Finite flat commutative group schemes embed locally into abelian schemes

Let \(G\) be a finite flat commutative group scheme over a fixed locally noetherian base scheme \(S\). In this brief note, I want to explain the proof of the following theorem due to Raynaud.

Theorem. There exists, Zariski-locally on \(S\), an abelian scheme \(A\) such that \(G\) embeds as a closed \(S\)-subgroup of \(A\).

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 \(S\) is a noetherian local ring. As in the previous post (see Section 3 there), there exists some projective space \(\bP_S^N\) equipped with an action of the Cartier dual \(G^{\vee}\) of \(G\), free away from a closed subscheme \(Z\) of codimension \(\geq 2\). The quotient \(Q = \bP^N/G^{\vee}\) in the category of locally ringed spaces is again a projective \(S\)-scheme, and it is smooth away from the image of \(Z\) in \(Q\). By a simple argument with Bertini’s theorem (plus some technical arguments in commutative algebra), there exists a smooth closed subscheme \(C\) of \(Q\) disjoint from \(Z\) with geometrically connected \(1\)-dimensional fibers over \(S\). A descent argument (see Lemma 1.4 of the previous post) shows that \(G\) is a closed \(S\)-subgroup of the Picard scheme \(\Pic_{C/S}\). Admit for a moment the following proposition.

Proposition. Let \(C\) be a smooth projective \(S\)-scheme with geometrically connected \(1\)-dimensional fibers. The torsion component of the Picard scheme \(\Pic_{C/S}^\tau\) is an abelian scheme over \(S\) (hence equal to \(\Pic_{C/S}^0\)) and the fppf quotient sheaf \(\Pic_{C/S}/\Pic_{C/S}^0\) is represented by the constant \(S\)-group scheme \(\bZ\).

Using this proposition, the theorem is fairly simple. Indeed, we may check that the map \(G \to \Pic_{C/S}\) factors through \(\Pic_{C/S}^0\) after fppf base change on \(S\), so we may assume that \(C(S) \neq \emptyset\). In this case, \(\Pic_{C/S}/\Pic_{C/S}^0\) is \(S\)-isomorphic to \(\bZ\), and it follows that the induced map \(G \to \Pic_{C/S}/\Pic_{C/S}^0\) is trivial: on fibers it is certainly trivial because \(G\) is finite (hence torsion by a theorem of Deligne) and \(\bZ\) is torsion-free. Since the identity section of \(\bZ\) is an open embedding, it follows that \(G \to \Pic_{C/S}/\Pic_{C/S}^0\) factors through the identity section, as desired. So the map \(G \to \Pic_{C/S}\) factors through the identity component \(\Pic_{C/S}^0\), which is an abelian scheme. It only remains to prove the proposition.

Proof of Proposition. First, we prove that \(\Pic_{C/S}\) is a smooth \(S\)-scheme using the infinitesimal lifting criterion. Let \(R\) be an Artin local ring and let \(I \subset R\) be a square zero ideal. We want to show that the natural map \(\pic(C_R) \to \pic(C_{R/I})\) is surjective. By base change, we may assume \(S = \Spec R\). Since \(C\) is smooth over \(S\) (and in particular flat), there is a short exact sequence of Zariski sheaves

\[ 0 \to \cO_C \otimes_R I \to \cO_C^{\times} \to \cO_{\overline{C}}^{\times} \to 0, \]

where \(\overline{C} = C \times_{\Spec R} \Spec R/I\). Passing to Cech cohomology gives

\[ H^1(C, \cO_C^{\times}) \to H^1(C, \cO_{\overline{C}}^{\times}) \to H^2(C, \cO_C \otimes_R I). \]

Since \(S\) is an Artin local scheme, \(C\) is a \(1\)-dimensional noetherian scheme. Hence by dimensional vanishing of quasicoherent cohomology for such schemes we find that \(H^2(C, \cO_C \otimes_R I) = 0\), and thus the map \(H^1(C, \cO_C^{\times}) \to H^1(C, \cO_{\overline{C}}^{\times})\) is surjective. Via the identifications \(H^1(C, \cO_C^{\times}) = \pic(C)\) and \(H^1(C, \cO_{\overline{C}}^{\times}) = \pic(\overline{C})\), we see that \(\Pic_{C/S}\) is smooth.

Next, we show that \(\Pic_{C/S}\) satisfies the valuative criterion of properness. Let \(R\) be a discrete valuation ring with fraction field \(K\) and residue field \(k\), and let \(\Spec R \to S\) be a morphism. We need to show that the map \(\eta: \pic(C_R) \to \pic(C_K)\) is bijective. For this, we may assume \(S = \Spec R\). Since \(C\) is smooth and \(R\) is regular, \(\Cl(C) = \Div(C)\) and \(\Cl(C_K) = \Div(C_K)\). We first show that \(\eta\) is surjective. Let \(D\) be a Weil divisor on \(C_K\) and let \(\overline{D}\) denote the closure of \(D\) in \(C\). This is a Weil divisor on \(C\) with \(\overline{D}|_{C_K} = D\), so that \(\eta\) is surjective.

To show that \(\eta\) is injective, let \(D\) be a Weil divisor on \(C\), say \(D = \sum_{i} n_i D_i\) for some irreducible Weil divisors \(D_i\) and \(n_i \in \bZ\). Note that \(C_k\) is the unique irreducible Weil divisor on \(C\) which is not \(R\)-flat, and \(C_k\) is principal. Thus we may assume that \(D_i\) is \(R\)-flat for each \(i\), in which case each \(D_i\) is the closure of its generic fiber \((D_i)_K\) in \(C\). Suppose \(D_K = \mathrm{div}_{C_K}(f)\) for some rational function \(f\) on \(C_K\). If \(\pi\) is a uniformizer of \(R\) and \(f\) has \(\pi\)-adic valuation \(0\) (as we may assume), then \(D = \mathrm{div}_C(f)\) for \(f\) considered as a rational function on \(C\), and it follows that \(\eta\) is injective. Thus \(\Pic_{C/S}\) does indeed satisfy the valuative criterion of properness. (This argument did not use the fact that \(C \to S\) has \(1\)-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), \(\Pic_{C/S}^\tau\) is an open and closed subscheme of \(\Pic_{C/S}\) of finite type over \(S\), and hence it is a smooth proper \(S\)-scheme by the above. To show that \(\Pic_{C/S}^{\tau}\) is an abelian scheme, it suffices to show that it has connected fibers, and by base change we may assume that \(S = \Spec k\) for a field \(k\). We now appeal to the standard (though rather non-trivial) fact that if \(C\) is a smooth, proper, geometrically connected curve over an algebraically closed field \(k\), then the component group \(\Pic_{C/k}/\Pic_{C/k}^0\) is isomorphic to \(\bZ\) via the degree map. In particular, \(\Pic_{C/k}^0 = \Pic_{C/k}^{\tau}\), i.e., \(\Pic_{C/k}^{\tau}\) is connected. So indeed \(\Pic_{C/S}^{\tau} = \Pic_{C/S}^0\) is an abelian scheme.

We can now prove the final claim. By a theorem of Artin (see Stacks Project, Tag 04S6), since \(\Pic_{C/S}^0\) is an fppf \(S\)-scheme, the fppf quotient sheaf \(F = \Pic_{C/S}/\Pic_{C/S}^0\) is an algebraic space. There is an \(S\)-homomorphism \(\Pic_{C/S}/\Pic_{C/S}^0 \to \bZ\) 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 \(S\) to assume that \(C(S) \neq \emptyset\). If \(x \in C(S)\), then there is a section \(\bZ \to \Pic_{C/S}/\Pic_{C/S}^0\) given by sending the integer \(n\) to the class of the divisor \(n[x]\), 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 \(\Pic_{C/S}\), it follows that this map is an isomorphism and thus indeed \(\Pic_{C/S}/\Pic_{C/S}^0 \cong \bZ\). 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 \(\Pic_{C/S}^{\tau} \to \Pic_{C/S}\) 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 \(S\) (see Neron Models, 9.3, Rem. 2) to reduce to the case that \(C \to S\) is projective and argue as above.

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Bogdan Zavyalov
2 years ago

A related question I am curious about.

Let G be a finitely presented flat commutative group scheme over S=\mathrm{Spec} A for a (strictly henselian) local (noetherian) ring A. Is there a (quasi-compact) smooth commutative group S-scheme H with a monomorphism G \to H?

I don’t know an answer even if A is an algebraically closed field.

Last edited 2 years ago by Bogdan Zavyalov
Andres Fernandez Herrero
Andres Fernandez Herrero
2 years ago

Over a field (of positive characteristic), you could consider the r^{th} Frobenius kernel K_r of G. By the post above, you can embed the finite commutative K_r into an Abelian variety A. Then G embeds into the pushout (G \times A)/K_r in the category of commutative algebraic groups (K_r acts diagonally by right and inverse left multiplication). Moreover, the pushout (G \times A)/K_r is an extension of G/K_r by A. If r is big enough, then the quotient G/K_r is smooth, and so the extension of smooth algebraic groups (G\times A)/K_r is also a smooth algebraic group by descent.