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Register a new userAn asymptotic bound for Castelnuovo-Mumford regularity of certain Ext modules over graded complete intersection rings
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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.
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Let $(A,mathfrak{m})$ be a local complete intersection ring and let $I$ be an ideal in $A$. Let $M, N$ be finitely generated $A$-modules. Then for $l = 0,1$, the values $depth Ext^{2i+l}_A(M, N/I^nN)$ become independent of $i, n$ for $i,n gg 0$. We also show that if $mathfrak{p}$ is a prime ideal in $A$ then the $j^{th}$ Bass numbers $mu_jbig(mathfrak{p}, Ext^{2i+l}_A(M,N/{I^nN})big)$ has polynomial growth in $(n,i)$ with rational coefficients for all sufficiently large $(n,i)$.
Let $A$ be a Noetherian standard $mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k$ such that [ mathrm{reg}(I_1^{n_1} cdots I_t^{n_t} M) leq (n_1 + cdots + n_t) k + k quadmbox{for all }~n_1,ldots,n_t in mathbb{N}. ]
This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$ over a general base ring. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$ of $R$ is eventually a linear function in $v$. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple characterization of the ideals whose powers have a linear resolution in terms of the regularity of the Rees ring.
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.
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.
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