Set $ A := Q/({f f}) $, where $ Q $ is a polynomial ring over a field, and$ {f f} = f_1, ldots, f_c $ is a homogeneous $ Q $-regular sequence. Let $ M$ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be ahomogeneous ideal of $ A $. For every $ l in { 0, 1 } $, we show that (1) $ mathrm{reg}left( mathrm{Ext}_A^{2i+l}(M, I^nN) ight) leqslantho_N(I) cdot n - w cdot i + e_{l} $ for all $ i, n geqslant 0 $, (2) $ mathrm{reg}left( mathrm{Ext}_A^{2i+l}(M,N/I^nN) ight) leqslantho_N(I) cdot n - w cdot i + e'_{l} $ for all $ i, n geqslant 0 $, where $e_{l}, e'_{l} $ are constants, $ w := min{ mathrm{deg}(f_j) : 1 leqslant jleqslant c } $, and $ ho_N(I) $ is an invariant defined in terms ofreduction ideals of $ I $ with respect to $ N $. There are explicit exampleswhich show that these inequalities are sharp.
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机译:设置$ a:= q /({ bf f})$,其中$ q $是一个字段上的多项式环,$ { bf f} = f_1, ldots,f_c $是一个同质$ q $ - 常规序列。让$ m $和$ n $合意生成渐变$ a $ -modules,$ i $ a $ a $ a $ a $ a。每$ l in {0,1 } $,我们展示了(1)$ mathrm {reg} left( mathrm {ext} _a ^ {2i + l}(m,i ^ nn)右) leqslant rho_n(i) cdot n - w cdot i + e_ {l} $ for for all $ i,n geqslant 0 $,(2)$ mathrm {reg} left( mathrm {ext } _a ^ {2i + l}(m,n / i ^ nn)右) leqslant rho_n(i) cdot n - w cdot i + e'_ {l} $ a for所有$ i,n geqslant 0 $,其中$ e_ {l},e'_ {l} $是常量,$ w:= min { mathrm {deg}(f_j):1 leqslant j leqslant c } $ $ rho_n(i)$是一个不变的,在$ i $的资料中定义了$ n $。有明确的例子显示这些不平等是尖锐的。
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