Answer by anonymous for Mathematically mature way to think about Mayer–Vietoris

2010-05-01T13:50:24Z

This answer is related to Tilman's: Let $U$ and $V$ be the open sets covering $X$. For $S$ an open subset of $X$, let $\mathbb{Z}_S$ be the pushforward to $X$ of the sheaf of locally constant integer valued functions on $S$. Then we have a short exact sequence of sheaves

$$0 \to \mathbb{Z}_X \to \mathbb{Z}_U \oplus \mathbb{Z}_V \to \mathbb{Z}_{U \cap V} \to 0$$

and the corresponding long exact sequence is the Mayer-Vietores sequence in cohomology.

This answer can be generalized easily to any open cover of $X$: you have a long exact sequence of sheaves:

$$0 \to \mathbb{Z}_X \to \bigoplus \mathbb{Z}_{U_i} \to \bigoplus \mathbb{Z}_{U_i \cap U_j} \to \cdots$$

which gives a spectral sequence

$$\bigoplus H^p(U_{i_1} \cap U_{i_2} \cap \cdots U_{i_q}) \to H^{p+q}(X).$$

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Answer by anonymous for Mathematically mature way to think about Mayer–Vietoris