The Stacks project

Exercise 111.60.5. Let $X$ be a topological space which is the union $X = Y \cup Z$ of two closed subsets $Y$ and $Z$ whose intersection is denoted $W = Y \cap Z$. Denote $i : Y \to X$, $j : Z \to X$, and $k : W \to X$ the inclusion maps.

  1. Show that there is a short exact sequence of sheaves

    \[ 0 \to \underline{\mathbf{Z}}_ X \to i_*(\underline{\mathbf{Z}}_ Y) \oplus j_*(\underline{\mathbf{Z}}_ Z) \to k_*(\underline{\mathbf{Z}}_ W) \to 0 \]

    where $\underline{\mathbf{Z}}_ X$ denotes the constant sheaf with value $\mathbf{Z}$ on $X$, etc.

  2. What can you conclude about the relationship between the cohomology groups of $X$, $Y$, $Z$, $W$ with $\mathbf{Z}$-coefficients?

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View Exercise 111.60.5 as pdf