Rigidity for pseudo pretheories (Q2509016)

\noindent A celebrated theorem of \textit{A. Suslin} says that algebraic \(K\)-theory with coefficients is rigid with respect to extensions of algebraically closed fields [Invent. Math. 73, 241--245 (1983; Zbl 0514.18008)]. In a more general setting \textit{A. Suslin} and \textit{V. Voevodsky} proved [Invent. Math. 123, No.~1, 61--94 (1996; Zbl 0896.55002)] that every homotopy invariant pseudo pretheory with torsion values is rigid with respect to extensions of algebraically closed fields. The authors prove the following: Theorem 1. Let \(K\) be a Hensel valued field with valuation \(v,\) and let \(k\subset K\) be a subfield which is dense in the \((K,v)\)-topology and is algebraically closed in \(K.\) If the exponential characteristic char\((k)>1\), assume \(K/k\) is separable. Suppose \(X\) is a smooth scheme of finite type over \(k\) and let \(X_{K}=X{\times}_{k}K.\) If \(F\) is a homotopy invariant pseudo pretheory on Sm\(_k\) with torsion values prime to char\,\(k\), base extension along the map \(k\rightarrow K\) induces an isomorphism \(F(X)\to F(X_K).\)