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

This is the authors' abstract. We don't add key points for this paper.

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 ∞ ∞ -category of non- A 1 A^1 -invariant motivic spectra, which turns out to be equivalent to the ∞ ∞ -category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this ∞ ∞ -category satisfies P 1 P^1 -homotopy invariance and weighted A 1 A^1 -homotopy invariance, which we use in place of A 1 A^1 -homotopy invariance to obtain analogues of several key results from A 1 A^1 -homotopy theory. These allow us in particular to define a universal oriented motivic E ∞ E_∞ -ring spectrum M G L MGL . We then prove that the algebraic K-theory of a qcqs derived scheme X X can be recovered from its M G L MGL -cohomology via a Conner–Floyd isomorphism \[ M G L ∗ ∗ ( X ) ⊗ L Z [ β ± 1 ] ≃ K ∗ ∗ ( X ) , MGL**(X)⊗ _{L{}}Z[β ± 1] K{}**(X), \] where L L{} is the Lazard ring and K p , q ( X ) = K 2 q − p ( X ) K{}p,q(X)=K{}2q-p(X) . Finally, we prove a Snaith theorem for the periodized version of M G L MGL .