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Homotopy invariant presheaves with framed transfers. (English) Zbl 1453.14066

Summary: The category of framed correspondences \(Fr_{\ast} (k)\), framed presheaves and framed sheaves were invented by V. Voevodsky in his unpublished notes [“Notes on framed correspondences”, unpublished, available at www.math.ias.edu/vladimir/files/framed.pdf]. Based on the notes [loc. cit.] a new approach to the classical Morel-Voevodsky motivic stable homotopy theory was developed in [the authors, J. Am. Math. Soc. 34, No. 1, 261–313 (2021; Zbl 1491.14034)]. This approach converts the classical motivic stable homotopy theory into an equivalent local theory of framed bispectra. The main result of the paper is the core of the theory of framed bispectra. It states that for any homotopy invariant quasi-stable radditive framed presheaf of Abelian groups \(\mathcal{F} \), the associated Nisnevich sheaf \(\mathcal{F}_{\mathrm{nis}}\) is strictly homotopy invariant and quasi-stable whenever the base field \(k\) is infinite perfect of characteristic different from 2.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
14F06 Sheaves in algebraic geometry
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
18N45 Categories of fibrations, relations to \(K\)-theory, relations to type theory

Citations:

Zbl 1491.14034
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