Rings that are almost Gorenstein

Huneke, Craig; Vraciu, Adela

doi:10.2140/pjm.2006.225.85

Cited by 25 publications

(30 citation statements)

References 2 publications

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“…. , x d ] that make R/I a local Artinian ring of type 2, in a purely combinatorial way, using only the poset structure of Z d , (ii) Comparing with [HV06, Example 4.3] we see that the analysis of type 2 monomial ideal done there is slightly different from ours from Corollary 5.9, in that there in [HV06] the authors describe when exactly the condition J 1 : J 2 + J 2 : J 1 ⊇ m = (x 1 , . .…”

Section: Remarksmentioning

confidence: 71%

On Monomial Ideals and Their Socles

Agnarsson

Epstein

2019

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For a finite subset M ⊂ [x1, . . . , x d ] of monomials, we describe how to constructively obtain a monomial ideal I ⊆ R = K[x1, . . . , x d ] such that the set of monomials in Soc(I) \ I is precisely M , or such that M ⊆ R/I is a K-basis for the the socle of R/I. For a given M we obtain a natural class of monomials I with this property. This is done by using solely the lattice structure of the monoid [x1, . . . , x d ]. We then present some duality results by using anti-isomorphisms between upsets and downsets of (Z d , ). Finally, we define and analyze zero-dimensional monomial ideals of R of type k, where type 1 are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in Z d .where no m i divides another m j , has a Gröbner basis M = {m 1 , . . . , m k }; exactly the minimum set of generators for I. For A ⊆ {1, . . . , k} let m A denote the least common multiple of the m i : i ∈ A.With this convention one can define the Scarf complex ∆(I) of the monomial ideal I as the simplicial complex consisting of all the subsets A ⊆ {1, . . . , k} with unique least common multiple m A , that isThe Scarf complex discussed in [BPS98] and [MS05] was first introduced in [Sca86]. It is easy to see that the facets F d−1 (∆(I)) of the Scarf complex ∆(I) are in bijective correspondence with the maximal monomials of R \ I (w.r.t. the divisibility partial order), which is exactly the set of monomials of the socle Soc(I) that are not in I (see definition in the following section.) The cardinality of this very set of monomials has many interesting combinatorial interpretations, two of which we will briefly describe here below.We say that an ideal I of R is co-generated by a set F of K-linear functionals R → K if I is the largest ideal of R contained in all the kernels of the functionals in F. A celebrated result by Macaulay from 1916 [Mac94] states that every ideal of R is finitely co-generated, so R is co-Noetherian in this sense. More specifically, it turns out that any monomial ideal I of R has at least |F d−1 (∆(I))| co-generators and can always be co-generated by |F d−1 (∆(I))| + 1 functionals (see [Agn00]). Further, for a given ideal I (not necessarily monomial) of R and a fixed term order, then I has a Gröbner basis where the head or leading terms of the basis elements form a corresponding monomial ideal L(I) of R, and I can then be co-generated by one functional if |F n−1 (∆(L(I)))| < d and by |F n−1 (∆(L(I)))| + 1 functionals otherwise [Agn00].The cardinality |F d−1 (∆(I))| is also linked to the number of edges in a simple graph on k vertices in the following way. For given d, k ∈ N let c d (k) denote the maximum number of facets |F d−1 (∆(I))| among all monomial ideals I of R that are minimally generated by k monomials. In general, the Scarf complex ∆(I) is always a sub-complex of the boundary complex of a simplicial polytope P (I) on k vertices where one facet is missing. When I is Artinian (or zero-dimensional) and generic (i.e. the powers of all x i...

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Section: Remarksmentioning

confidence: 71%

On Monomial Ideals and Their Socles

Agnarsson

Epstein

2019

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“…In [14] Huneke and Vraciu also define a notion of a ring R being 'almost' Gorenstein. They show that any Artinian Gorenstein ring modulo its socle is almost Gorenstein in their sense, for example, R = k[x, y]/(x 2 , xy, y 2 ).…”

Section: Non-extremalitymentioning

confidence: 99%

On the growth of the Betti sequence of the canonical module

Jorgensen

Leuschke

2007

Math. Z.

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Abstract. We study the growth of the Betti sequence of the canonical module of a Cohen-Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen-Macaulay ring possessing a canonical module to be Gorenstein.

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“…part of (3) follows by duality.Since (0 : R a) = (0 : R (aω)), (0 : R a) ⊆ a 2 gives (0 : ω a 2 ) ⊆ aω. Hence by(2) and(3), ker(h) ⊆ ker( f ), proving (4).…”

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confidence: 63%

“…When a = m, the above hypothesis says that R contains k and that soc(R) ⊆ m 2 . Huneke and Vraciu prove the theorem in this case in [2].…”

Section: Definition 43mentioning

confidence: 86%