Rigidity (K-theory)

In mathematics, rigidity of K-theory encompasses results relating algebraic K-theory of different rings.

Suslin rigidity

Suslin rigidity, named after Andrei Suslin, refers to the invariance of mod-n algebraic K-theory under the base change between two algebraically closed fields: Suslin (1983) showed that for an extension

of algebraically closed fields, and an algebraic variety X / F, there is an isomorphism

between the mod-n K-theory of coherent sheaves on X, respectively its base change to E. A textbook account of this fact in the case X = F, including the resulting computation of K-theory of algebraically closed fields in characteristic p, is in Weibel (2013).

This result has stimulated various other papers. For example Röndigs & Østvær (2008) show that the base change functor for the mod-n stable A1-homotopy category

is fully faithful. A similar statement for non-commutative motives has been established by Tabuada (2018).

Gabber rigidity

Another type of rigidity relates the mod-n K-theory of an henselian ring A to the one of its residue field A/m. This rigidity result is referred to as Gabber rigidity, in view of the work of Gabber (1992) who showed that there is an isomorphism

provided that n1 is an integer which is invertible in A.

If n is not invertible in A, the result as above still holds, provided that K-theory is replaced by the fiber of the trace map between K-theory and topological cyclic homology. This was shown by Clausen, Mathew & Morrow (2021).

Applications

Jardine (1993) used Gabber's and Suslin's rigidity result to reprove Quillen's computation of K-theory of finite fields.

References

  • Clausen, Dustin; Mathew, Akhil; Morrow, Matthew (2021), "K-theory and topological cyclic homology of henselian pairs", J. Amer. Math. Soc., 34: 411--473, arXiv:1803.10897
  • Gabber, Ofer (1992), "K-theory of Henselian local rings and Henselian pairs", Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126, pp. 59–70, doi:10.1090/conm/126/00509, MR 1156502
  • Jardine, J. F. (1993), "The K-theory of finite fields, revisited", K-Theory, 7 (6): 579–595, doi:10.1007/BF00961219, MR 1268594
  • Röndigs, Oliver; Østvær, Paul Arne (2008), "Rigidity in motivic homotopy theory", Mathematische Annalen, 341 (3): 651–675, doi:10.1007/s00208-008-0208-5, MR 2399164
  • Suslin, Andrei (1983), "On the K-theory of algebraically closed fields", Inventiones Mathematicae, 73 (2): 241–245, doi:10.1007/BF01394024, MR 0714090
  • Weibel, Charles A. (2013), The K-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, ISBN 978-0-8218-9132-2, MR 3076731
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