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: Template:Harvtxt showed that for an extension

$E/F$

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

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

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 Template:Harvtxt.

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

$\mathrm{S}\mathrm{H}(F,𝐙/n)\to \mathrm{S}\mathrm{H}(E,𝐙/n)$

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

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 Template:Harvtxt 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 Template:Harvtxt.

Applications

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

References

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