Author: Eiko

Tags: Cech Resolution, Homological Algebra, Sheaf Cohomology, Algebraic Geometry, Acyclic Resolution

Time: 2024-09-10 15:55:11 - 2024-09-10 15:55:11 (UTC)

Cech Resolution

Let $X$ be a separated scheme, $\mathcal{F}$ be a quasi-coherent sheaf of ${\mathcal{O}}_X$-modules. Let $\{U_i{\}}_I$ be an affine finite cover of $X$.

• Recall that a quasi-coherent $\mathcal{O}$-module is acyclic on affine schemes, i.e. $H^i(U,\mathcal{F})=0$ for $i>0$ and $U$ affine.

• Separated implies the intersection of two affine open sets is affine, making the covering an analogue of a good cover in manifold theory, (each non-empty intersection is contractible, and acyclic).

• Let $J\subset I$ be any non-empty subset with $U_J=\bigcap _{i\in J}{U}_i$. Then $U_J$ is affine, and $\mathcal{F}{|}_{U_J}$ is acyclic. Consider the open immersion $i_J:U_J\hookrightarrow X$, which is affine morphism and $(i_J{)}_{\ast }$ is exact. Then $i_J^{\ast }\mathcal{F}$ is acyclic and so is $(i_J{)}_{\ast }{i}_J^{\ast }\mathcal{F}$ as a sheaf on $X$.

Construction

These observations motivate the following construction, define the sheaf ${\mathcal{F}}^i$ as

$${\mathcal{F}}^i:=\underset{|J|=i+1}{⨁}(i_J{)}_{\ast }{i}_J^{\ast }\mathcal{F}$$

where the sum is over all non-empty subsets $J\subset I$ of cardinality $i+1$. The sheaf ${\mathcal{F}}^i$ is a sheaf of ${\mathcal{O}}_X$-modules, and the natural map $\mathcal{F}\to {\mathcal{F}}^0$ is injective.

The Cech differential is defined as the alternating sum of the restriction maps

$$d^i:{\mathcal{F}}^i\to {\mathcal{F}}^{i+1}$$

$$d^i(s{)}_J=\sum _{k=0}^{i+1}(-1{)}^k{s}_{J\mathrm{\setminus }{j}_k}$$

where $J$ is a subset of cardinality $i+2$. The Cech complex is then formed as

$$0\to {\mathcal{F}}^0\to {\mathcal{F}}^1\to \cdots$$

which is quasi-isomorphic to $\mathcal{F}$.

Computing Sheaf Cohomology using Acyclicity

Since each ${\mathcal{F}}^i$ is acyclic for the global section functor, the principle of acyclic resolution tells us that it can be used to compute the sheaf cohomology of $\mathcal{F}$,

$$H^i(X,\mathcal{F}):=R^i{\mathrm{\Gamma }}_X(\mathcal{F})=H^i({\mathrm{\Gamma }}_X({\mathcal{F}}^{\bullet })).$$

Note that

$${\mathrm{\Gamma }}_X({\mathcal{F}}^i)=\underset{J}{⨁}{\mathrm{\Gamma }}_X((i_J{)}_{\ast }{i}_J^{\ast }\mathcal{F})=\underset{J}{⨁}{\mathrm{\Gamma }}_{U_J}(i_J^{\ast }\mathcal{F})=\underset{J}{⨁}{\mathrm{\Gamma }}_{U_J}(\mathcal{F}{|}_{U_J})=\underset{J}{⨁}{\mathrm{\Gamma }}_{U_J}(\mathcal{F}),$$

denote the above group by $C^i(\mathcal{U},\mathcal{F})$, where $\mathcal{U}$ denotes the particular open cover being used. This tells us an explicit way to compute the sheaf cohomology,

$$H^i(X,\mathcal{F})=H^i(C^{\bullet }(\mathcal{U},\mathcal{F})).$$