Log purity, torsors on root stacks and log Nori fundamental group

Abstract

We generalize the logarithmic purity theorem of Fujiwara-Kato-Mochizuki to torsors which arise in the Kummer log flat topology under finite flat linearly reductive group schemes. We then give a stack-theoretic interpretation of our purity theorem via root stacks, relating it to the valuative criterion of properness for tame algebraic stacks of Bresciani-Vistoli. Finally, we construct a logarithmic Nori fundamental group scheme of a log regular log scheme classifying such torsors, and compare it with the classical Nori fundamental group and the tame fundamental group.