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Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence relating Cech to sheaf cohomology:

\[ E_2^{p,q}=\tilde{H}^p(U,H^q(-,F))\Rightarrow H^{p+q}(X,F) \]

I would like to see, how this implies the Mayer-Vietoris sequence for this easy cover $U$. Drawing the $E_2$-page, I get so far that only the first two columns $p=0,1$ are non-zero. Therefore this page equals the $E_\infty$-page.

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1 Answer 1

up vote 7 down vote accepted

Recall that the Cech-to-derived functor spectral sequence is constructed as follows. We start with a sheaf $F$ and an open cover $\mathfrak{U}$. Then we can write the Cech resolution of the sheaf; take an injective (or Godement or...) resolution thereof to get a double complex. Let $C^{\ast,\ast}$ be the resulting complex of global sections and take the filtration $F^i=\bigoplus C^{\geq i,\ast}$. See e.g. Godement, Th\'eorie des faisceaux, 5.2. The rows of the $E_1$ sheet are precisely the Cech cochain complexes constructed from the open cover $\mathfrak{U}$ and the presheaves $U\mapsto H^i(U,F)$ (see Godement, ibid, just before theorem 5.2.4).

If $\mathfrak{U}$ has just two elements, $U$ and $U''$, then the $E_1$ term has two columns, the 0-th and the 1-st ones. Applying e.g. theorem 4.6.1 from Godement, ibid, one gets the long exact sequence

$$\cdots\to E_1^{1,i-1}\to H^i(X,F)\to E_1^{0,i}\to E^{1,i}_1\to\cdots$$

where the last arrow is the $d_1$ differential, $E_1^{1,j}=H^j(U'\cap U'',F)$ and $E_1^{0,j}=H^j(U',F)\oplus H^j(U'',F)$.

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How does the $E_1$ page of the above spectral sequence look like? Does it exist or does one have to consider the double complex spectral sequences? –  user7316 Jul 5 '10 at 11:56
fs1504 -- I've added some details. –  algori Jul 5 '10 at 22:29

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