Garkusha, Grigory; Panin, Ivan Homotopy invariant presheaves with framed transfers. (English) Zbl 1453.14066 Camb. J. Math. 8, No. 1, 1-94 (2020). 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of framed correspondences \(Fr_{\ast} (k)\), framed presheaves and framed sheaves were invented by V. Voevodsky in his unpublished notes [“Notes on framed correspondences”, unpublished, available at www.math.ias.edu/vladimir/files/framed.pdf].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. Cited in 20 Documents 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 Keywords:motivic homotopy theory; framed presheaves Citations:Zbl 1491.14034 PDFBibTeX XMLCite \textit{G. Garkusha} and \textit{I. Panin}, Camb. J. Math. 8, No. 1, 1--94 (2020; Zbl 1453.14066) Full Text: DOI arXiv