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} .

我们制定并证明了任意 qcqs 导出方案的代数 K 理论的 Conner-Floyd 同构。为此，我们研究了非 A 1 \mathbb {A}^1 不变动机谱的稳定 ∞ \infty -类别，它等效于满足基本爆炸的基本动机谱的 ∞ \infty -类别切除，先前由第一作者和第三作者介绍过。我们证明这个 ∞ \infty -类别满足 P 1 \mathbb {P}^1 -同伦不变性和加权 A 1 \mathbb {A}^1 -同伦不变性，我们用它来代替 A 1 \mathbb {A} ^1 -同伦不变性，以获得 A 1 \mathbb {A}^1 -同伦理论的几个关键结果的类似物。这些特别允许我们定义一个通用的定向动机 E ∞ \mathbb {E}_\infty 环谱 M G L \mathrm {MGL} 。然后我们证明 qcqs 派生方案 X X 的代数 K 理论可以通过 Conner-Floyd 同构从其 M G L \mathrm {MGL} -上同调中恢复 \[ M G L ∗ ∗ ( X ) ⊗ L Z [ β ± 1 ] ≃ K * * ( X ) \mathrm {MGL}^{**}(X)\otimes _{\mathrm {L}{}}\mathbb {Z}[\beta ^{\pm 1}]\simeq \mathrm {K}{}^{**}(X), \] 其中 L \mathrm {L}{} 是拉扎德环，K p q ( X ) = K 2 q − p ( X ) \mathrm {K}{ }^{p,q}(X)=\mathrm {K}{}_{2q-p}(X) 。最后，我们证明了 M G L \mathrm {MGL} 的周期版本的斯奈斯定理。