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 a local system S on A. Is it true that the intermediate extension IC(U, S) is one of the perverse cohomologies of R (I.e. the cohomologies of R with respect to the perverse t-structure)?
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No. Let $X=U=\Bbb C$, $V=\emptyset$, $A=X\setminus\{0\}$, $S=\Bbb C_A[1]$, and $R=Rj_*S$, where $j\colon A\to X$ is inclusion. One can show that $R$ is perverse. However, $j_{!*}S=\Bbb C_X[1] \neq R$.
answered Aug 16, 2019 at 17:32
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$\begingroup$ Thank you very much, sorry, I meant to ask a different question -- do you know by any chance how to answer it? mathoverflow.net/questions/338604/… $\endgroup$ Commented Aug 18, 2019 at 12:11