vanishing and localization of $(d,frak{b})$-ideal transforms

Let $r$ be a commutative noetherian ring, $!m$ an $!r$-module and $d$ a non-negative integer. let $sigma$ denote the set of ideals $frak{i}$ of $r$ such that $mathrm{dim}(r/frak{i})!leq!d$. for an ideal $frak{b}$ of $r$, we define the $(d,ifrak{b})$-transform $d_{d,iifrak{b}}(m)$ and study its properties. then a criterion for $d_{d,frak{b}}(r)!=!bigcap_{frak{p}notin w(d,frak{b})}r_{frak{p}}$ will be given, where $w(d,frak{b})$ contains all ideals $frak{a}$ of $r$ such that $frak{i}subseteq frak{a}+frak{b}$ for some $frak{i}in sigma$. for each $igeq 0$, let $d^i_{d,frak{b}}(-)$ denote the $i$-th right derived functor of $d_{d,frak{b}}(m)$. we study the localization of the module $d^i_{d,frak{b}}(m)$ and prove that $d^i_{d,frak{b}}(m)_frak{p}cong d^i_{d-textrm{dim}(r/(frak{p}+frak{b})),frak{b}_frak{p}}(m_frak{p})$ for all $frak{p}inmathrm{spec}(r)$ and all $igeq 0$. finally, we establish vanishing theorems for $d^i_{d,frak{b}}(m)$.