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: showed that for an extension
 * $$E / F$$

of algebraically closed fields, and an algebraic variety X / F, there is an isomorphism
 * $$K_*(X, \mathbf Z/n) \cong K_*(X \times_F E, \mathbf Z/n), \ i \ge 0$$

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

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

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

provided that n&ge;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.