A corollary of the b-function lemma

Abstract

Let $X$ be an algebraic variety, $f$ a regular function, $j:U\subset X$ the complement to the locus of vanishing of $f$, and $M$ a holonomic D-module on $U$. Consider the $D_U[s]$-module $M\otimes "f^s"$. The goal of this note is to describe all $D_X[s]$-submodules $N\subset j_{\ast }(M\otimes "f^s")$ such that $j^{\ast }(N)\simeq M\otimes "f^s"$.