تقرير
Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory
| العنوان: | Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory |
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| المؤلفون: | Annala, Toni, Hoyois, Marc, Iwasa, Ryomei |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Algebraic Topology, Mathematics - K-Theory and Homology |
| الوصف: | 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 \[\mathrm{MGL}^{**}(X)\otimes_{\mathrm L}\mathbb Z[\beta^{\pm 1}]\simeq \mathrm K^{**}(X),\] 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 $\mathrm{MGL}$. Comment: 34 pages. Final version, to appear in JAMS |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2303.02051 |
| رقم الأكسشن: | edsarx.2303.02051 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |