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We prove that for a large class of A 1 -representable theories including all orientable theories it is possible to construct transfer maps and to prove rigidity theorems similar to those of Gabber for algebraic K-theory. This extends rigidity results of Panin and Yagunov from algebraically closed fields to arbitrary infinite ones. IntroductionThe aim of this paper is to establish rigidity results for graded cohomology type functors E on smooth varieties over an infinite base field k. This paper generalizes the results of [14] and [24] where the special case of orientable theories E resp. stably A 1 -representable theories on smooth varieties over algebraically closed fields have been studied.Consider some category of schemes (spaces) S over a base scheme (space) B together with a cohomology theory E * : S op → Ab. Then we say that E * satisfies rigidity if for every irreducible scheme X χ → B, any two sections σ 0 , σ 1 : B → X of the structure morphism χ induce the same homomorphism σ * 0 = σ * 1 : E * (X) → E * (B). In classical topology, the rigidity property is an obvious consequence of homotopy invariance of cohomology theories. However, in algebraic geometry A 1 -invariance does not always imply rigidity. It only holds under certain restrictions on S and the cohomology theory E * . In particular, rigidity fails for K 1 with integral coefficients.

We extend the classical rigidity results for K-theory to the equivariant setting of algebraic group actions. Following the work of Andrei Suslin on ordinary K-groups, these results can be considered as first steps toward an explicit computation of equivariant K-groups of algebraically closed fields.