(\mathcal{A}_k(\mathcal{G}))$; using this and the Dennis trace we obtain a map $\overline{D}_*:K_*(\mathcal{A}_k(\mathcal{G}))\to H_n(\mathcal{G},k)$. We study this map when $\mathcal{G}$ is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group $G$ on a graph, and compute $HH_*(\mathcal{A}_k(\mathcal{G}))$ and $H_*(\mathcal{G},k)$ in terms of the homology of $G$, and the $K$-theory of $\mathcal{A}_k(\mathcal{G})$ in terms of that of $k[G]$.

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arXiv2024

Operator algebras over the p-adic integers

We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^*$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^*$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and complex $C^*$-algebras, featuring limits, colimits, tensor products, crossed products and an enveloping construction permitting us to construct $p$-adic operator algebras from involutive algebras over $\mathbb{Z}_p$. In several cases of interest, the enveloping algebra construction recovers the $p$-adic completion of the underlying $\mathbb{Z}_p$-algebra. We then discuss an analogue of topological $K$-theory for Banach $\mathbb{Z}_p$-algebras, and compute it in basic examples such as the $p$-adic Cuntz algebra and rotation algebras. Finally, for a large class of $p$-adic operator algebras, we show that our $K$-theory coincides with the reduction mod $p$ of Quillen's algebraic $K$-theory.

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arXiv2023

Homological algebra for nonarchimedean completions of group algebras

Let $V$ be a complete discrete valuation ring, and let $G$ be either a word-hyperbolic group or a reductive $p$-adic group. We prove that the canonical morphism $V[G] \to V[G]^\dagger$ from the group algebra to its dagger completion is an isocohomological morphism (also known as an idempotent or a homotopy epimorphism). This allows us to deduce homological properties about the algebra $V[G]^\dagger$ in terms of the relatively simpler group algebra of $G$. These properties are then used towards Hochschild and cyclic homology computations.

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arXiv2023

Nonarchimedean bivariant K-theory

We introduce bivariant K-theory for nonarchimedean bornological algebras over a complete discrete valuation ring $V$. This is the universal target for dagger homotopy invariant, matricially stable and excisive functors, similar to bivariant K-theory for locally convex topological $\mathbb{C}$-algebras and algebraic bivariant K-theory. When the first variable is the ground algebra $V$, we get a version of Weibel's homotopy algebraic K-theory, which we call \textit{stabilised overconvergent analytic K-theory}. The resulting analytic K-theory satisfies dagger homotopy invariance, stability by completed matrix algebras, and excision.

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arXiv2022

Local cyclic homology for nonarchimedean Banach algebras

Let V be a complete discrete valuation ring with uniformiser p. We introduce an invariant of Banach V-algebras called local cyclic homology. This invariant is related to analytic cyclic homology for complete, bornologically torsionfree V-algebras. It is shown that local cyclic homology only depends on the reduction mod p of a Banach V-algebra and that it is homotopy invariant, matricially stable, and excisive.

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arXiv2022

A Quillen model structure of local homotopy equivalences

In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring $V$, the homotopy category of this model structure is the derived category of the quasi-abelian category $\overleftarrow{\mathsf{Ind}(\mathsf{Ban}_F)}$. This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree $V$-algebras and $\mathbb{F}$-algebras. When the base field is $\mathbb{C}$, the homotopy category is the target of local and analytic cyclic homology for pro-bornological $\mathbb{C}$-algebras, which includes the subcategory of pro-$C^*$-algebras.

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arXiv2021

Analytification, localization and homotopy epimorphisms

We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring $R$, along with a choice of Banach structure on $R$, a commutative monoid in the monoidal model category of simplicial ind-Banach $R$-modules. We show that several analytifications relevant to analytic geometry - such as Tate, overconvergent, Stein analytification, and formal completion - are homotopy epimorphisms. Another class of examples arises from Weierstrass, Laurent and rational localisations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions.

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arXiv2021

Analytic Hochschild-Kostant-Rosenberg Theorem

Let $R$ be a Banach ring. We prove that the category of chain complexes of complete bornological $R$-modules (and several related categories) is a derived algebraic context in the sense of Raksit. We then use the framework of derived algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which relates the circle action on the Hochschild algebra to the de Rham-differential-enriched-de Rham algebra of a simplicial, commutative, complete bornological algebra. This has a geometric interpretation in the language of derived analytic geometry, namely, the derived loop stack of a derived analytic stack is equivalent to the shifted tangent stack. Using this geometric interpretation we extend our results to derived schemes.

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arXiv2021

Analytic cyclic homology in positive characteristic

Let $V$ be a complete discrete valuation ring with residue field $\mathbb{F}$. We define a cyclic homology theory for algebras over $\mathbb{F}$, by lifting them to free algebras over $V$, which we enlarge to tube algebras and complete suitably. We show that this theory may be computed using any pro-dagger algebra lifting of an $\mathbb{F}$-algebra. We show that our theory is polynomially homotopy invariant, excisive, and matricially stable.

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arXiv2019

Non-Archimedean analytic cyclic homology

Let $V$ be a complete discrete valuation ring with fraction field $F$ of characteristic zero and with residue field $\mathbb{F}$. We introduce analytic cyclic homology of complete torsion-free bornological algebras over $V$. We prove that it is homotopy invariant, stable, invariant under certain nilpotent extensions, and satisfies excision. We use these properties to compute it for tensor products with dagger completions of Leavitt path algebras. If $R$ is a smooth commutative $V$-algebra of relative dimension $1$, then we identify its analytic cyclic homology with Berthelot's rigid cohomology of $R\otimes_{V}\mathbb{F}$.

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arXiv2019

Dagger completions and bornological torsion-freeness

We define a dagger algebra as a bornological algebra over a discrete valuation ring with three properties that are typical of Monsky-Washnitzer algebras, namely, completeness, bornological torsion-freeness and a certain spectral radius condition. We study inheritance properties of the three properties that define a dagger algebra. We describe dagger completions of bornological algebras in general and compute some noncommutative examples.

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arXiv2018

C*-Algebras for partial product systems over N

We define partial product systems over N. They generalise product systems over N and Fell bundles over Z. We define Toeplitz C*-algebras and relative Cuntz-Pimsner algebras for them and show that the section C*-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra. We describe the gauge-invariant ideals in the Toeplitz C*-algebra.

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