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Register a new userOn certain properties and invariants of graded rings and modules
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Authors
Fred Rohrer
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The behaviour under coarsening functors of simple, entire, or reduced graded rings, of free graded modules over principal graded rings, of superfluous monomorphisms and of homological dimensions of graded modules, as well as adjoints of degree restriction functors, are investigated.
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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 depth$G(M)geq d-3$. When $A = Q/(f)$ where $Q = k[[X_1,cdots, X_{d+1}]]$ then we give estimates for depth $G(M)$ in terms of minimal presentation of $M$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.
Set $ A := Q/({bf z}) $, where $ Q $ is a polynomial ring over a field, and $ {bf z} = z_1,ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be a homogeneous ideal of $ A $. We show that (1) $ mathrm{reg}left( mathrm{Ext}_A^{i}(M, I^nN) right) le rho_N(I) cdot n - f cdot leftlfloor frac{i}{2} rightrfloor + b mbox{ for all } i, n ge 0 $, (2) $ mathrm{reg}left( mathrm{Ext}_A^{i}(M,N/I^nN) right) le rho_N(I) cdot n - f cdot leftlfloor frac{i}{2} rightrfloor + b mbox{ for all } i, n ge 0 $, where $ b $ and $ b $ are some constants, $ f := mathrm{min}{ mathrm{deg}(z_j) : 1 le j le c } $, and $ rho_N(I) $ is an invariant defined in terms of reduction ideals of $ I $ with respect to $ N $. There are explicit examples which show that these inequalities are sharp.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called $S$-projective provided that the induced sequence $0rightarrow {rm Hom}_R(P,A)rightarrow {rm Hom}_R(P,B)rightarrow {rm Hom}_R(P,C)rightarrow 0$ is $S$-exact for any $S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $S$-projective modules are obtained. The notion of $S$-semisimple modules is also introduced. A ring $R$ is called an $S$-semisimple ring provided that every free $R$-module is $S$-semisimple. Several characterizations of $S$-semisimple rings are provided by using $S$-semisimple modules, $S$-projective modules, $S$-injective modules and $S$-split $S$-exact sequences.
Let $R$ be a commutative ring. We investigate $R$-modules which can be written as emph{finite} sums of {it {second}} $R$-submodules (we call them emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (emph{main}) emph{second attached prime ideals} related to a module with such a presentation.
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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