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$\mathbb{A}^1$-invariance of localizing invariants

Vladimir Sosnilo

Vol. 9 (2024), No. 1, 89–118
Abstract

Weibel proved that p-inverted K-theory is 𝔸1-invariant on 𝔽p-schemes and K-theory with p-coefficients is 𝔸1-invariant on [1 p]-schemes. We extend this result to all finitary localizing invariants of small stable -categories. Along the way we study the Frobenius and Verschiebung endofunctors defined by Tabuada and provide a categorical version of Stienstra’s projection formula.

Keywords
noncommutative motives, localizing invariants, $\mathbb{A}^1$-invariance, Witt vectors
Mathematical Subject Classification
Primary: 13F35, 14F42, 18F25
Milestones
Received: 7 January 2023
Revised: 19 February 2024
Accepted: 3 March 2024
Published: 25 May 2024
Authors
Vladimir Sosnilo
Faculty of Mathematics
Universität Regensburg
Regensburg
Germany