User anonymous - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:02:00Z http://mathoverflow.net/feeds/user/5758 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris/23179#23179 Answer by anonymous for Mathematically mature way to think about Mayer–Vietoris anonymous 2010-05-01T13:50:24Z 2010-05-01T13:50:24Z <p>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</p> <p><code>$$0 \to \mathbb{Z}_X \to \mathbb{Z}_U \oplus \mathbb{Z}_V \to \mathbb{Z}_{U \cap V} \to 0$$</code></p> <p>and the corresponding long exact sequence is the Mayer-Vietores sequence in cohomology.</p> <hr> <p>This answer can be generalized easily to any open cover of $X$: you have a long exact sequence of sheaves:</p> <p><code>$$0 \to \mathbb{Z}_X \to \bigoplus \mathbb{Z}_{U_i} \to \bigoplus \mathbb{Z}_{U_i \cap U_j} \to \cdots$$</code></p> <p>which gives a spectral sequence </p> <p><code>$$\bigoplus H^p(U_{i_1} \cap U_{i_2} \cap \cdots U_{i_q}) \to H^{p+q}(X).$$</code></p>