Base change theorems for log analytic spaces (Q6116783)

In this paper, for fs log analytic spaces, three kinds of base change theorems, that is, proper/smooth/proper smooth base change theorems in log Betti cohomology are discussed. A new feature is that the nonabelian coefficient cases are included. A large part of the abelian coefficient cases in the theorems in this article is already proved in [\textit{T. Kajiwara} and \textit{C. Nakayama}, J. Math. Sci., Tokyo 15, No. 2, 291--323 (2008; Zbl 1180.14016)] and [\textit{C. Nakayama} and \textit{A. Ogus}, Geom. Topol. 14, No. 4, 2189--2241 (2010; Zbl 1201.14007)], a systematic discussion of the abelian cases together with nonabelian cases is given in this paper. For an fs log analytic space \(X\), a certain topological space \(X^{\log}\) is defined. This construction \((-)^{\log}\) gives a functor from the category of fs log analytic spaces to that of topological spaces. For example, a smooth base change theorem in Theorem 3.1 asserts the following. Let \[ \begin{tikzcd} X' \arrow[r,"f'"]\arrow[d,"{g'}" '] & Y' \arrow[d,"g"]\\ X \arrow[r,"f"] & Y \end{tikzcd} \] be a Cartesian diagram of fs log analytic spaces with \(g\) being exact and log smooth. Let \(F\) be a sheaf of sets (resp.\ groups, resp.\ abelian groups) on \(X^{\log}\). Then the natural map \[ {g^{\log}}^{-1} R^q f^{\log}_{\ast} F \to R^q f'^{\log}_{\ast} {g'^{\log}}^{-1} F \] is an isomorphism for \(q=0\) (resp.\ \(q=0,1\), resp.\ \(q \in \mathbb{Z}\)). This is proved by reducing it to a topological smooth base change theorem. In the final section, it is shown that a real blowing-up of any log modification is a weak homotopy equivalence.