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The representability theorem [Demazure-Gabriel, III.2.7.1, p. 318] implies the following.

## Theorem

Let $A$ be a local artinian ring, let $G$ be a group over $A$ locally of finite type, and let $H\hookrightarrow G$ be a closed subgroup which is flat over $A$. Then the quotient $G/H$ in the category of fppf sheaves is a scheme, ; and the canonical morphism $G\rightarrow G/H$ is faithfully flat and of finite presentation.

Note that the group $G$ in the above theorem need not be either affine or flat over $A$; also, Demazure-Gabriel write in comprehensible language, unlike Weil.

The representability theorem [Demazure-Gabriel, III.2.7.1, p. 318] implies the following.

## Theorem

Let $A$ be a local artinian ring, let $G/A$ G$be a group over$A$locally of finite type, and let$H\rightarrow H\hookrightarrow G$be a closed subgroup which is flat over$A$. Then the quotient$G/H$in the category of fppf sheaves is a scheme, and the canonical morphism$G\rightarrow G/H$is faithfully flat and finite presentation. Note that the group$G/A$G$ in the above result theorem need not be either affine or flat ; plusover $A$; also, Demazure-Gabriel write in comprehensible language, unlike Weil! .

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The representability theorem [Demazure-Gabriel, III.2.7.1, p. 318] implies the following.

## Theorem

Let $A$ be a local artinian ring, let $G/A$ be a group locally of finite type, and let $H\rightarrow G$ be a closed subgroup which is flat over $A$. Then the quotient $G/H$ in the category of fppf sheaves is a scheme, and the canonical morphism $G\rightarrow G/H$ is faithfully flat and finite presentation.

Note that the group $G/A$ in the above result need not be either affine or flat; plus, Demazure-Gabriel write in comprehensible language, unlike Weil!