Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909330945605632 |
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| author | Krause, Achim Senger, Andrew |
| author_facet | Krause, Achim Senger, Andrew |
| contents | In this note, we prove that $K_{2i} (\mathbb{Z}/p^n) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in $π_* K(\mathbb{Z}/p^n)/p$ is equal to $\frac{p^n-1}{p-1}$.
Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_K /\varpi^n$ constructed by Antieau, Nikolaus and the first author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$ Krause, Achim Senger, Andrew K-Theory and Homology Algebraic Geometry Algebraic Topology In this note, we prove that $K_{2i} (\mathbb{Z}/p^n) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in $π_* K(\mathbb{Z}/p^n)/p$ is equal to $\frac{p^n-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_K /\varpi^n$ constructed by Antieau, Nikolaus and the first author. |
| title | Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$ |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2409.20523 |