On the rationalization of the $K(n)$-local sphere
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915487184584704 |
|---|---|
| author | Barthel, Tobias Schlank, Tomer M. Stapleton, Nathaniel Weinstein, Jared |
| author_facet | Barthel, Tobias Schlank, Tomer M. Stapleton, Nathaniel Weinstein, Jared |
| contents | We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze-Weinstein, integral $p$-adic Hodge theory, and an integral refinement of a theorem of Tate on the Galois cohomology of non-archimedean fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rationalization of the $K(n)$-local sphere Barthel, Tobias Schlank, Tomer M. Stapleton, Nathaniel Weinstein, Jared Algebraic Topology Algebraic Geometry Number Theory 55Q45, 14G22 We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze-Weinstein, integral $p$-adic Hodge theory, and an integral refinement of a theorem of Tate on the Galois cohomology of non-archimedean fields. |
| title | On the rationalization of the $K(n)$-local sphere |
| topic | Algebraic Topology Algebraic Geometry Number Theory 55Q45, 14G22 |
| url | https://arxiv.org/abs/2402.00960 |