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.
Let be a noetherian local ring and let be a finite flat commutative group scheme of rank over , as in the Main Theorem. There is a natural action of on the projective space and we claim that is an open subset of with complementary codimension on fibers. By Lemma 2.1, it is enough to show this when is the spectrum of an algebraically closed field. For a field , we have
naturally, and a point has nonzero stabilizer if and only if, lifting to some -morphism , there is a -homomorphism such that we have
for all -schemes and all . Note that satisfies this transformation property under left -translation if and only if the -morphism given functorially by is left -invariant. A -morphism is left -invariant if and only if it factors through the map defined by , as is easily seen by functorial reasoning.
For every character , there is a closed embedding induced by the map sending to , and the considerations in the previous paragraph show that
Note that the set is finite, being the set of -points of the finite -group . Thus is a closed subset of , and since is algebraically closed it follows that the above equality on -points is actually an equality of open subschemes, i.e.,
Since has dimension and has dimension , it follows that the closed complement of has codimension , as desired. Note that, in particular, this codimension is always at least .
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 3.1. 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.
Note that is cut out of by the preimages of the in , and it is easy to see from the lemma 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 .