# Quotients and exact sequences of algebraic group schemes

Now I study group schemes to understand fundamental properties about semi abelian varieties and generalized jacobian varieties.

Let $$G$$ be an algebraic group scheme over a field $$k$$ ($$=$$ $$k$$ group schemes of finite type), $$H$$ an algebraic group sub-scheme. In Milne's online note, the author defines that the quotient $$G/H$$ is the representable algebraic scheme of the sheaf associated to the presheaf $$S \mapsto G(S)/H(S)$$, in the "faithfully flat finite type site". (not fppf.) But in Conrad's "semistable reduction for abelian varieties", the author uses the different definition. (It seems for me that he uses the fppf site.)

And so these 2 references define exact sequences in the different way. The former says that a sequence $$1 \to G' \xrightarrow{f} G \xrightarrow{g} G'' \to 1$$ is exact if $$g$$ is faithfully flat and $$f$$ identifies $$G'$$ to $$\ker g$$. (The author shows that $$g$$ is faithfully flat $$\iff$$ $$g$$ is surjective $$\iff$$ $$g$$ induces $$G / \ker g \cong G''$$.) Are these two definitions same?

Next, In Serre's "algebraic groups and class fields", the author says that if $$k$$ is algebraically closed, a sequence of group varieties (= algebraic groups that are varieties = smooth algebraic groups) $$1 \to G' \xrightarrow{f} G \xrightarrow{g} G'' \to 1$$ is exact iff this is exact on the rational points as abstract groups and it induces the exact sequence of the tangent spaces at $$1$$. Is this true? I don't know the Weil's foundation, so I don't understand whether this is true for schemes.

And please suggest me some references.

Thank you very much!

I don't have Milne on me, but I'm pretty sure that "faithfully flat finite type" is just fppf. Recall that fppf is just french for faithfully flat and finite presentation, but our objects are Noetherian.

As to your second point, the only question is whether or not if $$f:G\to H$$ is faithfully flat then $$H$$ is isomorphic to the fppf sheafification of $$G/\ker f$$. It sounds like you know that the fppf sheafification is representable by some group scheme $$Q$$. Note moreover that we get a map $$Q\to H$$ of groups since we have a map of presheaves $$(G/\ker f)^\mathrm{pre}\to H$$ and thus we get a map by the definition of sheafification since $$H$$ is a sheaf. We want to show that this map of groups is an isomorphism. The fact that it's injective is clear. To show surjectivity it suffices to show that $$f:G\to H$$ is a surjective map of fppf sheaves. But, this almost follows by definition/the fact that we have effective fppf descent for morphisms.

For your final point, if I'm understanding you correctly. There are two points here:

1. If $$f:G\to H$$ is a map of finite type algebraic groups over $$k$$, then $$\ker f$$ is trivial iff $$(\ker f)(\overline{k})=1$$ and the map $$(df)_e:T_e G\to T_e H$$ is injective. Moreover, both of these are equivalent to the fact that $$f$$ is a closed immersion.
2. If $$f:G\to H$$ is a map of smooth algebraic groups over $$k$$, then $$f$$ is fatithfully flat if and only if $$f(\overline{k}):G(\overline{k})\to H(\overline{k})$$ is a surjection.
3. If $$f:G\to H$$ is a surjective map of smooth groups, and $$\ker f$$ is smooth, then the induced maps on tangent spaces is a short exact sequence.

The first fact can be found here as proposition 10.2.1 with proof in subsequent pages. The second fact follows from miracle flatness and basic facts about surjectivity of maps of schemes. The last part is clear by noting that since $$\ker f$$ is smooth that $$f$$ itself is smooth by fppf descent (since $$G\times_H G\cong G\times_{\mathrm{Spec}(k)}\ker f$$) and thus surjectivity on the tangetn spaces is clear, and the short exact sequence part is an exercise I leave to you.

• Thank you very much, this is great! But I don't understand the argument about fppf descent. I think the map of sheaves $G \to H$ is surjective iff for every $k$ algebraic scheme $S$ and $k$-morphism $S \to H$, there exists a faithfully flat of finite type $S' \to S$ such that $S' \to H$ factors through $G \to H$. But since $G \to H$ itself is fppf, tanking $S' = S \times_H G$, we have this, hence the map of sheaves $G \to H$ is surjective. – agababibu Apr 11 at 7:09