تقرير
Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory
العنوان: | Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory |
---|---|
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. |