Local Cohomology

This paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a Zgraded module M over a Z-graded commutative Noetherian ring R, the graded Cousin complex for M introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.

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“…has a natural structure as an R (p) -module, and it follows from Theorem 3.5(v) and [1,Corollary 4.3.3] that, as such,…”

Section: (In This Notation Each T P Denotes a Homogeneous Element Ofmentioning

confidence: 99%

Comparison of Multigraded and Ungraded Cousin Complexes

Aghcheghloo¹,

Enshaei²,

Goto³

et al. 2001

Proceedings of the Edinburgh Mathematical Society

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“…There are other equivalent definitions of local cohomology, which make some of its properties evident, for example viaČech complexes. For more information on local cohomology see [12,9,16,24]. Most authors do not treat the graded case explicitly, but the first two of these sources do.…”

Section: Castelnuovo-mumford Regularitymentioning

confidence: 99%

“…Following [9], given an integer and a graded R-module M , we define the regularity at and below level to be…”

Section: The Strong Conjecturementioning

confidence: 99%

On the Castelnuovo-Mumford regularity of the cohomology ring of a group

Symonds

2010

J. Amer. Math. Soc.

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We record here one easily stated consequence. Proposition 0.3. For any non-trivial finite group, the cohomology ring His generated by elements in degree at most |G| − 1 and the relations between them (as a graded commutative algebra) are generated in degrees at most 2(|G| − 1). This bound is weak, although it can be improved somewhat at the cost of a more complicated formulation, but previously no such bound was known.We go on to prove some other conjectures of Benson on the regularity of the cohomology of other classes of groups [3], specifically for compact Lie groups and virtual Poincaré duality groups.The proof uses standard techniques in equivariant cohomology based on work of Quillen [30] and a paper of Duflot [14].David Green and Simon King have verified Benson's Conjecture computationally for all groups of order less than 256 [20].There is a survey of the properties of group cohomology from the point of view of commutative algebra by Benson [3]. Castelnuovo-Mumford regularityWe will work in the category of Z-graded rings and modules, so maps will respect the grading, elements will be homogeneous, etc., without specific mention. For a module M = i∈Z M i , we will write M ≥d = i≥d M i and similarly for other

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“…, a n+1 ∈ A. The above sequence is a complex for A and M. The n-th cohomology group ofC(A, E) is said to be n-th Hochschild cohomology group and denoted by H n (A, M), for more details see (Brodmann & Sharp, 1998), (Rotman, 2009) …”

Section: Introductionmentioning

confidence: 99%