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Let $S$ be a scheme and let $N$, $G$ be affine flat group schemes of finite presentation over $S$. If we assume that $N$ is a closed normal subgroup of $G$, we may form the fppf quotient sheaf $G/N$, which is a sheaf of groups. Formally, by By using descent and Artin's representability result for algebraic spaces, it follows that $G/N$ is an algebraic space which is flat, separated and of finite presentation over $S$. By a classic result in the case where $S$ is the spectrum of a field, it also follows formally that $G/N$ has affine fibres. My question is:

Is $G/N$ always affine over $S$?

Phrased in another way: Is the category of affine fppf group schemes over $S$ a semi-abelian subcategory of the semi-abelian category of fppf group sheaves over $S$?

Another related question is if a flat, affine, finitely presented group scheme always may be embedded in the automorphism group of a locally free coherent sheaf on the base. There is a remark about this in SGAIII_1 EXPOSE VI_B 11.11.1, where it is stated without proof that this is supposed to be true for some base schemes under certain regularity conditions.

EDIT: A third related question is if someone knows of an example of a separated, flat, finitely presented group algebraic space with affine fibres which is not affine.

2 edited body

Let $S$ be a scheme and let $N$, $G$ be affine flat group schemes of finite presentation over $S$. If we assume that $N$ is a closed normal subgroup of $G$, we may form the fppf quotient sheaf $G/N$, which is a sheaf of groups. Formally, by using descent, it follows that $G/N$ is an algebraic space which is flat, separated and of finite presentation over $S$. By a classic result in the case where $S$ is the spectrum of a field, it also follows formally that $G/N$ has affine fibres. My question is:

Is $G/N$ always affine over $S$?

Phrased in another way: Is the category of affine fppf group schemes over $S$ a semi-abelian subcategory of the semi-abelian category of fppf group sheaves over $S$?

Another related question is if a flat, affine, finitely presented group scheme always may be embedded in the automorphism group of a locally free coherent sheaf on the base. There is a remark about this in SGAIII_2 SGAIII_1 EXPOSE VI_B 11.11.1, where it is stated without proof that this is supposed to be true for some base schemes under certain regularity conditions.

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# Is the category of affine fppf groups closed under normal quotients?

Let $S$ be a scheme and let $N$, $G$ be affine flat group schemes of finite presentation over $S$. If we assume that $N$ is a closed normal subgroup of $G$, we may form the fppf quotient sheaf $G/N$, which is a sheaf of groups. Formally, by using descent, it follows that $G/N$ is an algebraic space which is flat, separated and of finite presentation over $S$. By a classic result in the case where $S$ is the spectrum of a field, it also follows formally that $G/N$ has affine fibres. My question is:

Is $G/N$ always affine over $S$?

Phrased in another way: Is the category of affine fppf group schemes over $S$ a semi-abelian subcategory of the semi-abelian category of fppf group sheaves over $S$?

Another related question is if a flat, affine, finitely presented group scheme always may be embedded in the automorphism group of a locally free coherent sheaf on the base. There is a remark about this in SGAIII_2 EXPOSE VI_B 11.11.1, where it is stated without proof that this is supposed to be true for some base schemes under certain regularity conditions.