اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the finite generation of a family of Ext modules
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تأليف
Tony J. Puthenpurakal
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Let $Q$ be a Noetherian ring with finite Krull dimension and let $mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $projdim_Q M$ finite. Set $R = bigoplus_{ngeq 0}I^n$, the Rees-Algebra of $I$. Let $N = bigoplus_{j geq 0}N_j$ be a finitely generated graded $R$-module. We show that [bigoplus_{jgeq 0}bigoplus_{igeq 0} Ext^{i}_{A}(M,N_j) ] is a finitely generated bi-graded module over $Sc = R[t_1,...,t_c]$. We give two applications of this result to local complete intersection rings.
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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 $(n,i)$.
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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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