Differential Forms for T-Algebras in Kahler Categories

Abstract

A Kahler category axiomatizes the algebraic geometric theory of Kahler Differentials in an abstract categorical setting. To facilitate this, a Kahler category is equipped with an algebra modality, which endows each object in the image of a specified monad with an associative algebra structure; universal derivations are then required to exist naturally for each of these objects. Moreover, it can be demonstrated that for each T-algebra of said monad there is a natural associative algebra structure.

In this paper I will show that under certain conditions on the Kahler category, the universal derivations for the algebras arising from T-algebras exist and arise via a coequalizer. Furthermore, this result is extended to provide an alternative construction for universal derivations for a more general class of algebras, including all algebras in a Kahler category. A prospective categorical formulation of the theory of noncommutative Kahler differentials is then given, and the above said results are shown to apply in this context. Finally, another class of algebras is constructed via a colimit, and the modules of differential forms for these algebras is computed.