# Intermediate extension and irreducible subquotients of perverse cohomologies

Let a set X be the union of two locally closed subsets U and V such that U does not lie in the closure of V. Let the restriction of a complex R of constructible sheaves on X to a smooth open subset A of U be an irreducible local system S on A. Is it true that the intermediate extension IC(U, S) is one of the irreducible perverse subquotients in one of the perverse cohomologies of R (I.e. the cohomologies of R with respect to the perverse t-structure)?

Yes. Let $$A' = A - (A \cap \overline{V})$$. Then $$A'$$ is an open subset of $$U$$ disjoint from the closure of $$V$$, hence an open subset of $$X$$. So the $$i$$'th perverse cohomology of the restriction of $$R$$ to $$A'$$ is the restriction to $$A'$$ of the $$i$$'th perverse cohomology of $$R$$.
Taking $$i= \dim A$$ (or something else if $$S$$ appears in nonzero degree), we see that the restriction to $$A'$$ of the $$i$$'th perverse cohomology of $$R$$ is $$S$$. This $$i$$'th perverse cohomology has a finite filtration into irreducible perverse sheaves, and their restrictions remain perverse.
Because $$A$$ is smooth, it is normal, so $$S$$ remains irreducible on restriction to $$A'$$. Hence the restrictions of all members of this filtration but one must vanish, and the restriction of the last one must equal $$S$$.
An irreducible perverse sheaf whose restriction to $$A'$$ is $$S$$ must equal the intermediate extension from $$A'$$ to $$X$$ of $$S$$, which is also the intermediate extension from $$A$$ to $$X$$ of $$S$$.