# spectral sequence of cohomology with compact support

Let k be a field, and A, B is subscheme of C over k, F is a sheaf over C.

Question 1. what is the relation between $H_c(A\cap B,k)$, $H_c(B,F)$, and $H_c(A\cap B,F)$ ?

Question 2. Let A be a strata of a stratification over C, F be a perverse sheaf over C. How can we construct a spectral sequence $H_c^m(A,H^k(F)) \to H_c^{m+k}(A,F)$, where $H^k(F)$ is the hyper-cohomology.

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If $A,B$ are closed, there should be a restriction map from the second to third group in Q1, but essentially no relation to the first (for general $F$). Unless there is a typo? The spectral sequence does exist in Q2 for general reasons; it's a standard hypercohomology spectral sequence. – Donu Arapura Sep 26 '11 at 14:27
thank you, Thierry Zell and Donu Arapura – yingjin bi Sep 27 '11 at 3:05