On the arithmetic rank of certain Segre products
Abstract
We compute the arithmetic ranks of the defining ideals of homogeneous coordinate rings of certain Segre products arising from elliptic curves. The cohomological dimension of these ideals varies with the characteristic of the field, though the arithmetic rank does not. We also study the related set-theoretic Cohen-Macaulay property for these ideals.
Primary 13C40, 14M10; Secondary 13D45, 14B15
In [12] Lyubeznik writes: Part of what makes the problem about the number of defining equations so interesting is that it can be very easily stated, yet a solution, in those rare cases when it is known, usually is highly nontrivial and involves a fascinating interplay of Algebra and Geometry.
In this note we present one of these rare cases where a solution is obtained: for a smooth elliptic curve , we determine the arithmetic rank of the ideal defining the Segre embedding , and exhibit a natural generating set for up to radical. The ideal is not a set-theoretic complete intersection and, in the case of characteristic zero, we use reduction modulo methods to prove moreover that is not set-theoretically Cohen-Macaulay.
1 The Segre embedding of
Let and be -graded rings over a field . The Segre product of and is the ring
which is a subring, in fact a direct summand, of the tensor product . The ring has a natural -grading in which . If and are projective varieties with homogeneous coordinate rings and respectively, then the Segre product is a homogeneous coordinate ring for the Segre embedding .
Let be a smooth elliptic curve over a field . Then can be embedded in , and so where is a homogeneous cubic polynomial. The Segre product has an embedding in with homogeneous coordinate ring
i.e., is the subring of generated by the six monomials for and . The relations amongst these generators arise from the equations
defining , and the four equations
The ring is normal since it is a direct summand of a normal ring. The local cohomology of with support at its homogeneous maximal ideal may be computed by the Künneth formula, which shows that is isomorphic to
In particular, is a normal domain of dimension which is not Cohen-Macaulay.
Let be a polynomial ring. Then has a -algebra surjection onto where
We use to denote the partial derivative of with respect to . The Euler identity implies that
and multiplying by and we obtain, respectively, the equations
and
Consequently if is a field of characteristic other than , then the kernel of the surjection is the ideal generated by the seven polynomials , , , and
We prove that the ideal has arithmetic rank four, and that the last four polynomials above generate up to radical:
Theorem 1.1.
Let be a smooth elliptic curve and let be the defining ideal of the homogeneous coordinate ring of the Segre embedding . If the characteristic of the field does not equal , then the arithmetic rank of the ideal is and
Proof.
Let be a root of the four polynomials above which, we claim, generate up to radical. Then
,
,
,
.
We claim that the size two minors of the matrix
must be zero. If not, then and are distinct points of . The first and fourth equations imply that the points and lie on the elliptic curve . The second equation implies that lies on the tangent line to at point , and similarly the third equation implies that lies on the tangent line to at . But then the secant line joining and meets the elliptic curve with multiplicity , which is not possible. Alternatively, consider the group law on using an inflection point as the identity element of the group. Then the second and third equations imply that in the group law, and so and are the same point of . This proves the claim, and it follows that is a zero of all polynomials in the ideal . By Hilbert’s Nullstellensatz, is generated by the four polynomials up to radical once we tensor with , the algebraic closure of . Since is faithfully flat over , the same is true over as well and, in particular, .
Since the ideal has height , the proof of the theorem will be complete once we show that . In the next section we use étale cohomology to prove, more generally, that the defining ideal of any projective embedding of is not a set-theoretic complete intersection, Theorem 2.4. However, in characteristic , we can moreover show that the local cohomology module is nonzero, from which it follows that . (In positive characteristic, if the elliptic curve is supersingular.) We proceed with the characteristic zero case, and our main tool here is the connection between the local cohomology modules supported at , and topological information about the affine variety defined by .
If , then each of the generators of has a power belonging to an ideal . This gives us seven equations, each with finitely many coefficients, and hence we may replace by the extension of obtained be adjoining these finitely many coefficients. A finitely generated field extension of can be identified with a subfield of , and so it suffices to prove the desired result in the case .
Let be the complement of in and let be the cone over . The de Rham cohomology of can be computed from the Čech-de Rham complex [1, Chapter II] corresponding to any affine cover of . The de Rham functor on an -dimensional affine smooth variety, when applied to any module, can only produce cohomology up to degree (in our case, ). It follows that the de Rham cohomology of will be zero beyond the sum of and the index of the highest nonvanishing local cohomology module .
On the other hand, there is a Leray spectral sequence, [1, Thm. 14.18],
corresponding to the fibration with fiber . Since arises through the removal of a variety of codimension two from the simply connected space , it follows that is simply connected as well. Hence in the above spectral sequence the local coefficients are in fact constant coefficients. The nonzero terms in the -page of the spectral sequence have . Hence if denotes the index of the top nonzero de Rham cohomology group of , then is the index of the top nonzero de Rham cohomology group of and .
We now claim that in nonzero. Note that , so the Künneth formula gives . Since sheaf and de Rham cohomology agree, using Alexander duality [10, V.6.6] and the compactness of , we obtain . There is a long exact sequence of sheaf cohomology
which, since , implies that . It follows that is nonzero as well, and so the local cohomology module must be nonzero. ∎
2 Elliptic curves in positive characteristic
Let be a ring of prime characteristic. We say that is -pure if the Frobenius homomorphism is pure, i.e., if is injective for all -modules . By [8, Proposition 6.11], a local ring is -pure if and only if the map
is injective where is the injective hull of the residue field .
Let be a smooth elliptic curve over a field of characteristic . The Frobenius induces a map
on the one-dimensional cohomology group . The elliptic curve is supersingular (or has Hasse invariant ) if the map above is zero, and is ordinary (Hasse invariant ) otherwise. If , then the map above is precisely the action of the Frobenius on the socle of the injective hull of the residue field of , and hence is ordinary if and only if is an -pure ring.
Let be a cubic polynomial defining a smooth elliptic curve . Then the Jacobian ideal of is -primary in . Hence after localizing at an appropriate nonzero integer , the Jacobian ideal of in contains high powers of , and . Consequently, for all but finitely many prime integers , the polynomial defines a smooth elliptic curve . If the elliptic curve has complex multiplication, then it is a classical result [3] that the density of the supersingular prime integers , i.e.,
is , and that this density is if does not have complex multiplication. However, even if does not have complex multiplication, the set of supersingular primes is infinite by [5]. It is conjectured that if does not have complex multiplication, then the number of supersingular primes less than grows asymptotically like , where is a positive constant, [11].
Hartshorne and Speiser observed that the cohomological dimension of the defining ideal of varies with the prime , [7, Example 3, p. 75]. Their arguments use the notion of -depth, and we would like to point out how their results also follow from a recent theorem of Lyubeznik:
Theorem 2.1.
[14, Theorem 1.1] Let be a regular local ring containing a field of positive characteristic, and be an ideal of . Then if and only if there exists an integer such that is the zero map, where denotes the -th iteration of the Frobenius morphism.
Corollary 2.2.
Let be a cubic polynomial defining a smooth elliptic curve , and let be the ideal defining the Segre embedding . Then
Proof.
The ring may be identified with the Segre product where
Let be a prime for which is smooth, in which case the ring and hence its direct summand are normal. For the Künneth formula gives us
Hence
and the Frobenius action on the one-dimensional vector space may be identified with the Frobenius
which is the zero map precisely when is supersingular. Consequently every element of is killed by the Frobenius (equivalently, by an iteration of the Frobenius) if and only if is supersingular. The assertion now follows from Theorem 2.1. ∎
Example 2.3.
The cubic polynomial defines a smooth elliptic curve in any characteristic . It is easily seen that is supersingular for primes , and is ordinary if . Let . The defining ideal of is the ideal of generated by
If is a prime integer, then if and only if , and consequently
As the above example shows, the cohomological dimension varies with the characteristic , so we cannot use local cohomology to complete the proof of Theorem 1.1 in arbitrary prime characteristic. We instead use étale cohomology to show that the defining ideal of any projective embedding of cannot be a set-theoretic complete intersection which, in particular, completes the proof of Theorem 1.1.
Theorem 2.4.
Let be a smooth elliptic curve. Then the defining ideal of any projective embedding of is not a set-theoretic complete intersection.
Proof.
Consider an embedding . Then the defining ideal has height . We need to prove that , and for this we may replace the field by its separable closure. Let be a prime integer different from the characteristic of . We shall use the étale cohomology groups , i.e., with coefficients in .
If , then the complement of in can be covered by the affine open sets for . Each is an affine smooth variety of dimension , and so for all by [16, Thm. VI.7.2]. The Mayer-Vietoris principle [16, III.2.24] now implies that . We shall show that this leads to a contradiction.
Since is nonzero, the Künneth formula [16, VI.8.13] implies that is nonzero as well. Since is proper, the first compactly supported étale cohomology of is , [16, III.1.29]. By [13, (1.4a)] and [16, Cor. VI.11.2] there is a natural isomorphism between and the dual of , so it follows that is nonzero. By [16, III.1.25] we have an exact sequence
But by [16, VI.5.6], which gives a contradiction. ∎
3 The set-theoretically Cohen-Macaulay property
Given an affine variety , it is an interesting question whether supports a Cohen-Macaulay scheme, i.e., whether there exists a Cohen-Macaulay ring such that is isomorphic to . More generally, let be a regular local ring. We say that an ideal is set-theoretically Cohen-Macaulay if there exists an ideal with for which the ring is Cohen-Macaulay. A homogeneous ideal of a polynomial ring is set-theoretically Cohen-Macaulay if is a set-theoretically Cohen-Macaulay ideal of , where is the homogeneous maximal ideal of .
There is a well-known example of a determinantal ideal which is not a set-theoretic complete intersection, but is Cohen-Macaulay (and hence set-theoretically Cohen-Macaulay). For an integer , let be an matrix of variables over a field , and let be the localization of the polynomial ring at its homogeneous maximal ideal. Let be the ideal of generated by the minors of the matrix . If has characteristic zero, then is nonzero by an argument due to Hochster, [9, Remark 3.13], so . If has positive characteristic it turns out that , but nevertheless the ideal has arithmetic rank , [17, 2]. In particular, is not a set-theoretic complete intersection though is Cohen-Macaulay.
We next show that for a smooth elliptic curve , the defining ideal of is not set-theoretically Cohen-Macaulay. We begin with a lemma of Huneke, [4, page 599]. We include a proof here for the convenience of the reader.
Lemma 3.1 (Huneke).
Let be an ideal of a regular local ring of characteristic . If the ring is -pure and not Cohen-Macaulay, then the ideal is not set-theoretically Cohen-Macaulay.
Proof.
Note that is a radical ideal since is -pure. Let be an ideal of with , and choose such that their images form a system of parameters for and . Since is is not Cohen-Macaulay, there exist and such that
Let be a prime power such that . Then
If is Cohen-Macaulay, then
The hypothesis that is -pure implies that , which is a contradiction. ∎
For the remainder of this section, will denote a polynomial ring over the integers, and we use the notation and .
Lemma 3.2.
Let be an ideal of , and consider the multiplicative set . If is Cohen-Macaulay, then the rings are Cohen-Macaulay for all but finitely many prime integers .
Proof.
Let be elements whose images form a system of parameters for . Since this ring is Cohen-Macaulay, there exists an element in the multiplicative set such that is a regular sequence on where . Moreover, we may choose in such a way that
These conditions are preserved if we enlarge the ring by inverting finitely many nonzero integers. By the result on generic freeness, [15, Theorem 24.1], we may assume (after replacing by a nonzero integer multiple and by its localization at the element ) that , , and each of
are free -modules. In particular, for all , we have short exact sequences of free -modules,
Let be any prime integer not dividing , and apply to the sequences above. The resulting exact sequences show that is a regular sequence on
and hence on as required. ∎
Theorem 3.3.
Let be a smooth elliptic curve. Then the defining ideal of the Segre embedding is not set-theoretically Cohen-Macaulay.
Proof.
Let be an ideal such that is the defining ideal of . There exist infinitely many prime integers such that is a smooth ordinary elliptic curve. For these infinitely many primes, the ring is -pure and not Cohen-Macaulay, and hence the ideal is not set-theoretically Cohen-Macaulay by Lemma 3.1. It now follows from Lemma 3.2 that the ideal is not set-theoretically Cohen-Macaulay. ∎
Remark 3.4.
If is an ordinary elliptic curve, then the defining ideal of the Segre embedding is not set-theoretically Cohen-Macaulay by Lemma 3.1, since the corresponding homogeneous coordinate ring is -pure and not Cohen-Macaulay. If is supersingular, we do not know whether the defining ideal is set-theoretically Cohen-Macaulay.
Remark 3.5.
Lemma 3.1 can be strengthened by Lyubeznik’s theorem as follows: Let be a regular local ring of positive characteristic, and and be ideals of such that is -pure and . If the local cohomology module is nonzero for some integer , then Theorem 2.1 implies that
is nonzero as well. Using Theorem 2.1 once again, it follows that is nonzero. This implies in particular that
Consequently if is an ideal of a regular local ring of positive characteristic such that is -pure, then
This is false without the assumption that is -pure: let where is a field of positive characteristic, and consider the ideal
Then which is not Cohen-Macaulay. Hartshorne proved that is a set-theoretic complete intersection, i.e., that for elements , [6]. Hence, in this example,
Acknowledgments
It is a pleasure to thank Gennady Lyubeznik for several helpful discussions.
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