On filtered algebraic $K$-theory of stacks I: characteristic zero
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| Format: | Preprint |
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2025
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| _version_ | 1866916651393351680 |
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| author | Elmanto, Elden Kubrak, Dmitry Sosnilo, Vladimir |
| author_facet | Elmanto, Elden Kubrak, Dmitry Sosnilo, Vladimir |
| contents | Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of $X$. We prove an analog of this result for algebraic $K$-theory over a field of characteristic 0. In our setting $G$ is a reductive group that acts on a derived algebraic space $X$ with the assumption that all stabilizer groups are nice (in the sense of Alper). Our main result identifies the value $R^{\mathrm{dAff}}K([X/G])$ of right Kan extension of the $K$-theory functor from schemes to stacks with the completion of $K$-theory of the category $\mathrm{Perf}([X/G])$ at the augmentation ideal of $K_0(\mathrm{Rep}(G))$. The main novelty of our results is that $X$ is allowed to be singular or even derived. This generality is achieved by employing and improving analogous versions of completion theorem for negative cyclic homology (after Ben-Zvi--Nadler and Chen) and for homotopy $K$-theory (after van den Bergh--Tabuada). We also show that in the singular setting the completion theorem does not necessarily hold without the nice stabilizer assumption. We view our results as a part of the general paradigm of extending the motivic filtration on algebraic $K$-theory of schemes to algebraic $K$-theory of stacks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On filtered algebraic $K$-theory of stacks I: characteristic zero Elmanto, Elden Kubrak, Dmitry Sosnilo, Vladimir Algebraic Geometry Algebraic Topology K-Theory and Homology Representation Theory Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of $X$. We prove an analog of this result for algebraic $K$-theory over a field of characteristic 0. In our setting $G$ is a reductive group that acts on a derived algebraic space $X$ with the assumption that all stabilizer groups are nice (in the sense of Alper). Our main result identifies the value $R^{\mathrm{dAff}}K([X/G])$ of right Kan extension of the $K$-theory functor from schemes to stacks with the completion of $K$-theory of the category $\mathrm{Perf}([X/G])$ at the augmentation ideal of $K_0(\mathrm{Rep}(G))$. The main novelty of our results is that $X$ is allowed to be singular or even derived. This generality is achieved by employing and improving analogous versions of completion theorem for negative cyclic homology (after Ben-Zvi--Nadler and Chen) and for homotopy $K$-theory (after van den Bergh--Tabuada). We also show that in the singular setting the completion theorem does not necessarily hold without the nice stabilizer assumption. We view our results as a part of the general paradigm of extending the motivic filtration on algebraic $K$-theory of schemes to algebraic $K$-theory of stacks. |
| title | On filtered algebraic $K$-theory of stacks I: characteristic zero |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology Representation Theory |
| url | https://arxiv.org/abs/2503.09928 |