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Register a new userCohomological dimensions of specialization-closed subsets and subcategories of modules
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Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.
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Let a be an ideal of a commutative Noetherian ring R with identity. We study finitely generated R-modules M whose a-finiteness and a-cohomological dimensions are equal. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one.
As a stable analogue of degenerations, we introduce the notion of stable degenerations for Cohen-Macaulay modules over a Gorenstein local algebra. We shall give several necessary and/or sufficient conditions for the stable degeneration. These conditions will be helpful to see when a Cohen-Macaulay module degenerates to another.
Let $lambda in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_{lambda}$, where $W_{lambda}$ is the stabilizer of $lambda$ in a finite Weyl group $W$. In this paper, we give a module $mathbb{K}_{w}(lambda)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q,,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $mathbb{K}_{w}(lambda)$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $V_{w}^{-}(lambda)$ by the sum of the submodules $V_{z}^{-}(lambda)$ for all those coset representatives $z$ of minimal length for the cosets in $W/W_{lambda}$ such that $z > w$ in the Bruhat order $<$ on $W$.
Let fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd_{fa}(M,N) denote the supremum of the is such that H^i_{fa}(M,N) eq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd_{fa}(M,N). Next, over a Cohen-Macaulay local ring (R,fm), we show that cd_{fm}(M,N)=dim R-grade(Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.
For a wide class of Cohen--Macaulay modules over the local ring of the plane curve singularity of type $T_{36}$ we describe explicitly the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.
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