In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field.

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. in [H] and [L].)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.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 exampleLet 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.

We describe an algorithm for computing parameter-test-ideals in certain local CohenMacaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp's notion of 'special ideals'. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article

This paper is concerned with ideals in a commutative Noetherian ring R of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of R generated by regular sequences exhibit a desirable type of 'uniform' behaviour. The principal technical tool used is a result, proved by R. Hartshorne and R. Speiser in the case where R is local and contains its residue field which is perfect, and subsequently extended to all local rings of prime characteristic by G. Lyubeznik, about a left module over the skew polynomial ring R[x, f ] (associated to R and the Frobenius homomorphism f , in the indeterminate x) that is both x-torsion and Artinian over R.  2005 Elsevier Inc. All rights reserved.

This paper gives an explicit formula for the Hilbert series of algebras of Veronese type.

This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting. 2000 Mathematics Subject Classification. 14B05, 13A35.

We prove that the F -jumping coefficients of a principal ideal of an excellent regular local ring of characteristic p > 0 are all rational and form a discrete subset of R.

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component, and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings