# Kunneth formular with coefficients the constructible sheaves

Let $X$, $Y$ be complex algebraic varieties (possibly open and singular) and $A$ (resp. $B$) be a constructible sheaf of $\mathbb{Q}$-vector spaces on $X$ (resp. $Y$). Do we have the Kunneth type isomorphisms: $$H^k(X\times Y,A\boxtimes B)\simeq\bigoplus_{i+j=k}H^i(X,A)\otimes H^j(Y,B)?$$ Here $A\boxtimes B$ denotes $p_1^\ast A\otimes p_2^\ast B$.

P.S. If we consider the cohomology with compact supports, then we have $$H^k_{\rm cpt}(X\times Y,A\boxtimes B)\simeq\bigoplus_{i+j=k}H^i_{\rm cpt}(X,A)\otimes H^j_{\rm cpt}(Y,B)$$ following from the projection formula for pushforward with compact supports.