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Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map that assigne to every vertex $s\in Y$ a finite subcomplex $X_{s}$ of $X$ such that the collection $\mathcal{Y}=(X_{s})_{s\in Y}$ is a recouvrement of $X$ and the nerve of this recouvrement $\mathcal{Y}$ is isomorphic to the simplicial complexe $Y$.

My question is : It is true that $H_{c}^{p}(X,\mathbb{Z})\simeq\bigoplus_{s\in Y}H^{p}(X_{s},\mathbb{Z})\oplus H_{c}^{p}(Y,\mathbb{Z})$ ?

where $H_{c}^{\bullet}$ is the cohomoloy with compact support.

Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite subcomplex $X_{s}$ of $X$ such that the collection $\mathcal{Y}=(X_{s})_{s\in Y}$ is a covering by closed subsets of $X$ and the nerve of this covering $\mathcal{Y}$ is isomorphic to the simplicial complexe $Y$.

My question is : It is true that $H_{c}^{p}(X,\mathbb{Z})\simeq\bigoplus_{s\in Y}H^{p}(X_{s},\mathbb{Z})\oplus H_{c}^{p}(Y,\mathbb{Z})$ ?

where $H_{c}^{\bullet}$ is the cohomoloy with compact support.