Rigidity (K-theory) explained

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

Suslin rigidity

E / F of algebraically closed fields, and an algebraic variety X / F, there is an isomorphismK _{*(X,} Z/n) \cong K _{*(X} x _{F} E, Z/n), i \ge 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 .

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

SH (F, Z/n) \to SH (E, Z/n) is fully faithful. A similar statement for non-commutative motives has been established by .

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 who showed that there is an isomorphism

K _{*(A,} Z/n) = K _{*(A} / m, 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 .

Applications

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