arXiv subjects

Artem Prikhodko

Publications and source records attributed to Artem Prikhodko.

6 recordsLinked to original sources

arXiv2026

Local coefficients for genuine equivariant cohomology I

We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories -- such as Atiyah--Segal $K$-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner--Meier -- analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space $X$ and a coefficient system $\mathscr{A} \colon \mathrm{Orb}^{\mathrm{op}} \to \mathrm{Pr}^{\mathrm{L}}$, the categories $\mathrm{LS}^{\mathrm{glo}}(X,\mathscr{A})$ and $\mathrm{LS}^{\mathrm{gen}}(X,\mathscr{A})$ of globally equivariant and genuine local systems, the latter generalizing the genuine stable category $\mathrm{Sp}^G$ of a compact Lie group $G$ to non-constant coefficients. When the coefficients come from an oriented abelian group stack $A$ over a locally complex periodic base, we also construct the category $\mathrm{LS}^\mathrm{temp}(X,A)$ of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah--Segal completion theorem. We show that global sections induce an equivalence $\mathrm{LS}^{\mathrm{temp}}(BU(1),A) \simeq \mathrm{QCoh}(A)$ and we prove that $\mathrm{LS}^\mathrm{temp}(X,A)$ is a smashing localization of $\mathrm{LS}^\mathrm{gen}(X,A)$ if $X$ is an orbispace.

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arXiv2022

Rational $p$-adic Hodge theory for $d$-de Rham-proper stacks

In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in our previous work. As a corollary, we get that if a smooth Artin stack over $K$ has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more general setting of smooth $d$-de Rham-proper stacks over $\mathcal O_K$: here we only require first $d$ de Rham cohomology groups be finitely-generated over $\mathcal O_K$. As an application, we deduce a certain purity-type statement for étale $\mathbb Q_p$-cohomology of Raynaud generic fiber, as well as crystallinity of a first several étale cohomology groups in the presence of a Cohen--Macauley model over $\mathcal O_K$ in the schematic setting.

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arXiv2021

$p$-adic Hodge theory for Artin stacks

This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the étale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between étale cohomology of the algebraic and the Raynaud generic fibers is often an equivalence even outside of the proper case. In particular, we show that this holds for global quotients $[X/G]$ where $X$ is a smooth proper scheme and $G$ is a reductive group. As applications we deduce Totaro's conjectural inequality and also set up a theory of $A_{\mathrm{inf}}$-characteristic classes.

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arXiv2021

Hodge-to-de Rham degeneration for stacks

We introduce a notion of a Hodge-proper stack and extend the method of Deligne-Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic $0$ to characteristic $p$, we need to find a good integral model of a stack (a so-called spreading), which, unlike in the case of schemes, need not to exist in general. To address this problem we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hodge-properly spreadable. As a corollary we deduce a (non-canonical) Hodge decomposition of the equivariant cohomology for certain classes of varieties with an algebraic group action.

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arXiv2019

Categorical proof of Holomorphic Atiyah-Bott formula

Given a symmetric monoidal $(\infty,2)$-category $\mathscr E$ we promote the trace construction to a functor. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

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arXiv2019

Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory

We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on $X$ and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme $\mathcal LX$. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.

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