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- A 1 \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 P 1 \mathbb {P}^1 -homotopy invariance and weighted A 1 \mathbb {A}^1 -homotopy invariance, which we use in place of A 1 \mathbb {A}^1 -homotopy invariance to obtain analogues of several key results from A 1 \mathbb {A}^1 -homotopy theory. These allow us in particular to define a universal oriented motivic E ∞ \mathbb {E}_\infty -ring spectrum M G L \mathrm {MGL} . We then prove that the algebraic K-theory of a qcqs derived scheme X X can be recovered from its M G L \mathrm {MGL} -cohomology via a Conner–Floyd isomorphism \[ M G L ∗ ∗ ( X ) ⊗ L Z [ β ± 1 ] ≃ K ∗ ∗ ( X ) , \mathrm {MGL}^{**}(X)\otimes _{\mathrm {L}{}}\mathbb {Z}[\beta ^{\pm 1}]\simeq \mathrm {K}{}^{**}(X), \] where L \mathrm {L}{} is the Lazard ring and K p , q ( X ) = K 2 q − p ( X ) \mathrm {K}{}^{p,q}(X)=\mathrm {K}{}_{2q-p}(X) . Finally, we prove a Snaith theorem for the periodized version of M G L \mathrm {MGL} .