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 \[ \mathrm {MGL}^{**}(X)\otimes _{\mathrm {L}{}}\mathbb {Z}[\beta ^{\pm 1}]\simeq \mathrm {K}{}^{**}(X), \] 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 $\mathrm {MGL}$.