Cambridge Journal of Mathematics

Volume 8 (2020)

Number 1

Homotopy invariant presheaves with framed transfers

Pages: 1 – 94



Grigory Garkusha (Department of Mathematics, Swansea University, Swansea, United Kingdom)

Ivan Panin (V. A. Steklov Institute of Mathematics, St. Petersburg, Russia; and Department of Mathematics, University of Oslo, Norway)


The category of framed correspondences $Fr_{\ast} (k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [20]. Based on the notes [20] a new approach to the classical Morel–Voevodsky motivic stable homotopy theory was developed in [8]. 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 $2$.


motivic homotopy theory, framed presheaves

2010 Mathematics Subject Classification

Primary 14F42. Secondary 14F05.

In memory of Andrei Suslin

The authors acknowledge support by the RCN Frontier Research Group Project no. 250399 “Motivic Hopf Equations”.

Received 17 February 2018

Published 25 February 2020