Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917589358215168 |
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| author | Annala, Toni Hoyois, Marc Iwasa, Ryomei |
| author_facet | Annala, Toni Hoyois, Marc Iwasa, Ryomei |
| contents | We formulate and prove a Conner-Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra, which turns out to be equivalent to the $\infty$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\infty$-category satisfies $\mathbb P^1$-homotopy invariance and weighted $\mathbb A^1$-homotopy invariance, which we use in place of $\mathbb A^1$-homotopy invariance to obtain analogues of several key results from $\mathbb A^1$-homotopy theory. These allow us in particular to define a universal oriented motivic $\mathbb E_\infty$-ring spectrum $\mathrm{MGL}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$ can be recovered from its $\mathrm{MGL}$-cohomology via a Conner-Floyd isomorphism \[\mathrm{MGL}^{**}(X)\otimes_{\mathrm L}\mathbb Z[β^{\pm 1}]\simeq \mathrm K^{**}(X),\] where $\mathrm L$ is the Lazard ring and $\mathrm K^{p,q}(X)=\mathrm K_{2q-p}(X)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{MGL}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02051 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory Annala, Toni Hoyois, Marc Iwasa, Ryomei Algebraic Geometry Algebraic Topology K-Theory and Homology We formulate and prove a Conner-Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra, which turns out to be equivalent to the $\infty$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\infty$-category satisfies $\mathbb P^1$-homotopy invariance and weighted $\mathbb A^1$-homotopy invariance, which we use in place of $\mathbb A^1$-homotopy invariance to obtain analogues of several key results from $\mathbb A^1$-homotopy theory. These allow us in particular to define a universal oriented motivic $\mathbb E_\infty$-ring spectrum $\mathrm{MGL}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$ can be recovered from its $\mathrm{MGL}$-cohomology via a Conner-Floyd isomorphism \[\mathrm{MGL}^{**}(X)\otimes_{\mathrm L}\mathbb Z[β^{\pm 1}]\simeq \mathrm K^{**}(X),\] where $\mathrm L$ is the Lazard ring and $\mathrm K^{p,q}(X)=\mathrm K_{2q-p}(X)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{MGL}$. |
| title | Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2303.02051 |