Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory

Abstract

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 $\mathrm{MGL}$.

Theorems & Definitions (119)

• Theorem 1.1: Universality of $\mathrm{MGL}$, Theorem \ref{['thm:universality']}
• Theorem 1.2: Conner--Floyd isomorphism, Theorem \ref{['thm:Conner-Floyd']}
• Theorem 1.3: Snaith theorem for $\mathrm{PMGL}$, Theorem \ref{['thm:Snaith']}
• Theorem 1.4: $\mathbb P$-homotopy invariance, Theorem \ref{['thm:euler']}
• Theorem 1.5
• Theorem 1.6: Bass delooping, Corollary \ref{['cor:delooping']}
• Remark 1.7
• Remark 1.8
• Definition 2.1
• Proposition 2.2