The algebraic K-theory of k[SL₂(\mathbbF_q)]

2026/06/19 by Isaac Moselle · 1 voice
Mathematics · #math.KT #math.AT #math.RT

paper · pdf

Abstract

We compute via trace methods the higher algebraic K-theory of the group ring k[SL₂(\mathbbF_q)], as well as the related groups PSL₂(\mathbbF_q), PGL₂(\mathbbF_q), and GL₂(\mathbbF_q), where k is a perfect field of characteristic p and q=p^r. At the core of the computation is the algebraic K-theory of the group ring of the Sylow p-subgroup, k[C_p^r], which we determine via a theorem of Lück--Reich--Rognes--Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus--Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.

Citations

• Derived induction theory for the K-theory of modular group algebras
• On curves in K-theory and TR
• Descent and vanishing in chromatic algebraic K-theory via group actions
• Topological cyclic homology
• On the K-theory of truncated polynomial algebras, revisited
• Cartier modules and cyclotomic spectra
• K-theory and topological cyclic homology of henselian pairs
• Derived induction and restriction theory
• Nilpotence and descent in equivariant stable homotopy theory
• Spectral Mackey functors and equivariant algebraic K-theory (II)
• Spectral Mackey functors and equivariant algebraic K-theory (I)
• On the algebraic K-theory of truncated polynomial algebras in several variables
• On the relative and bi-relative K-theory of rings of finite characteristic
• Higher Topos Theory
• On the K-theory of the coordinate axes in the plane
• Higher limits via subgroup complexes

Discussions

• Isaac Moselle: The algebraic K-theory of $k[\operatorname{SL}_2(\mathbb{F}_q)]$ https://arxiv.org/abs/2606.21421 https://arxiv.org/pdf/2606.21421 https://arxiv.org/html/2606.21421 [bsky, 0 points, 0 comments]

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