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41.
The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I t (Γ) when Γ is a tree. We first show that I t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank. 相似文献
42.
Tadakazu Sawada 《代数通讯》2013,41(5):1633-1639
A vector bundle on a hypersurface is arithmetically Cohen–Macaulay if its intermediate cohomologies vanish. On projective spaces, such bundles coincide with those which split into a direct sum of line bundles, but this fails on hypersurfaces of higher degree in general. In this article, we give an inequality which gives a sufficient condition for splitting of arithmetically Cohen–Macaulay bundles on general hypersurfaces. 相似文献
43.
Wolfgang Rump 《代数通讯》2013,41(8):2808-2824
Let R be a complete discrete valuation ring with quotient field K, and let Λ be an R-order in a semisimple K-algebra. For an indecomposable Λ-lattice E, a sublattice Bi E satisfying Rad E ? Bi E is defined, and it is shown that the middle term H of an almost split sequence τE ? H ? E can be obtained by an amalgamation of E/Bi E with E′/τE for a suitable overlattice E′ of τE. The method is bound to dim R = 1. 相似文献
44.
Let G be a finite simple graph on a vertex set V(G) = {x 11,…, x n1}. Also let m 1,…, m n ≥ 2 be integers and G 1,…, G n be connected simple graphs on the vertex sets V(G i ) = {x i1,…, x im i }. In this article, we provide necessary and sufficient conditions on G 1,…, G n for which the graph obtained by attaching the G i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs. 相似文献
45.
《Quaestiones Mathematicae》2013,36(3):399-410
Abstract An old result of J.S. Cohen states that each p* -summing operator on a L p -space has a p* -summing dual. Also, another old result of S. Kwapień states that each p* -summing operator on a Banach space X has a p* -summing dual if and only if X is isomorphic to a quotient of some L p (µ). In this paper we prove some multilinear and polynomial variants of these results. 相似文献
46.
Josep Àlvarez Montaner 《代数通讯》2013,41(9):3695-3704
We prove that sequentially Cohen–Macaulay rings in positive characteristic, as well as sequentially Cohen–Macaulay Stanley–Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring. 相似文献
47.
《代数通讯》2013,41(11):5085-5094
Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext t R (M,R) ≠ 0, see [11]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext t R (M, R) ≠ 0, see [3]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results. 相似文献
48.
Rajsekhar Bhattacharyya 《代数通讯》2013,41(12):5373-5380
In this article, we show that almost Cohen–Macaulay algebras are solid. Moreover, we seek for the conditions when (a) an almost Cohen–Macaulay algebra is a phantom extension and (b) when it maps into a balanced big Cohen–Macaulay module. 相似文献
49.
Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen–Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem. 相似文献
50.