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

${\displaystyle E/F}$

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

${\displaystyle K_{*}(X,\mathbf {Z} /n)\cong K_{*}(X\times _{F}E,\mathbf {Z} /n),\ i\geq 0}$

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

${\displaystyle \mathrm {SH} (F,\mathbf {Z} /n)\to \mathrm {SH} (E,\mathbf {Z} /n)}$

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

${\displaystyle K_{*}(A,\mathbf {Z} /n)=K_{*}(A/m,\mathbf {Z} /n)}$

provided that n≥1 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

• Tabuada, Gonçalo (2018), "Noncommutative rigidity", Mathematische Zeitschrift, 289 (3–4): 1281–1298, arXiv:1703.10599, doi:10.1007/s00209-017-1998-5, MR 3830249

• 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