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تسجيل مستخدم جديدGraded components of local cohomology modules over invariant rings-II
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تأليف
Tony J. Puthenpurakal
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Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,ldots, X_m]$. Consider $R$ as standard graded with $deg A = 0$ and $deg X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. 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 emph{arbitrary} homogeneous ideal in $S$. We prove stronger results when $G subseteq GL_m(K)$. Some of our results are new even in the case when $A$ is a field.
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Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,ldots, X_m]$ and $R= B[X_1, ldots, X_m]$ be standard
graded with $ deg A=0$, $ deg B=0$ and $ deg X_i=1$ for all $i$. Extend the action of $G$ on $A$ to $S$ by fixing $X_i$. Note $S^G=R$. Let $I$ be an arbitrary homogeneous ideal in $R$. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those proven in a previous paper of the first author for $H_J^i(S)$ where $J$ is an arbitrary homogeneous ideal in $S$.
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ded components of local cohomology modules $H_I^i(R)$ where $I$ is an arbitrary homogeneous ideal in $R$. We mostly restrict our attention to the Vanishing, Tameness and Rigidity problems.
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R)$ for some $i, j$. We show that $Ext_{A_n(K)}^{ u}(M,N)$ is concentrated in degree zero for all $ u geq 0$, i.e., $Ext_{A_n(K)}^{ u}(M,N)_l=0$ for $l eq0$. This proves a conjecture stated in part I of this paper.
Let $(A,mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with reduction no.2 and $mu(M)=2$ or $3$ then we have proved that depth$G(M)geq d-mu(M)+1$. If $e(A)=3$ and $mu(M)=4$ then in this case we have proved that dept
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It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t
^{-1}]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a corollary.
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