sheaf cohomology
Let $X$ be a topological space^{}. The category of sheaves of abelian groups^{} on $X$ has enough injectives. So we can define the sheaf cohomology ${H}^{i}(X,\mathcal{F})$ of a sheaf $\mathcal{F}$ to be the right derived functors^{} of the global sections functor $\mathcal{F}\to \mathrm{\Gamma}(X,\mathcal{F})$.
Usually we are interested in the case where $X$ is a scheme, and $\mathcal{F}$ is a coherent sheaf. In this case, it does not matter if we take the derived functors in the category of sheaves of abelian groups or coherent sheaves.
Sheaf cohomology can be explicitly calculated using Čech cohomology^{} (http://planetmath.org/CechCohomologyGroup2). Choose an open cover $\{{U}_{i}\}$ of $X$. We define
$${C}^{i}(\mathcal{F})=\prod \mathcal{F}({U}_{{j}_{0}\mathrm{\cdots}{j}_{i}})$$ |
where the product^{} is over $i+1$ element subsets of $\{1,\mathrm{\dots},n\}$ and ${U}_{{j}_{0}\mathrm{\cdots}{j}_{i}}={U}_{{j}_{0}}\cap \mathrm{\cdots}\cap {U}_{{j}_{i}}$. If $s\in \mathcal{F}({U}_{{j}_{0}\mathrm{\cdots}{j}_{i}})$ is thought of as an element of ${C}^{i}(\mathcal{F})$, then the differential
$$\partial (s)=\prod _{\mathrm{\ell}}\left({\prod _{k={j}_{\mathrm{\ell}}+1}^{{j}_{\mathrm{\ell}+1}-1}{(-1)}^{\mathrm{\ell}}s|}_{{U}_{{j}_{0}\mathrm{\cdots}{j}_{\mathrm{\ell}}k{j}_{\mathrm{\ell}+1}\mathrm{\cdots}{j}_{i}}}\right)$$ |
makes ${C}^{*}(\mathcal{F})$ into a chain complex^{}. The cohomology of this complex is denoted ${\stackrel{\u02c7}{H}}^{i}(X,\mathcal{F})$ and called the Čech cohomology of $\mathcal{F}$ with respect to the cover $\{{U}_{i}\}$. There is a natural map ${H}^{i}(X,\mathcal{F})\to {\stackrel{\u02c7}{H}}^{i}(X,\mathcal{F})$ which is an isomorphism^{} for sufficiently fine covers. (A cover is sufficiently fine if ${H}^{i}({U}_{j},\mathcal{F})=0$ for all $i>0$, for every $j$ and for every sheaf $\mathcal{F}$). In the category^{} of schemes, for example, any cover by open affine schemes^{} has this property. What this means is that if one can find a finite fine enough cover of $X$, sheaf cohomology becomes computable by a finite process. In fact in [2], this is how the cohomology of projective space^{} is explicitly calculated.
References
- 1 Grothendieck, A. Sur quelques points d’algèbre homologique, Tôhoku Math. J., Second Series, 9 (1957), 119–221.
- 2 Hartshorne, R. Algebraic Geometry^{}, Springer-Verlag Graduate Texts in Mathematics 52, 1977
Title | sheaf cohomology |
---|---|
Canonical name | SheafCohomology |
Date of creation | 2013-03-22 13:50:59 |
Last modified on | 2013-03-22 13:50:59 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 14 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 14F25 |
Related topic | EtaleCohomology |
Related topic | LeraysTheorem |
Related topic | AcyclicSheaf |
Related topic | DeRhamWeilTheorem |
Defines | sufficiently fine |