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)?

## 1 Answer

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 $S$.