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Review of the Morel-Voevodsky Localization Theorem
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Profile image of Milen PetrovMilen Petrov

2018

Abstract

We prove the analog of the Morel-Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes. Contents 1. Review of the Morel-Voevodsky localization theorem 2 2. The localization theorem for framed motivic spaces 4 3. The reconstruction theorem over a general base scheme 6 4. Application to motivic cohomology 7 References 9 In this article we show that the theory of framed motivic spaces introduced in [EHK + 19b] satisfies localization: if i : Z → S is a closed immersion of schemes, j : U → S is the complementary open immersion, and F ∈ H fr (S) is a framed motivic space over S, then there is a cofiber sequence j j * F → F → i * i * F (see Theorem 10). Consequently, the theory of framed motivic spectra satisfies Ayoub's axioms [Ayo08], which implies that it admits a full-fledged formalism of six operations. Using this formalism, we show that the equivalence SH fr (S) SH(S), proved in [EHK + 19b] for S the spectrum of a perfect field, holds for any scheme S (see Theorem 18). The ∞-category H fr (S) of framed motivic spaces consists of A 1-invariant Nisnevich sheaves on the ∞-category Corr fr (Sm S) of smooth S-schemes and framed correspondences. A framed correspondence between S-schemes X and Y is a span Z X Y f over S, where f is a finite syntomic morphism equipped with a trivialization of its cotangent complex in the K-theory of Z. Our result stands in contrast to the case of finite correspondences in the sense of Voevodsky, where the analog of the Morel-Voevodsky localization theorem remains unknown. The essential ingredient in our proof is the fact that the Hilbert scheme of framed points [EHK + 19b, Definition 5.1.7] is smooth.

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What are the implications of framed motivic spaces satisfying localization?add

The study demonstrates that framed motivic spaces allow for a cofiber sequence involving closed immersions, as shown in Theorem 10. This finding enhances the formalism for six operations in algebraic structures involving these spaces.

How is the reconstruction theorem for framed motivic spaces extended?add

The paper establishes the reconstruction theorem for framed motivic spaces, now applicable to any scheme S, as addressed in Theorem 18. This theorem ensures the functorial behavior of the category SH fr (S) under smooth morphisms.

What methods were utilized for proving the localization theorem?add

The proof employs cofiber sequences and a specific focus on smooth S-schemes to validate the localization theorem for framed spaces. Essential tools include Novel adjunctions and the application of Ayoub's axioms.

When was the localization theorem for framed motivic spaces proven?add

The findings regarding the localization theorem for framed motivic spaces were detailed in September 2020, as indicated in the study. This theorem significantly advances the categorical framework of motivic cohomology.

What defines a framed correspondence in the study's context?add

A framed correspondence is characterized as a span Z ⟶ X ⟶ Y with a finite syntomic morphism and a trivial cotangent complex, crucial in discussions on framed motivic spaces. This concept underpins many results related to motivic cohomology and morphisms within the framework.

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