The torsion component of the Picard scheme
This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there.
Main Theorem. Let be a noetherian local ring and let
be a finite flat commutative group scheme over
. There exists a smooth projective scheme
over
with geometrically connected
-dimensional fibers such that
. If the Cartier dual
is etale (i.e.,
is of multiplicative type), then we may take
to have
-dimensional fibers instead. Moreover,
can be taken to be the quotient of a complete intersection under the free action of
.
A major motivation for the Main Theorem is that it can be used to construct examples of smooth projective schemes over a DVR of equicharacteristic such that the Hodge numbers of the special and generic fibers are not equal; for this phenomenon, see the final section. This answers a question asked in the Stanford Number Theory Learning Seminar in a way not easily findable in the literature.
The basic construction of as in the Main Theorem will be very similar to the one given in Cotner’s previous post, but there are a few major simplifications owing to the use of descent techniques, as well as some technical difficulties to overcome coming from the new relative setting. For example, there are some small intermediate arguments with algebraic spaces owing to the fact that certain Picard functors are not obviously representable (though all the relevant ones for us are a posteriori representable). We also remove all mention of Igusa’s theorem from the previous post. Our basic strategy in the general case is as follows. (See the rest of the post for notation and definitions.)
- Show that any complete intersection of dimension
has trivial
.
- Show that if
is a
-torsor, then the pullback map
has kernel
, the Cartier dual of
.
- Construct a projective space
over
on which
acts, freely outside of a codimension
closed subset.
- Use Bertini’s theorem to slice the quotient
by hypersurfaces to obtain a smooth projective scheme
of dimension
so that the pullback
of
in
is a complete intersection on which
acts freely.
- Conclude that
is a
-torsor and thus
by points 1 and 2.
1. Complete intersections and Picard schemes
Definition. Let be a scheme and let
be a closed subscheme of
. We say that
is a complete intersection of dimension
if it is a flat finitely presented
-scheme with fibers of pure dimension
which are complete intersections.
Lemma 1.1. Let be a complete intersection of dimension
over a field
and let
be an integer. We have
for all
,
, and
for all
. In particular,
is geometrically connected. Moreover, if
is smooth and
then
.
Proof. We induct on the codimension of
inside
. If
, then this follows from the familiar computations of the cohomology of projective space. In general, by the definition of a complete intersection there is some complete intersection
and hyperplane
of degree
such that
and moreover there is a short exact sequence
where is the natural inclusion. Tensoring by
and passing to cohomology gives exact sequences
By induction, if then the two outside terms vanish. If
, then we have the exact sequence
If , this shows
. If
, then again
. Since
, it follows from the Stein factorization that
is geometrically connected.
Now suppose that is smooth and
. To compute
, recall first the Euler exact sequence for
-dimensional projective space:
From this, we see that . In general, if a smooth
is the intersection of hypersurfaces whose degrees sum to
, then we look at the conormal exact sequence
for the natural inclusion. Passing to cohomology gives the short exact sequence
and it suffices by the above to show that . This now follows from the short exact sequence
since and
. This completes the cohomology calculations.
Recall the definition of the Picard functor: if is a morphism of schemes then we define
to be the fppf sheafification of the functor sending an
-scheme
to
. The formation of this sheaf commutes with base change on S, i.e.,
for all
-schemes
. In this generality, this functor is essentially useless, but Grothendieck and Artin proved the following remarkable representability theorems, see FGA Explained, Theorem 9.4.8, and Artin, Algebraization of Formal Moduli: I, Theorem 7.3.
Theorem 1.2. (i) (Grothendieck) Suppose that is flat, finitely presented, projective Zariski-locally on
, with geometrically integral fibers. Then
is represented by a separated, locally finitely presented
-scheme. If
, then
naturally for all
-schemes
.
(ii) (Artin) If is flat, proper, finitely presented, and cohomologically flat in dimension
(i.e.,
holds universally), then
is a quasi-separated algebraic space locally of finite presentation over
.
Algebraic spaces only intervene for us in a rather technical way, as one can see in the proof of Lemma 1.3; ultimately everything we discuss will be a scheme, but to apply geometric methods we will need to know a priori that is an algebraic space.
There are two further important notions for us: the identity component and the torsion component
. The identity component is defined as the subfunctor of
whose set of
-points consists of those
-points
of
such that for every
there exists an algebraically closed extension
of
, a connected
-scheme
, points
such that
is an extension of
, and a
-point
of
such that
in
and
. Note that if
is representable then this is the same as the set-theoretic union
, where
is the identity component of the locally finite type
-group scheme
. The torsion component is defined as
, where
denotes the multiplication by
map; this is also a subgroup functor of
.
For our purposes, will not be a very useful functor: in general, when
has non-reduced geometric fibers,
need not be an open subscheme of
, though it is on fibers. (We will see examples of this phenomenon later.) However, the morphism of functors
is always representable by open immersions when
is proper over
; if
is projective over
then this morphism is also representable by closed immersions. For both of these assertions, see SGA6, Exp. XIII, Thm. 4.7.
Lemma 1.3. Let be a complete intersection of dimension
over a scheme
. If
is smooth or
then
. (In particular,
is representable.)
Proof. First suppose for an algebraically closed field
. By Theorem 1.2(ii) and Lemma 1.1,
is a quasi-separated algebraic space locally of finite type over
. By Lemma 4.2 of Artin’s paper cited above, it follows in this case that in fact
is a
-group scheme. (Note that quasi-separatedness is part of the definition of an algebraic space in Artin’s paper.) As
, we see that
by Lemma 1.1. So it suffices to show that
has no torsion. If
, then a Lefschetz theorem (see SGA2, Exp. XII, Cor. 3.7, and note that no smoothness hypotheses on
are necessary) states that
, so indeed
is torsion-free in this case. Now suppose
is smooth. Recall that
for all
since
is algebraically closed. If
does not divide
then
, so we have
By a Lefschetz theorem (see SGA2, Exp. XII, Cor. 3.5, and note again the lack of smoothness hypotheses), we have , so that this Hom set is trivial. To prove that there is no
-torsion, note that there is an exact sequence of Zariski sheaves
where the first map is given by and the second map is given by
. The
th power map is clearly injective since
is reduced, and the composition of the two maps is evidently
. Exactness in the middle is more involved: the idea is to use normality and local freeness of
to reduce to proving that for a finitely generated field extension
and
,
implies that
is a
th power. After checking this for purely transcendental extensions, one uses the existence of a separating transcendence basis to deduce the general case. Since the complete argument is rather long, we omit it.
Now let be the image of the map
in the exact sequence above, so there is a corresponding short exact sequence for
and we obtain, passing to cohomology, an exact sequence
where the right map is multiplication by . Since
by Lemma 1.1, it follows that
, and indeed
.
Now we work in the case of general . Using Lemma 1.1, a simple argument using cohomology and base change shows that
is cohomologically flat in dimension
, so
is an algebraic space by Theorem 1.2(ii) and
is also an algebraic space since
is representable by open immersions. Since formation of
commutes with arbitrary base change, we see that
for all
by the first paragraph. Thus the identity section
is an isomorphism on fibers, and since
is trivially flat over itself, the fibral isomorphism theorem (see EGA IV, Cor. 17.9.5) implies that
. (Note that the fibral isomorphism criterion holds for a morphism from a scheme to an algebraic space: morphisms may be checked to be isomorphisms after etale base change, so this follows immediately from the fact that algebraic spaces admit etale covers which are schemes and the relative diagonal of an algebraic space is representable.) This completes the proof.
Question. The above proof shows that all complete intersections in dimension (smooth or not) have trivial
and no
-torsion for any prime
. Do there exist (non-smooth) complete intersections
of dimension
in characteristic
such that
?
Lemma 1.4. Let be a commutative finite locally free group scheme over a scheme
and let
be an fppf
-torsor, where
is cohomologically flat in dimension
and
satisfies the hypotheses of the Theorem. The pullback map
has kernel isomorphic to
, the Cartier dual of
. The same is therefore true of
.
Proof. We work with the presheaf defined by
for all
-schemes
. There is a pullback map
, and we claim that it has kernel
. After fppf sheafification, this easily gives the result. To prove this claim, it suffices by base change on
to show that the kernel of the map
can be canonically identified with the group
, and we will do this below.
Recall that fppf descent for quasicoherent sheaves says that a line bundle on is equivalent to the data of a line bundle
on
along with an isomorphism
, where
are the two canonical projections, satisfying the cocycle condition
, where
is the projection onto the
and
coordinates. Given a line bundle
on
, we obtain a datum of this form (a descent datum) via taking
and using the canonical isomorphism of functors
. So a line bundle on
which becomes trivial after pullback to
is the same as the datum of an isomorphism
satisfying the cocycle condition. Since
canonically, this is the same as an automorphism of
, i.e., an element of
, satisfying the cocycle condition. We will show below that the cocycle condition can be described more concretely in terms of
.
Since is a
-torsor, there is a canonical isomorphism
given functorially by
. There is also an identification
given functorially by
. Under these identifications, the maps
,
, and
are identified with maps
via
Suppose that is such that the cocycle condition holds. Functorially, this means
for all and
, where
ranges over the category of
-schemes. Since
is cohomologically flat in dimension
, we have
naturally via pullback of units, so
may be regarded as a morphism
, and the cocycle condition is precisely saying that
is a group homomorphism, i.e.,
. This completes the proof.
2. Group actions
Let be a scheme,
a finite locally free
-group scheme, and
a separated
-scheme. An action of
on
is an
-morphism
such that
is a group action for every
-scheme
. (As usual, this is equivalent to the commutativity of various diagrams like those in the ordinary definition of a group action.)
Definition. If is an
-scheme and
, then the stabilizer of
in
is the functor
sending a
-scheme
to
. The free locus of the action is the functor
sending an
-scheme
to
.
Lemma 2.1. If acts on
and
for some
-scheme
, then
is representable by a closed
-subgroup scheme of
, and for any
-scheme
we have
. The functor
is represented by an open subscheme of
, and
for every
-scheme
.
Proof. The claims about base change are simple from the functorial definition, and will be omitted. For the first representability claim, note that there is a Cartesian diagram
where denotes the action map
. Since
is separated,
is a closed embedding, and it follows by base change that
is a closed
-subgroup of
, hence in particular a finite
-scheme. Now if
is any
-morphism, we see that there is a Cartesian diagram
so that is a finite
-group scheme. The morphism
is a proper monomorphism, hence a closed embedding, and we obtain the claim.
Recall that if is a finite morphism, then the function
is an upper semicontinuous function: this follows directly from Nakayama’s lemma. If
is moreover an
-group scheme then
for all
(since there exists a section
), and
is trivial if and only if the fiber
is trivial for all
(apply the fibral isomorphism criterion to the identity section). It follows from these considerations applied to
that if
then
is an open subscheme of
and
represents
. This proves the lemma.
We will also need the following Theorem proved in SGA3, Exp. V, Thm. 4.1 and Rem. 5.1. With notation as in the first paragraph of this section, recall that a morphism of -schemes
is a
-torsor if
acts on
by
-automorphisms and the natural map
given by
is an isomorphism.
Theorem 2.2. If is a quasi-projective
-scheme and
is a finite locally free
-group scheme acting on
then the ringed space quotient
exists as a quasi-projective
-scheme, and the natural morphism
is finite and open. If
acts freely on
, then
is a
-torsor and
represents the fppf sheaf quotient for the equivalence relation on
defined by the action of
. In particular, in this latter case the formation of the quotient
commutes with all base change on
.
The relevance of quasi-projectivity in this statement is that it implies that any finite collection of points lying over an affine open of
all lie in an affine open of
. This permits one, using some nontrivial formal arguments, to reduce to the case that
and
are affine, in which case one proves that
can be formed as the spectrum of the ring of invariants for the action of
on
. If
is an open
-stable subscheme then
exists as an open subscheme of
. If
is an open subscheme then also
naturally.
If acts freely on
then many properties of
follow from properties of
. For example, if
is flat, then
is flat: in general, elementary commutative algebra arguments show that it is true that flatness may be checked on any faithfully flat cover. Moreover, if
is smooth then
is also smooth: this follows from the fact that “smooth = locally finite presentation + flat with geometrically regular fibers”, reduction to the Noetherian setting, and Matsumura, Commutative Ring Theory, Thm. 23.7. It is simple to see that properness of
is equivalent to that of
, so that
is projective whenever
is projective.
3. Construction
Let be a noetherian local ring and let
be a finite flat commutative group scheme of rank
over
, as in the Main Theorem. For any integer
there is a natural action of
on the projective space
. The following lemma is established in Raynaud’s paper
-torsion du schéma de Picard, paragraph preceding Lemme 4.2.2.
Lemma 3.1. The free locus is an open subset of
with complementary codimension
on fibers.
Proof. We offer a very brief sketch of the idea and refer to loc. cit. for a detailed argument. First, pass to geometric fibers to assume that S is the spectrum of an algebraically closed field, say of characteristic . If
has nontrivial stabilizer, then it contains a subgroup isomorphic to one of
,
, or
for some prime number
. One first shows that the locus fixed by any of these subgroups in
has large codimension. In general,
contains only finitely many subgroups of the form
or
for
, but it may contain infinitely many subgroups of the form
and
. However, such subgroups are determined by their Lie algebras, so the collection of subgroups of each of these forms is parameterized by some projective space. One shows then that this projective space is of relatively small dimension to obtain the lemma.
By Theorem 2.2, the ringed space quotient exists as a projective
-scheme, so because
is local it is an
-closed subscheme of some projective space
. By the discussion directly following Theorem 2.2,
is a smooth open subscheme of
; its closed complement
has fibers of codimension
because dimension is insensitive to finite surjective maps. By Bertini’s theorem [see Poonen, Bertini theorems over finite fields, or Gabber, On space filling curves and Albanese varieties for a proof over finite fields], if
then there exists
hypersurfaces
in the special fiber
such that
is a smooth integral
-dimensional closed subscheme of
disjoint from the image of
. Lift each
to a hypersurface
in
and let
. Let
be the (schematic) preimage of
in
.
Lemma 3.2. The scheme is
-smooth with equidimensional fibers of dimension
.
Proof. We prove first that is
-flat. Note that the special fiber
is integral of dimension
, and Chevalley’s theorem on upper-semicontinuity of fiber dimension (see EGA IV, 13.1.3 for a proof for general
) implies that in fact every fiber of
over
has dimension at most
. But every fiber of
over
is integral of dimension
, and intersection with a hyperplane can cut the dimension down by at most
; this follows from Krull’s height theorem. Since every zerodivisor in a noetherian local ring
is contained in a minimal prime ideal, if
is equidimensional and
is a nonunit then
is a nonzerodivisor if and only if
. By induction on
, using the above considerations and the fact that
has integral fibers, we see that
has equidimensional fibers of dimension
. If each
is defined locally on
by some function
, it follows that
is a regular sequence since this may be checked on fibers over
. So if
is an affine open on which
is defined by
, there is an exact sequence
, where the first map is given by multiplication by
. This map is injective after base change to any residue field of
(as we noted above), so that
is flat over
: we see that
for all
, so flatness follows from Stacks Project, 00M5 and a Zorn’s lemma-style argument. These same considerations show by induction on
that
is
-flat.
Now recall that the -smooth locus in
is open, and
is
-smooth at all points of
since
is
-smooth at a point
if and only if it is flat over
at
and
is smooth in its fiber. Since
is proper, it follows immediately that
is smooth over
, completing the proof of the Lemma.
Now we take in Lemmas 3.1 and 3.2. Note that
is cut out of
by the preimages of the
in
, and it is easy to see from Lemma 3.2 that
is a complete intersection of dimension
. So if
then
is smooth and projective over
with
-dimensional geometrically integral fibers and
is a complete intersection of dimension
. By Lemma 1.3 we see that
, and it follows from Lemma 1.4 that
. If
is etale and
, then
is smooth, being a
-torsor over the smooth
, so the same argument shows again that
. This completes the proof of the Main Theorem.
Question. If is not of multiplicative type, then does there necessarily exist a
-dimensional smooth projective
with
? (This is related to the previous question: if the answer to this question is “no”, then the construction above will yield a
-dimensional complete intersection with nontrivial
-torsion in its Picard group.) What is certainly true is that the above argument can be modified in a simple way to show that if
is connected over a field
then there does exist a
-dimensional smooth projective
with
.
4. Jumping Hodge numbers
We now give two examples of “pathologies” we can deduce from the Main Theorem. Recall that if are integers then the
Hodge number of a proper scheme
over a field
is defined by
. Classically (i.e., over the complex numbers), the Hodge numbers satisfy various magical properties: for example, if
is smooth and projective then one always has
. As mentioned in Cotner’s previous post, taking
for a field
of characteristic
and
, we find an example of a variety in characteristic
which does not satisfy Hodge symmetry: namely,
and
. Another magical property of Hodge numbers over
(or, more generally, over a field of characteristic
) is that they are constant in smooth projective families; this is proved via analytic methods. In the following two examples, we will see that this fails away from equicharacteristic
.
Example. Jumping Hodge numbers in mixed characteristic .
Let
be a DVR of mixed characteristic
and
, so that the generic fiber
is etale and the special fiber
is connected. Let
be as in the Main Theorem, so
is smooth and projective over
, and
. Recall that for any scheme
over a field
such that
is representable, we have
. Thus we have
and
, i.e., “the Hodge number jumps”. Notice also that
has special fiber
and generic fiber
, so that this is not open in
.
Example. Jumping Hodge numbers in equicharacteristic Let
be a DVR of equicharacteristic
and let
be a totally ramified generically separable extension of degree
. Let
denote the Weil restriction
, so that
has special fiber
and generic fiber
. Let
denote the kernel of Frobenius on
, so that
is a finite flat commutative group scheme over
with special fiber
and generic fiber
. So if
is the Cartier dual of
then we have
and
Let
be the smooth projective
-scheme whose existence guaranteed by the Main Theorem, so that
. So as in the previous section we have
and
.
Insane post.