Voevodsky's motives and zeroth slice of motivic sphere

Abstract

Let $S$ be a noetherian scheme of finite Krull dimension and let $R$ be a commutative ring in which every positive residue characteristic of $S$ is invertible. We prove that Voevodsky's stable Nisnevich category of motives with $R$-coefficients is canonically equivalent to modules over the zeroth slice of the motivic sphere with derived $R$-coefficients. The comparison identifies the corresponding commutative coefficient algebras and is compatible with arbitrary pullback. The theorem applies to singular and nonreduced schemes, including mixed-characteristic bases, and requires no excellence or finite-type hypothesis. Its special cases include rational coefficients over arbitrary bases, arbitrary coefficients over $\mathbb Q$-schemes, and $\mathbb Z[1/p]$-algebra coefficients over $\mathbb Z_{(p)}$-schemes. The proof establishes base change for finite-cycle spectra using permutation quotients, families of cycles, and arithmetic power homotopies, then passes to general bases by universal homeomorphism invariance and continuity.