RIGIDITY FOR HENSELIAN LOCAL RINGS AND A 1 -REPRESENTABLE THEORIES JENS HORNBOSTEL AND SERGE YAGUNOV 1 Abstract. We prove that for a large class of A 1 -representable theories includ- ing all orientable theories it is possible to construct transfer maps and to prove rigidity theorems in the style of Gabber. This generalizes results of Panin and Yagunov obtained over algebraically closed fields to arbitrary infinite ones. Introduction The aim of this paper is to establish rigidity results for graded cohomology type functors E on smooth varieties over a base field k. This paper generalizes the results of [PY] and [Ya] where the special case of orientable theories E resp. stably A 1 -representable theories on smooth varieties over algebraically closed fields have been studied. In algebraic geometry the rigidity theorems correspond to the homotopy invari- ance for cohomology theories on topological spaces. They have been first studied for algebraic K-theory by Suslin, Gabber and others (see [Su1, Ga, GT], and also [Ka, Ja1] for hermitian K-theory). The rigidity theorem and one of its corollaries in Gabber’s paper [Ga] are the following: Theorem 0.1. Suppose that R is a henselian local ring and l ∈ Z is invertible in R, f : M → Spec(R) is a smooth affine morphism of (pure) relative dimension 1, s 0 ,s 1 : Spec(R) → M two sections of f such that s 0 (p)= s 1 (p), where p is the closed point of Spec(R). Then for every homomorphism R → F , F a field, the two composed maps K ∗ (M, Z/l) si → K ∗ (R, Z/l)→K ∗ (F, Z/l) are equal (i =0, 1). The second morphism is known to be injective in many cases at least with integral coefficients if R is regular. In particular, if F = Frac(R) and R contains a field, this is the Gersten conjecture for algebraic K-theory as proved by Quillen [Qu] and Panin [Pa] in this case. We will give a prove for the Gersten conjecture with finite coefficients in Proposition 2.3. Corollary 0.2. If M is a smooth scheme over a field k with l invertible in k, P ∈ M (k), R = O h M,P , then K ∗ (R, Z/l) ∼ = → K ∗ (k, Z/l) is an isomorphism. The proof of Theorem 0.1 relies on the existence of transfer maps fulfilling cer- tain properties and on homotopy invariance (i.e. K ∗ (X ) ∼ = K ∗ (X × A 1 ) if X 1 The author was supported in part by RTN-HPRN-CT-2002-00287, INTAS 00-566 and 03-51- 3251 grants, and the “Russian Science Support Foundation”. 1