2,415 research outputs found
F-stable submodules of top local cohomology modules of Gorenstein rings
This paper applies G. Lyubeznik's notion of F-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring
The Hilbert series of algebras of Veronese type
This paper gives a fairly explicit formula for the Hilbert series of algebras of Veronese type
The support of top graded local cohomology modules
Let be any domain, let , where are indeterminates of some positive degrees, and is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of contain a certain non-zero ideal of called the
``content'' of . It follows that the support of is simply V(\content(I)R + R_+) (Corollary 1.8) and, in particular, vanishes if and only if is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes-- this paper provides further evidence in favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results
On ideals of minors of matrices with indeterminate entries
This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main
result in the first part of this paper is a theorem which gives sufficient conditions for the ideal of minors of a matrix to be prime. This theorem is general enough to include interesting examples, such as the ideal of maximal
minors of catalecticant matrices and their generalisations discussed in the second part of the paper.
The second aim of this paper is to settle a specific problem raised by David Eisenbud and Frank-Olaf Schreyer on the primary decomposition of an ideal of maximal minors. We solve this problem by applying the theorem above together with some ad-hoc techniques
The Betti numbers of forests
This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals,
which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs
On the arithmetic of tight closure
We provide a negative answer to an old question in tight closure theory by showing that the containment x^3y^3 \in (x^4,y^4,z^4)^* in K[x,y,z]/(x^7+y^7-z^7) holds for infinitely many but not for almost all prime
characteristics of the field K. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal (x,y,z) \subset K[x,y,z,u,v,w]/(x^7+y^7-z^7, ux^4+vy^4+wz^4+x^3y^3) has then
the property that the cohomological dimension fluctuates arithmetically between 0 and 1
The Betti numbers of forests
This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals,
which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs
Counting monomials
This paper presents two enumeration techniques based on Hilbert functions. The paper illustrates these techniques by solving two chessboard problems
Bipartite graphs whose edge algebras are complete intersections
Let R be monomial sub-algebra of generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are and whose edges are . Conversely, for any graph G with vertices we define the {\it edge algebra associated with G} as the sub-algebra of generated by the monomials . We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections
- …