Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion of the open cone $Y = \left(L \times [0, 1)\right) / \left(L \times \{0\}\right)$.


Let $\boldsymbol{IC}^{\bullet}\_{\overline{p}}(Y)$ denote the perversity $\overline{p}$ intersection chain complex of the open cone $Y$ (using $\mathbb{Q}$-coefficients) as introduced in M. Goresky and R.D. MacPherson's paper [Intersection homology II, Invent. Math. 71 (1983), 77-129](http://math.mit.edu/conferences/geometryworkshop/references/GM83.pdf), then it is well-known that the perversity $\overline{p}$ intersection homology of $Y$ can be computed to be the following cotruncation of $H\_{\ast}(L)$:


$IH\_{i}^{\overline{p}}(Y) \cong \mathcal{H}^{-i}(Y; \boldsymbol{IC}^{\bullet}\_{\overline{p}}(Y)) \cong \begin{cases}
0, \quad i \leq d - \overline{p}(d+1), \\
H\_{i-1}(L), \quad i > d - \overline{p}(d+1).
\end{cases}$


**My question**:
Is there a similar formula for $\mathcal{H}^{\ast}(X; i\_{\ast}\boldsymbol{IC}^{\bullet}\_{\overline{p}}(Y))$, i.e. the hypercohomology of the push-forward of $\boldsymbol{IC}^{\bullet}\_{\overline{p}}(Y)$ to $X$?
And if so, how can it be derived?



What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?
I just realized I made an error in my previous answer here, so I've updated it:


Do you mean $Ri\_\*$ or literally $i\_\*$? $Ri\_\*$ would be the more usual thing to ask about in this context. Then, in general, if $f:X\to Y$ we have $\mathcal H^i(Y;Rf\_\*S^\*)\cong \mathcal H^i(X;S^\*)$. So in this case you'd get again the same hypercohomology groups.


