On infinitesimally flat affine group schemes with connected fibers over rings

Source abstract

This paper explores Csima and Kottwitz’s comodule theory for representations of multiplicative affine group schemes, framed through comodules over their coordinate algebras. We extend this framework to broader classes of group schemes and provide concise proofs of Jantzen’s classical results concerning flat group schemes with infinitesimally flat structures and connected fibers. This establishes a comprehensive characterization of their representation categories. Using N\'eron blowup techniques, we reconstruct smooth group schemes with connected fibers and explicitly derive their coordinate algebras. The versatility of our approach is demonstrated through additive group scheme examples, mirroring Csima--Kottwitz’s multiplicative models.