Finite flat commutative group schemes embed locally into abelian schemes
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.
A related question I am curious about.
Let
be a finitely presented flat commutative group scheme over
for a (strictly henselian) local (noetherian) ring
. Is there a (quasi-compact) smooth commutative group
-scheme
with a monomorphism
?
I don’t know an answer even if
is an algebraically closed field.
Over a field (of positive characteristic), you could consider the
Frobenius kernel
of
. By the post above, you can embed the finite commutative
into an Abelian variety
. Then
embeds into the pushout
in the category of commutative algebraic groups (
acts diagonally by right and inverse left multiplication). Moreover, the pushout
is an extension of
by
. If
is big enough, then the quotient
is smooth, and so the extension of smooth algebraic groups
is also a smooth algebraic group by descent.