Spectral sequence generalizing Čech cohomology

Question

Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.

For a subset $A\subset I$ denote $$U_A:=\cap_{i\in A}U_i.$$

Does there exist a spectral sequence which converges to the cohomology $H^*(X,\mathcal{F})$ and which starts at $$\bigoplus_{|A|=p}H^q(U_A,\mathcal{F}_{U_A})?$$

Here $\mathcal{F}_{U_A}$ denotes the restriction of $\mathcal{F}$ to $U_A$.

A reference will be helpful.

Answer 1

Yes, such a spectral sequence exists and is called the Čech-to-derived spectral sequence or perhaps the Mayer-Vietoris spectral sequence. For any topological space $X$ and any open cover $\mathfrak{U}=(U_i)_{i\in I}$ (no technical restrictions necessary), the functor $\smash{\check{H}}^0(\mathfrak{U},-)\colon\mathrm{PSh}(X)\rightarrow\mathbf{Ab}$ that takes a presheaf to the abelian group of compatible collections of sections over the elements of $\mathfrak{U}$ has Čech cohomology as right-derived functors. For a presheaf $\mathcal{F}$, Čech cohomology can be explicitly computed via the cochain complex $$0\rightarrow\prod_{i_0}\mathcal{F}(U_{i_0})\rightarrow\cdots\rightarrow\prod_{i_0,\dotsc,i_n}\mathcal{F}(U_{i_0\dotsc i_n})\rightarrow\cdots.$$ [This is essentially obtained from the Čech nerve of $\mathfrak{U}$.] Now, if $\iota\colon\mathrm{Sh}(X)\rightarrow\mathrm{PSh}(X)$ is the inclusion of sheaves into presheaves, the composite $\smash{\check{H}}^0(\mathfrak{U},-)\circ\iota=\Gamma(X,-)$ is the global sections functor, pretty much by definition. This gives rise to a (cohomological, first quadrant, convergent) Grothendieck spectral sequence $$E_2^{pq}=\smash{\check{H}}^p(\mathfrak{U},\mathcal{H}^q(\mathcal{F}))\Rightarrow H^{p+q}(X,\mathcal{F}),\qquad\text{naturally in }\mathcal{F}.$$ Here, $\mathcal{H}^q$ is the $q$-th derived functor of $\iota$, but since evaluating on an open is exact as a functor on presheaves, $\mathcal{H}^q(\mathcal{F})$ is just the presheaf $V\mapsto H^q(V,\mathcal{F})$. This spectral sequence starts on page $2$, but if you plug these presheaves into the complex computing Čech cohomology, you can reconstruct the page $1$ you want.

There is a direct argument not using the Grothendieck spectral sequence at the Stacks Project. The analogous spectral sequence for $\smash{\check{H}}^{\bullet}(X,-)$ (and the argument is the same) is in Section 5.9. of Godement's Topologie algébrique et théorie des faisceaux.