Regularity in Logarithmic Algebraic Geometry: a different viewpoint

Abstract

Logarithmic algebraic geometry is the right context to deal with some problems in algebraic geometry. For instance, certain morphisms of schemes which are not smooth can be considered as morphisms of logarithmic schemes for some logarithmic structure making them log smooth. Thus, as it happens in algebraic geometry, log smoothness and log regularity are important in this field. The aim of this dissertation is to study regularity in logarithmic algebraic geometry from a different viewpoint. This allows us to obtain new results and also to remove unnecessary hypotheses from some known results. For that purpose, we will develop the necessary results on the logarithmic cotangent complex in order to characterize log regularity (and also log smoothness) in terms of this complex.