Abstract
Let (R, m) be a local Noetherian ring, let I ⊂ R be any ideal and let M be a finitely generated R-module. It has been long conjectured that the local cohomology modules H If R is not required to be local these sets of associated primes may be infinite, as shown by Anurag Singh in [S], where he constructed an example of a local cohomology module of a finitely generated module over a finitely generated Z-algebra with infinitely many associated primes. This local cohomology module has p-torsion for all primes p ∈ Z.
Abstract
However, the question of the finiteness of the set of associated primes of local cohomology modules defined over local rings and over k-algebras (where k is a field) has remained open until now. In this paper I settle this question by constructing a local cohomology module of a local finitely generated k-algebra with an infinite set of associated primes, and I do this for any field k.
Abstract
Let k be any field, let R 0 = k[x, y, s, t] and let S = R 0 [u, v]. Define a grading on S by declaring deg(x) = deg(y) = deg(s) = deg(t) = 0 and deg(u) = deg(v) = 1. Let f = sx 2 v 2 −(t+s)xyuv+ty 2 u 2 and let R = S/f S. Notice that f is homogeneous and hence R is graded. Let S + be the ideal of S generated by u and v and let R + be the ideal of R generated by the images of u and v.
Introduction
Let (R, m) be a local Noetherian ring, let I ⊂ R be any ideal and let M be a finitely generated R-module. It has been long conjectured that the local cohomology modules H i I (M ) have finitely many associated primes for all i (see Conjecture 5.1 in [H] and [L] .)
Introduction
If R is not required to be local these sets of associated primes may be infinite, as shown by Anurag Singh in [S] , where he constructed an example of a local cohomology module of a finitely generated module over a finitely generated Z-algebra with infinitely many associated primes. This local cohomology module has p-torsion for all primes p ∈ Z.
Introduction
However, the question of the finiteness of the set of associated primes of local cohomology modules defined over local rings and over k-algebras (where k is a field) has remained open until now. In this paper I settle this question by constructing a local cohomology module of a local finitely generated k-algebra with an infinite set of associated primes, and I do this for any field k.
The example
Let k be any field, let R 0 = k[x, y, s, t] and let S = R 0 [u, v] . Define a grading on S by declaring
The example
and let R = S/f S. Notice that f is homogeneous and hence R is graded. Let S + be the ideal of S generated by u and v and let R + be the ideal of R generated by the images of u and v.
The example
Consider the local cohomology module H 2 R+ (R): it is homogeneously isomorphic to H 2 S+ (S/f S) and we can use the exact sequence
The example
of graded R-modules and homogeneous homomorphisms (induced from the exact sequence . We will study the graded components of H 2 S+ (S/f S) by considering the cokernels of the R 0 -homomorphisms
The example
given by multiplication by f . In order to represent these R 0 -homomorphisms between free R 0 -modules by matrices, we specify an ordering for each of the above-mentioned bases by declaring that
The example
(where α 1 , β 1 , α 2 , β 2 < 0 and α 1 + β 1 = α 2 + β 2 ) precisely when α 1 > α 2 . If we use this ordering for both the source and target of each f d , we can see that each
The example
is given by multiplication on the left by the tridiagonal d − 1 by d + 1 matrix
The example
We also define
The example
1.1. Lemma. (ii) Let S be an infinite set of positive integers. Suppose that either k has characteristic zero or that k has prime characteristic p and S contains infinitely many integers of the form
The example
Proof. We prove the first statement by induction on i. Since
The example
the lemma holds for i = 1 and i = 2. Assume now that i ≥ 3. Expanding the determinant of B i by its first row and applying the induction hypothesis we obtain
The example
We now prove the second statement. Define σ i = t i + t i−1 + · · · + t + 1 and notice that it is enough to show that the set of irreducible factors of {σ i } i∈S is infinite. Let I be the set of irreducible factors We also introduce a bigrading on the free R 0 -modules R n 0 by declaring deg(x α y β s a t b e j ) = (α + β, β + j) for all non-negative integers α, β, a, b and all 1 ≤ j ≤ n. Notice that R n 0 is a bigraded R 0 -module when R 0 is equipped with the bigrading mentioned above. (Coker A d−1 ) (D,j) .
Consider the k[s, t]-module (Coker
It is generated by the images of
Consider the k[s, t]-module (Coker
and the relations among these generators are given by k[s, t]-linear combinations of
Consider the k[s, t]-module (Coker
where c 1 , . . . , c d+1 are the columns of A d−1 . So we have
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
Coker A d−1 has τ d−1 torsion, and so does Coker A d−1 itself. Lemma 1.1(ii) applied with S = N now shows that there exist infinitely many irreducible homogeneous polynomials {p i ∈ k[s, t] : i ≥ 1} each one of them contained in some associated prime of the R 0 -module ⊕ d≥2 Coker A d−1 . Clearly, if i = j then any prime ideal P ⊂ R 0 which contains both p i and p j must contain both s and t.
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
Since the localisation of (Coker
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
which do not contain s and such that p i ⊂ P i , p j ⊂ P j , and the previous paragraph shows that
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
The second statement now follows from the fact that
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
1.3. Corollary. Let T be the localisation of R at the irrelevant maximal ideal m = s, t, x, y, u, v .
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
Then H 2 (u,v)T (T ) has infinitely many associated primes.
Using Lemma 1.1(i) we deduce that for all d ≥ 2 the direct summand (Coker
Proof.
A connection with associated primes of Frobenius powers
In this section we apply a technique similar to the one used in section 1 to give a proof of a slightly more general statement of Theorem 12 in [K] . it gives a connection between associated primes of Frobenius powers of ideals and of local cohomology modules, at least on a purely formal level.
A connection with associated primes of Frobenius powers
Let k be any field, let S = k[x, y, s, t], let F = xy(x − y)(sx − ty) = sx 3 y − (t + s)x 2 y 2 + txy 3 and let R = S/F S.
A connection with associated primes of Frobenius powers
2.1. Theorem. Let S be an infinite set positive integers and suppose that either k has characteristic zero or that k has characteristic p and that S contains infinitely many powers of p. The set n∈S Ass R R x n , y n is infinite.
A connection with associated primes of Frobenius powers
Proof. We introduce a grading in S by setting deg(x) = deg(y) = 1 and deg(s) = deg(t) = 0. Since F is homogeneous, R is also graded. 
A connection with associated primes of Frobenius powers
When d = n, T d is isomorphic to the cokernel of the submatrix of M d obtained by deleting the first and last rows which correspond to the generators y n , x n of T n .
A connection with associated primes of Frobenius powers
When d = n + 1, T d is isomorphic to the cokernel of the submatrix of M d obtained by deleting the first two rows and and last two rows which correspond to the generators y n+1 , xy n , x n y, x n+1 of T n+1 , and the resulting submatrix is B n−2 defined in Lemma 1.1; the result now follows from that lemma.
A connection with associated primes of Frobenius powers
This technique for finding associated primes of non-finitely generated graded modules and of sequences of graded modules has been applied in [BKS] and [KS] to yield further new and surprising properties of top local cohomology modules.

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