I'm not sure why this hasn't been mentioned yet, but the category of abelian group objects in smooth proper geometrically integral schemes is not abelian, already if $S = \operatorname{Spec} k$. Indeed, if $A$ is any abelian variety, then the map $[n] \colon A \to A$ for $n > 1$ is a monomorphism and an epimorphism:

• As the OP explained here, it is an epimorphism of schemes since it is surjective on points and injective on sections (being a dominant morphism of integral schemes). Thus it is also an epimorphism of group schemes.
• If $f \colon G \to A$ is a map such that $G \to A \stackrel{[n]}\to A$ is trivial, then $f(G)$ lands in $A[n]$ scheme-theoretically. But if $G$ is geometrically integral, this implies $f(G)$ lands in the trivial subgroup $\boldsymbol 1$, again scheme-theoretically.

Thus, $[n] \colon A \to A$ is both a monomorphism and an epimorphism, but it clearly does not have an inverse (even as schemes) when $n > 1$.

Dropping "geometrically integral" doesn't help: if $\operatorname{char} k = p > 0$ and $A$ is supersingular, the same example with $n = p$ shows that looking at smooth abelian group schemes is not enough. Dropping both does work if you further impose finite type hypotheses: the category of abelian group schemes of finite type over a field $k$ is abelian; see this post and the references therein. Then the proper ones form a Serre subcategory, hence an abelian category.

It would be interesting to see what versions are true over a general base. It seems hard to me to make quotients if there are no flatness assumptions, but maybe kernels or cokernels of flat group schemes don't want to be flat. I should probably try to read some SGA3...

I'm not sure why this hasn't been mentioned yet, but the category of abelian group objects in smooth proper geometrically integral schemes is not abelian, already if $S = \operatorname{Spec} k$. Indeed, if $A$ is any abelian variety, then the map $[n] \colon A \to A$ for $n > 1$ is a monomorphism and an epimorphism:

• As the OP explained at What are the epimorphisms in the category of schemes?, it is an epimorphism of schemes since it is surjective on points and injective on sections (being a dominant morphism of integral schemes). Thus it is also an epimorphism of group schemes.
• If $f \colon G \to A$ is a map such that $G \to A \stackrel{[n]}\to A$ is trivial, then $f(G)$ lands in $A[n]$ scheme-theoretically. But if $G$ is geometrically integral, this implies $f(G)$ lands in the trivial subgroup $\boldsymbol 1$, again scheme-theoretically.

Thus, $[n] \colon A \to A$ is both a monomorphism and an epimorphism, but it clearly does not have an inverse (even as schemes) when $n > 1$.

Dropping "geometrically integral" doesn't help: if $\operatorname{char} k = p > 0$ and $A$ is supersingular, the same example with $n = p$ shows that looking at smooth abelian group schemes is not enough. Dropping both does work if you further impose finite type hypotheses: the category of abelian group schemes of finite type over a field $k$ is abelian; see Is the category of commutative group schemes abelian? and the references therein. Then the proper ones form a Serre subcategory, hence an abelian category.

It would be interesting to see what versions are true over a general base. It seems hard to me to make quotients if there are no flatness assumptions, but maybe kernels or cokernels of flat group schemes don't want to be flat. I should probably try to read some SGA3….

I'm not sure why this hasn't been mentioned yet, but the category of abelian group objects in smooth proper geometrically integral schemes is not abelian, already if $S = \operatorname{Spec} k$. For example, the map $(-)^n \colon \mathbf G_m \to \mathbf G_m$ for $n > 1$ is a monomorphism and an epimorphism:

• As the OP explained here, it is an epimorphism of schemes since it is surjective on points and injective on sections. Thus it is also an epimorphism of group schemes.
• If $f \colon G \to \mathbf G_m$ is a map such that $G \to \mathbf G_m \stackrel{(-)^n}\to \mathbf G_m$ is trivial, then $f(G)$ lands in $\boldsymbol\mu_n$ scheme-theoretically. But if $G$ is geometrically integral, this implies $f(G)$ lands in the trivial subgroup $\boldsymbol 1$, again scheme-theoretically.

Thus, $(-)^n \colon \mathbf G_m \to \mathbf G_m$ is both a monomorphism and an epimorphism, but it clearly does not have an inverse (even as schemes) when $n > 1$.

Dropping "geometrically integral" doesn't help: the same example with $n = p$ for $\operatorname{char} k = p > 0$ shows that looking at smooth abelian group schemes is not enough. Dropping both does work if you further impose finite type hypotheses: the category of abelian group schemes of finite type over a field $k$ is abelian; see this post and the references therein.

It would be interesting to see what versions are true over a general base. It seems hard to me to make quotients if there are no flatness assumptions, but maybe kernels or cokernels of flat group schemes don't want to be flat. I should probably try to read some SGA3...

I'm not sure why this hasn't been mentioned yet, but the category of abelian group objects in smooth proper geometrically integral schemes is not abelian, already if $S = \operatorname{Spec} k$. Indeed, if $A$ is any abelian variety, then the map $[n] \colon A \to A$ for $n > 1$ is a monomorphism and an epimorphism:

• As the OP explained here, it is an epimorphism of schemes since it is surjective on points and injective on sections (being a dominant morphism of integral schemes). Thus it is also an epimorphism of group schemes.
• If $f \colon G \to A$ is a map such that $G \to A \stackrel{[n]}\to A$ is trivial, then $f(G)$ lands in $A[n]$ scheme-theoretically. But if $G$ is geometrically integral, this implies $f(G)$ lands in the trivial subgroup $\boldsymbol 1$, again scheme-theoretically.

Thus, $[n] \colon A \to A$ is both a monomorphism and an epimorphism, but it clearly does not have an inverse (even as schemes) when $n > 1$.

Dropping "geometrically integral" doesn't help: if $\operatorname{char} k = p > 0$ and $A$ is supersingular, the same example with $n = p$ shows that looking at smooth abelian group schemes is not enough. Dropping both does work if you further impose finite type hypotheses: the category of abelian group schemes of finite type over a field $k$ is abelian; see this post and the references therein. Then the proper ones form a Serre subcategory, hence an abelian category.

It would be interesting to see what versions are true over a general base. It seems hard to me to make quotients if there are no flatness assumptions, but maybe kernels or cokernels of flat group schemes don't want to be flat. I should probably try to read some SGA3...