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
- [math]\displaystyle{ E / F }[/math]
of algebraically closed fields, and an algebraic variety X / F, there is an isomorphism
- [math]\displaystyle{ K_*(X, \mathbf Z/n) \cong K_*(X \times_F E, \mathbf Z/n), \ i \ge 0 }[/math]
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) show that the base change functor for the mod-n stable A1-homotopy category
- [math]\displaystyle{ \mathrm{SH}(F, \mathbf Z/n) \to \mathrm{SH}(E, \mathbf Z/n) }[/math]
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
- [math]\displaystyle{ K_*(A, \mathbf Z/n) = K_*(A / m, \mathbf Z/n) }[/math]
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).
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
- 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., 126, pp. 59—70, doi:10.1090/conm/126/00509
- Jardine, J. F. (1993), "The K-theory of finite fields, revisited", K-Theory 7 (6): 579–595, doi:10.1007/BF00961219
- Röndigs, Oliver; Østvær, Paul Arne (2008), "Rigidity in motivic homotopy theory", Mathematische Annalen 341 (3): 651—675, doi:10.1007/s00208-008-0208-5
- Suslin, Andrei (1983), "On the K-theory of algebraically closed fields", Inventiones Mathematicae 73 (2): 241—245, doi:10.1007/BF01394024
- Tabuada, Gonçalo (2018), "Noncommutative rigidity", Mathematische Zeitschrift 289 (3-4): 1281—1298, doi:10.1007/s00209-017-1998-5
- Weibel, Charles A. (2013), The K-book, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, ISBN 978-0-8218-9132-2, http://www.math.rutgers.edu/~weibel/Kbook.html
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