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Journals and Conferences
We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We then compare this bound to a bound given by Faltings. Finally, we study a family of clutters associated to graphs and compute domination parameters for certain classes of these clutters.
We study the relationship between the projective dimension of a squarefree monomial ideal and the domination parameters of the associated graph or clutter. In particular, we show that the projective dimensions of graphs with perfect dominating sets can be calculated combinatorially. We also generalize the wellknown graph domination parameter τ to clutters,… (More)
In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free… (More)
We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections.
Let R be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules (M, N) over R we study applications of rigidity of Tor(M, N), based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like… (More)
Let R be a Cohen-Macaulay ring and M a maximal CohenMacaulay R-module. Inspired by recent striking work by Iyama, BurbanIyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on… (More)
We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs. In particular, we draw heavily from the topic of dominating sets. Through Hochster’s Formula, we recover and strengthen existing results on the homological connectivity of graph independence… (More)
Let R be a local complete intersection and M,N are R-modules such that l(Tor i (M,N)) < ∞ for i ≫ 0. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of l(Tor i (M,N)) using Eisenbud operators and show that they have well-behaved growth. We define and study a function η(M,N) which generalizes Serre’s intersection… (More)
Let R be a local ring and M, N be finitely generated R-modules. The complexity of (M, N), denoted by cxR(M, N), measures the polynomial growth rate of the number of generators of the modules ExtR(M, N). In this paper we study several basic equalities and inequalities involving complexities of different pairs of modules.
Let R be a commutative Noetherian ring of prime characteristic p. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of H I(R) is finite for any ideal I and for any t ≥ 0 when R has finite F -representation type or finite singular locus. This extends a previous result by Takagi-Takahashi and gives… (More)