Pushout of quasi-finite and flat group schemes over a Dedekind ring

Abstract

Let $G$, $G_1$ and $G_2$ be quasi-finite and flat group schemes over a complete discrete valuation ring $R$, ${\phi }_1:G\to G_1$ any morphism of $R$-group schemes and ${\phi }_2:G\to G_2$ a model map. We construct the pushout $P$ of $G_1$ and $G_2$ over $G$ in the category of $R$-affine group schemes. In particular when ${\phi }_1$ is a model map too we show that $P$ is still a model of the generic fibre of $G$. We also provide a short proof for the existence of cokernels and quotients of finite and flat group schemes over any Dedekind ring.

18 pages, preliminary version