Let A be a regular ring containing a field K of characteristic zero and let R = A[X 1 , . . . , Xm]. Consider R as standard graded with deg A = 0 and deg X i = 1 for all i. Let G be a finite subgroup of GLm(A). Let G act linearly on R fixing A. Let S = R G . In this paper we present a comprehensive study of graded components of local cohomology modules H i I (S) where I is an arbitrary homogeneous ideal in S. We prove stronger results when G ⊆ GLm(K). Some of our results are new even in the case when A is a field.
Let A be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let R = A[X 1 , . . . , X n ] be a polynomial ring and I = (a 1 U 1 , . . . , a c U c ) ⊆ R an ideal, where a j ∈ A (not necessarily units) and U j 's are monomials in X 1 , . . . , X n . We call such an ideal I as a C-monomial ideal. Consider the standard multigrading on R. We produce a structure theorem for the multigraded components of the local cohomology modules H i I (R) for i ≥ 0. We further analyze the torsion part and the torsion-free part of these components. We show that if A is a PID then each component can be written as a direct sum of its torsion part and torsion-free part. As a consequence, we obtain that their Bass numbers are finite.The second author is grateful to the Infosys Foundation for providing partial financial support.
Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for N sgraded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for N-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds. Dedicated to Le Tuan Hoa on the occasion of his sixtieth birthday 1. Introduction The objective of this paper is to prove Eakin-Sathaye type theorems [4] for joint reductions and good filtrations of ideals. Recall that an ideal J contained in an ideal I in a commutative ring R is called a reduction of I if there is a non-negative integer n such that JI n = I n+1 . The concept of reduction of an ideal was introduced by Northcott and Rees [15]. It has become an important tool in many investigations in commutative algebra and algebraic geometry such as Hilbert-Samuel functions [18], blow-up algebras [6] singularities of hypersurfaces [23], number of defining equations of algebraic varieties [13] and many others. Research in this paper is motivated by the following result of Paul Eakin and Avinash Sathaye [4]. Let µ(I) denote the minimum number of generators of an ideal I in a local ring. Theorem 1.1. Let I an ideal of a local ring R with infinite residue field. If µ(I n ) < n+r r for some positive integers n and r then there is a reduction J of I generated by r elements such that JI n−1 = I n . The case of n = 2, r = 1 was proved by J. D. Sally [22]. The EST (Eakin-Sathaye Theorem) has been revisited by G. Caviglia [2], N. V. Trung [24] and Liam O'Carroll [16]. Caviglia used Green's hyperplane section theorem to give a new proof of the EST. The EST was generalised by Liam O'Carroll for complete reductions [16]. The versions of the EST proved in this paper follow the approach used by O'Carroll. In order to state his result we recall necessary definitions Date: June 21, 2018. Both of the first author and the second author are supported by UGC fellowship, Govt. of India.
We give Macaulay2 algorithms for computing mixed multiplicities of ideals in a polynomial ring. This enables us to find mixed volumes of lattice polytopes and sectional Milnor numbers of a hypersurface with an isolated singularity. The algorithms use the defining equations of the multi-Rees algebra of ideals. We achieve this by generalizing a recent result of David A.Cox, Kuei-Nuan Lin, and Gabriel Sosa in [4]. One can also use a Macaulay2 command 'reesIdeal' to calculate the defining equations of the Rees algebra. We compare the computation time of our scripts with the scripts already available.
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