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Noncommutative rigidity

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Abstract

In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin’s rigidity theorem, as well as of Yagunov-Østvær’s equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the way, we introduce the category of n-adic noncommutative mixed motives.

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Notes

  1. Recall that a field extension l / k is called primary if the algebraic closure of k in l is purely inseparable over k. Whenever k is algebraically closed, every field extension l / k is primary.

  2. For the Tannakian formalism, we invite the reader to consult [3,4,5]. Roughly speaking, the Tannakian formalism provides a characterization of the symmetric monoidal abelian categories of linear representations of algebraic groups.

  3. Recall that a commutative ring k is called connected if \(\mathrm {Spec}(k)\) is a connected topological space or, equivalently, if k does not contain non-trivial idempotent elements.

  4. Recall that the smooth compactification \(\overline{C}\) is unique up to isomorphism.

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Acknowledgements

The author is grateful to Joseph Ayoub and Ivan Panin for useful discussions, to Oliver Röndigs and Paul Arne Østvær for references, and to Charles Vial for comments on a previous version. The author is also thankful to the anonymous referees for their comments and to the Mittag-Leffler Institute for its hospitality.

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Correspondence to Gonçalo Tabuada.

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The author was supported by the NSF CAREER Award #1350472 and by the Portuguese Foundation for Science and Technology Grant PEst-OE/MAT/UI0297/2014.

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Tabuada, G. Noncommutative rigidity. Math. Z. 289, 1281–1298 (2018). https://doi.org/10.1007/s00209-017-1998-5

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