Let be a finite flat commutative group scheme over a fixed locally noetherian base scheme . In this brief note, I want to explain the proof of the following theorem due to Raynaud.
Theorem. There exists, Zariski-locally on , an abelian scheme such that embeds as a closed -subgroup of .
This theorem is rather useful in reducing statements of a cohomological nature about finite flat commutative group schemes to analogous statements about abelian schemes, where often one has more tools at one’s disposal (e.g., the fact proved in the appendix to Grothendieck’s Brauer III that étale cohomology and fppf cohomology coincide for smooth commutative group schemes). For an example application, see Proposition 3.1 in Bhargav Bhatt’s paper Annihilating the cohomology of group schemes. It will turn out that the theorem follows with a bit of extra work from the results described in my previous post with Bogdan Zavyalov, The torsion component of the Picard scheme.
To begin the proof, first note that by standard principles of spreading out we may assume that is a noetherian local ring. As in the previous post (see Section 3 there), there exists some projective space equipped with an action of the Cartier dual of , free away from a closed subscheme of codimension . The quotient in the category of locally ringed spaces is again a projective -scheme, and it is smooth away from the image of in . By a simple argument with Bertini’s theorem (plus some technical arguments in commutative algebra), there exists a smooth closed subscheme of disjoint from with geometrically connected -dimensional fibers over . A descent argument (see Lemma 1.4 of the previous post) shows that is a closed -subgroup of the Picard scheme . Admit for a moment the following proposition.
Proposition. Let be a smooth projective -scheme with geometrically connected -dimensional fibers. The torsion component of the Picard scheme is an abelian scheme over (hence equal to ) and the fppf quotient sheaf is represented by the constant -group scheme .
Using this proposition, the theorem is fairly simple. Indeed, we may check that the map factors through after fppf base change on , so we may assume that . In this case, is -isomorphic to , and it follows that the induced map is trivial: on fibers it is certainly trivial because is finite (hence torsion by a theorem of Deligne) and is torsion-free. Since the identity section of is an open embedding, it follows that factors through the identity section, as desired. So the map factors through the identity component , which is an abelian scheme. It only remains to prove the proposition.
Proof of Proposition. First, we prove that is a smooth -scheme using the infinitesimal lifting criterion. Let be an Artin local ring and let be a square zero ideal. We want to show that the natural map is surjective. By base change, we may assume . Since is smooth over (and in particular flat), there is a short exact sequence of Zariski sheaves
where . Passing to Cech cohomology gives
Since is an Artin local scheme, is a -dimensional noetherian scheme. Hence by dimensional vanishing of quasicoherent cohomology for such schemes we find that , and thus the map is surjective. Via the identifications and , we see that is smooth.
Next, we show that satisfies the valuative criterion of properness. Let be a discrete valuation ring with fraction field and residue field , and let be a morphism. We need to show that the map is bijective. For this, we may assume . Since is smooth and is regular, and . We first show that is surjective. Let be a Weil divisor on and let denote the closure of in . This is a Weil divisor on with , so that is surjective.
To show that is injective, let be a Weil divisor on , say for some irreducible Weil divisors and . Note that is the unique irreducible Weil divisor on which is not -flat, and is principal. Thus we may assume that is -flat for each , in which case each is the closure of its generic fiber in . Suppose for some rational function on . If is a uniformizer of and has -adic valuation (as we may assume), then for considered as a rational function on , and it follows that is injective. Thus does indeed satisfy the valuative criterion of properness. (This argument did not use the fact that has -dimensional fibers.)
By SGA6, Exp. XIII, Prop. 3.2(iii) and Thm. 4.7 (and Grothendieck’s representability theorem for the Picard scheme, see FGA Explained, Theorem 9.4.8), is an open and closed subscheme of of finite type over , and hence it is a smooth proper -scheme by the above. To show that is an abelian scheme, it suffices to show that it has connected fibers, and by base change we may assume that for a field . We now appeal to the standard (though rather non-trivial) fact that if is a smooth, proper, geometrically connected curve over an algebraically closed field , then the component group is isomorphic to via the degree map. In particular, , i.e., is connected. So indeed is an abelian scheme.
We can now prove the final claim. By a theorem of Artin (see Stacks Project, Tag 04S6), since is an fppf -scheme, the fppf quotient sheaf is an algebraic space. There is an -homomorphism given by the degree map (see Bosch-Lütkebohmert-Raynaud, Neron Models, 9.1, Prop. 2). To check that it is an isomorphism, we may pass to an fppf cover of to assume that . If , then there is a section given by sending the integer to the class of the divisor , and on fibers this is an isomorphism. By the fibral isomorphism criterion (valid for morphisms from a scheme to an algebraic space by the argument at the end of the proof of Lemma 1.3 in the previous post) and the above-cited fact concerning the component group of the fibers of , it follows that this map is an isomorphism and thus indeed . This completes the proof.
Remark. The projectivity hypothesis in the proposition is not necessary and can be weakened to properness, but more care is needed in that case because one can no longer appeal to Grothendieck’s representability theorem for the Picard scheme. One way to prove it is to instead use Artin’s representability theorem for the Picard functor as an algebraic space (see Artin, Algebraization of Formal Moduli: I, Theorem 7.3) and a result due to Raynaud which states that any proper smooth group algebraic space with geometrically connected fibers is an abelian scheme (see Faltings-Chai, Degenerations of Abelian Varieties, Chap. I, Thm. 1.9). Since is representable by open embeddings (see the above reference to SGA6), we need only show smoothness and properness. For this we may pass to an etale cover of (see Neron Models, 9.3, Rem. 2) to reduce to the case that is projective and argue as above.