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

TL;DR

This paper constructs pushouts of quasi-finite, flat group schemes over a Dedekind ring, and proves the existence of cokernels and quotients for finite flat group schemes, advancing the understanding of their categorical properties.

Contribution

It introduces a method to construct pushouts of certain group schemes and provides a simplified proof for cokernels and quotients over Dedekind rings.

Findings

Constructed pushouts in the category of R-affine group schemes.

Proved that pushouts preserve the generic fiber when certain conditions hold.

Provided a short proof for the existence of cokernels and quotients of finite flat group schemes.

Abstract

Let $G$, $G_1$ and $G_2$ be quasi-finite and flat group schemes over a complete discrete valuation ring $R$, $\varphi_1:G\to G_1$ any morphism of $R$-group schemes and $\varphi_2:G\to G_2$ 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 $\varphi_1$ 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.