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We study the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})from an\mathbf{A}^{1}-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})is\mathbf{A}^{1}-equivalent to the Grassmannian of(d-1)-planes in\mathbf{A}^{\infty}. We then describe the\mathbf{A}^{1}-homotopy type of\mathrm{Hilb}_{d}(\mathbf{A}^{n})in a certain range, fornlarge compared tod. For example, we compute the integral cohomology of\mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C})in a range. We also deduce that the forgetful map\mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect}from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an\mathbf{A}^{1}-equivalence after group completion. This implies that the moduli stack\mathcal{FF}\mathrm{lat}, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum\mathrm{kgl}representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the\mathrm{kgl}-homology of smooth proper schemes over a perfect field. 

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 $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-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 ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$.