On pro-cdh descent on derived schemes
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909995905318912 |
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| author | Kelly, Shane Saito, Shuji Tamme, Georg |
| author_facet | Kelly, Shane Saito, Shuji Tamme, Georg |
| contents | Grothendieck's formal functions theorem states that the coherent cohomology of a Noetherian scheme can be recovered from that of a blowup and the infinitesimal thickenings of the center and of the exceptional divisor of the blowup. In this article, we prove an analogous descent result, called ``pro-cdh descent'', for certain cohomological invariants of arbitrary quasi-compact, quasi-separated derived schemes. Our results in particular apply to algebraic $K$-theory, topological Hochschild and cyclic homology, and the cotangent complex.
As an application, we deduce that $K_n(X) = 0$ when $n < -d$ for quasi-compact, quasi-separated derived schemes $X$ of valuative dimension $d$. This generalises Weibel's conjecture, which was originally stated for Noetherian (non-derived) $X$ of Krull dimension $d$, and proved in this form in 2018 by Kerz, Strunk, and the third author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On pro-cdh descent on derived schemes Kelly, Shane Saito, Shuji Tamme, Georg K-Theory and Homology Algebraic Geometry Grothendieck's formal functions theorem states that the coherent cohomology of a Noetherian scheme can be recovered from that of a blowup and the infinitesimal thickenings of the center and of the exceptional divisor of the blowup. In this article, we prove an analogous descent result, called ``pro-cdh descent'', for certain cohomological invariants of arbitrary quasi-compact, quasi-separated derived schemes. Our results in particular apply to algebraic $K$-theory, topological Hochschild and cyclic homology, and the cotangent complex. As an application, we deduce that $K_n(X) = 0$ when $n < -d$ for quasi-compact, quasi-separated derived schemes $X$ of valuative dimension $d$. This generalises Weibel's conjecture, which was originally stated for Noetherian (non-derived) $X$ of Krull dimension $d$, and proved in this form in 2018 by Kerz, Strunk, and the third author. |
| title | On pro-cdh descent on derived schemes |
| topic | K-Theory and Homology Algebraic Geometry |
| url | https://arxiv.org/abs/2407.04378 |