Morphisms of Idealized Logarithmic Schemes
In the field of algebraic geometry, schemes are a powerful tool for generalizing geometric objects such as curves and surfaces. In particular, we may study smooth morphisms between schemes, which correspond to the existence of thickenings between schemes. Moreover, a smooth morphism uniquely factors into an affine chart. Such morphisms provide insight into how schemes behave under compactification and degeneration, as smoothness is not generally preserved by these transformations. Equipping schemes with an additional monoidal sheaf allows for the formation of a logarithmic structure on the scheme, which generates a natural conception of logarithmically smooth morphisms that are invariant under these transformations. Logarithmically smooth morphisms also correspond to a chart on the spectrums of monoidal algebras, which generalizes the aforementioned affine charts. In our work, we investigate a further generalization called idealized logarithmic schemes and determine chart properties for corresponding idealized logarithmically smooth morphisms.