# What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?

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, 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?

## 1 Answer

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.

• Dear Greg, thanks a lot for your answer! Yes, let's take $R i_{\ast}$, and then it is good to know that the hypercohomology will transform as you say! – Rahmpilz Nov 6 '18 at 8:32