Relative assembly maps and the K-theory of Hecke algebras in prime characteristic
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917738249715712 |
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| author | Lueck, Wolfgang |
| author_facet | Lueck, Wolfgang |
| contents | We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic $K$-theory of twisted group rings of a group G with coefficients in a regular ring R or, more generally, with coefficients in a regular additive category. They are known to be isomorphisms rationally. We show that it suffices to invert only those primes p for which G contains a non-trivial finite p-group and p is not invertible in R. The key ingredient is the detection of Nil-terms of a twisted group ring of a finite group F after localizing at p in terms of the p-subgroups of F using Verschiebungs and Frobenius operators. We construct and exploit the structure of a module over the ring of big Witt vectors on the Nil-terms. We analyze the algebraic K-theory of the Hecke algebras of subgroups of reductive p-adic groups in prime characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative assembly maps and the K-theory of Hecke algebras in prime characteristic Lueck, Wolfgang K-Theory and Homology Representation Theory 19D35 (Primary), 19D50, 20C08 (Secondary) We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic $K$-theory of twisted group rings of a group G with coefficients in a regular ring R or, more generally, with coefficients in a regular additive category. They are known to be isomorphisms rationally. We show that it suffices to invert only those primes p for which G contains a non-trivial finite p-group and p is not invertible in R. The key ingredient is the detection of Nil-terms of a twisted group ring of a finite group F after localizing at p in terms of the p-subgroups of F using Verschiebungs and Frobenius operators. We construct and exploit the structure of a module over the ring of big Witt vectors on the Nil-terms. We analyze the algebraic K-theory of the Hecke algebras of subgroups of reductive p-adic groups in prime characteristic. |
| title | Relative assembly maps and the K-theory of Hecke algebras in prime characteristic |
| topic | K-Theory and Homology Representation Theory 19D35 (Primary), 19D50, 20C08 (Secondary) |
| url | https://arxiv.org/abs/2402.05278 |