Topological logarithmic structures

A logarithmic structure on a commutative ring A is a commutative monoid M with a homomorphism to the underlying multiplicative monoid of A. This determines a localization AŒM  of A. In algebro-geometric terms, we might say that M cuts out a divisor D from Spec.A/, and AŒM  is the ring of regular functions on the open complement. In general the logarithmic structure carries more information than the localization. For example, the Kahler differentials of A form an A–module  A , generated by differentials of the form da, which are globally defined over Spec.A/. The Kahler differentials of the localization form the AŒM –module  AŒM  , which also contains differentials of the form m da, having poles of arbitrary degree along D . The logarithmic structure specifies an intermediate A–module of logarithmic Kahler differentials,  .A;M / , generated by differentials of the form da and d log mDm dm, having only poles of simple, or logarithmic, type along D . The logarithmic structure is therefore a more moderate way of specifying a localization than the actual localized ring. See Kato [35] and Illusie [34] for introductions to logarithmic algebraic geometry.