Let $X$, $F$ be manifolds and $f:X\times F\to X$ be a natural projection. Then is the first higher direct image of a constant sheaf $R^1f_*(\mathbb{Z}_{X\times F})$ also a constant sheaf on $X$ with stalk $H^1(F,\mathbb{Z})$?
Here is my attempt: $R^1f_*(\mathbb{Z}_{X\times F})$ is the sheafification of the presheaf $V\mapsto H^1(f^{-1}(V),\mathbb{Z})$. So I tried to compute it. By using the Künneth formula,
$H^1(f^{-1}(V),\mathbb{Z})=H^1(V\times F,\mathbb{Z})=\{H^0(V,\mathbb{Z})\otimes H^1(F,\mathbb{Z})\}\oplus \{H^1(V,\mathbb{Z})\otimes H^0(F,\mathbb{Z})\}$.
Thus $H^1(f^{-1}(V),\mathbb{Z})=H^1(F,\mathbb{Z})$ when $V$ is a constarctible open subset of $X$. But, in general, there appears $H^1(V,\mathbb{Z})$ and so on. In the end, I could not figure what the sheafification of the presheaf is.
Thank you in advance.