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Graded Components of Local Cohomology Modules II
Journal
Vietnam Journal of Mathematics
ISSN
2305221X
Date Issued
2024-03-01
Author(s)
Puthenpurakal, Tony J.
Roy, Sudeshna
Abstract
Let A be a commutative Noetherian ring containing a field of characteristic zero. Let R = A[X1,…,Xm] be a polynomial ring and Am(A) = A〈X1,…,Xm, ∂1,…,∂m〉 be the m th Weyl algebra over A, where ∂i = ∂/∂Xi. Consider standard gradings on R and Am(A) by setting degz= 0 for all z ∈ A, degXi= 1 , and deg∂i= − 1 for i = 1,…,m. We present a few results about the behavior of the graded components of local cohomology modules HIi(R), where I is an arbitrary homogeneous ideal in R. We mostly restrict our attention to the vanishing, tameness, and rigidity properties. To obtain this, we use the theory of D-modules and show that generalized Eulerian Am(A)-modules exhibit these properties. As a corollary, we further get that components of graded local cohomology modules with respect to a pair of ideals display similar behavior.
Subjects