On jets, extensions and characteristic classes I

Abstract

In this paper, we give general definitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors $\operatorname{Exan}_k(A,I)$, where $A$ is any $k$-algebra, and $I$ is any left and right $A$-module and use this to construct affine non-commutative jets. We also study the Kodaira-Spencer class $\operatorname{KS}(\mathcal{L})$ and relate it to the Atiyah class.