p-torsion elements in local cohomology modules. II

n
all relations
i=0
Fi Gi = 0 with F1 , . . . , Fn ∈ R.
Remark 1.2. The above conjecture is easily established if n = 2 since in this case we have F1 G1 + F2 G2 = 0 and so (F1 G1 )q + (F2 G2 )q (F1 G1 )q + (−F1 G1 )q = = p p
Consider the ideal a = (G1 , . . . , Gn )R and the local cohomology module R n Ha (R) = lim k − → (G , . . . , Gk n )R
1
where the maps in the direct limit system are induced by multiplication by the element G1 · · · Gn . For a prime integer p and prime power q = pe , let λq = (F1 G1 )q + · · · + (Fn Gn )q . p
q +k q +k λq (G1 · · · Gn )k ∈ (G1 , . . . , Gnhe following conjecture: Conjecture 1.1. Let R be a polynomial ring over the integers and let Fi , Gi ∈ R such that F1 G1 + · · · + Fn Gn = 0. Then for every prime integer p and prime power q = pe , there exists k ∈ N such that (F1 G1 )q + · · · + (Fn Gn )q q +k q +k (G1 · · · Gn )k ∈ (G1 , . . . , Gn )R. p We shall say that Conjecture 1.1 holds for G1 , . . . , Gn ∈ R, if it holds for
q which is an element of (Gq 1 , G2 )R.
(F1 G1 )q 0
if p = 2, if p = 2,
p-TORSION ELEMENTS IN LOCAL COHOMOLOGY MODULES II
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Example 1.3. The hypersurface example. Conjecture 1.1 is false if the condition that R is a polynomial ring over the integers is replaced by the weaker condition that the ring R is a hypersurface over the integers. To see this, let R = Z[U, V, W, X, Y, Z ]/(U X + V Y + W Z ) and a denote the ideal (x, y, z )R. (We use lowercase letters here to denote the images of the corresponding variables.) Consider the relation ux + vy + wz = 0 where, in the notation of Conjecture 1.1, F1 = u, F2 = v, F3 = w and G1 = x, G2 = y, G3 = z . By means of a multi-grading argument, it is established in [Si, §4] that (ux)p + (vy )p + (wz )p (xyz )k ∈ / (xp+k , y p+k , z p+k )R for all k ∈ N. p Consequently Conjecture 1.1 does not hold here even for the choice of the prime power q = p. This is, of course, the example from [Si] of a local cohomology module with infinitely many associated prime ideals: more precisely, for every prime integer p, (ux)p + (vy )p + (wz )p + (xp , y p , z p )R p is a nonzero p-torsion element. Conjecture 1.4. We record another formulation of Conjecture 1.1. Let Fi , Gi be elements of a polynomial ring R over the integers such that F1 G1 + · · · + Fn Gn = 0. Let a denote the ideal (G1 , . . . , Gn ) of R. For an arbitrary prime integer p and prime power q = pe , we have (F1 G1 )q + · · · + (Fn Gn )q ≡ 0 mod p denotes which is a relation on the elements G1 , . . . , Gn ∈ R/pR, where the image of an element of R in the ring R/pR. This relation may be n−1 (R/pR). Conjecture 1.1 is equivalent to the viewed as an element µq ∈ Ha conjecture that this element µq is in the image of the natural homomorphism n−1 (R) → H n−1 (R/pR). Ha a To see the equivalence of these conjectures, suppose
1
It is immediately seen that pηq = 0 and so if ηq is a nonzero element of n (R), then it must be a p-torsion element. Hence if the local cohomology Ha n (R) has only finitely many associated prime ideals then, for all module Ha but finitely many prime integers p, the elements ηq as constructed above must be zero i.e., for some k ∈ N, we have
February 14, 2001 This manuscript is based on work supported in part by the National Science Foundation under Grant No. DMS 0070268.
1
2
ANURAG K. SINGH
Note that has λq has integer coefficients, i.e., that λq ∈ R. Consider R q n ηq = [λq + (Gq lim k . 1 , . . . , Gn )R] ∈ Ha (R) = − → (G , . . . , Gk n )R
arXiv:math/0406355v1 [math.AC] 17 Jun 2004
p-TORSION ELEMENTS IN LOCAL COHOMOLOGY MODULES II
ANURAG K. SINGH
1. Introduction Gennady Lyubeznik conjectured that if R is a regular ring and a is an i (R) have only finitely ideal of R, then the local cohomology modules Ha many associated prime ideals, [Ly1, Remark 3.7 (iii)]. While this conjecture remains open in this generality, several results are now available: if the regular ring R contains a field of prime characteristic p > 0, Huneke and Sharp i (R) is finite. If showed in [HS] that the set of associated prime ideals of Ha R is a regular local ring containing a field of characteristic zero, Lyubeznik i (R) has only finitely many associated prime ideals, see [Ly1] showed that Ha and also [Ly2, Ly3]. Recently Lyubeznik has also proved this result for unramified regular local rings of mixed characteristic, [Ly4]. In [Hu] Craig Huneke first raised the following question: for a Noetherian ring R, an ideal a ⊂ R, and a finitely generated R-module M , is the number i (M ) always finite? For some of the work on of associated primes ideals of Ha this problem, we refer the reader to the papers [BL, BRS, He] in addition to those mentioned above. In [Si] we constructed an example of a hypersurface 3 (R) has p-torsion elements for R for which a local cohomology module Ha every prime integer p, and consequently has infinitely many associated prime ideals. Since this is the only known source of infinitely many associated prime ideals so far, it is worthwhile to investigate whether similar techniques may yield an example of a regular ring R for which a local cohomology i (R) has p-torsion elements for every prime integer p. This leads module Ha to some very intriguing questions as we shall see in this paper. Our results thus far support Lyubeznik’s conjecture that local cohomology modules of all regular rings have only finitely many associated prime ideals. Let R be a polynomial ring over the integers and Fi , Gi be elements of R for which F1 G1 + F2 G2 + · · · + Fn Gn = 0.