Rigidity for henselian local rings and \(\mathbb{A}^1\)-representable theories
Let \(R\) be a henselian local ring essentially smooth over an infinite field \(k\). Assume that \(E=E^{**}\) is a contravariant bigraded functor on the category \(S\) of smooth schemes of finite type over \(k\), \(E:S\rightarrow \mathbf{Ab}\), that is representable in the stable \({\mathbb A}^1\)-homotopy theory and satisfies \( lE=0\) for \( l\in {\mathbb Z}\) invertible in \(R\). Assume that \(E\) is so called normalized with respect to the fraction field of \(R\). Let \(f:M\rightarrow \mathrm{Spec } R\) be a smooth affine morphism of (pure) relative dimension \(d\) and \(s_0,s_1:\mathrm{Spec } R \rightarrow M\) two sections of \(f\) such that \(s_0(P)=s_1(P)\), where \(P\) is the closed point of \(\mathrm{Spec } R\). Then the induced maps \(s_i^*:E(M)\rightarrow E(\mathrm{Spec }R)\), \(i=1,2\) are equal. This is a rigidity theorem, which extends the results of \textit{I. Panin} and the second author [J. Pure Appl. Algebra 172, No. 1, 49--77 (2002; Zbl 1056.14027)] obtained for an algebraically closed field \(k\) and it is similar to the rigidity theorem of \textit{O. Gabber} [Contemp. Math. 126, 59--70 (1992; Zbl 0791.19002)] obtained for algebraic \(K\)-theory.
stable \({\mathbb A}^1\)-homotopy category
scientific article; zbMATH DE number 5706074
scientific article