Čech cohomology

Construction

Let X be a topological space, and let ${\displaystyle \mathcal{F}}$ be a presheaf of abelian groups on X. Let ${\displaystyle \mathcal{U}}$ be an open cover of X.

Simplex

A q-simplex σ of ${\displaystyle \mathcal{U}}$ is an ordered collection of q+1 sets chosen from ${\displaystyle \mathcal{U}}$, such that the intersection of all these sets is non-empty. This intersection is called the support of σ and is denoted |σ|.

Now let ${\displaystyle \sigma =(U_i{)}_{i\in \{0,\dots ,q\}}}$ be such a q-simplex. The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:

${\displaystyle {\mathrm{\partial }}_j\sigma :=(U_i{)}_{i\in \{0,\dots ,q\}\setminus \{j\}}.}$

The boundary of σ is defined as the alternating sum of the partial boundaries:

${\displaystyle \mathrm{\partial }\sigma :=\sum _{j=0}^q(-1{)}^{j+1}{\mathrm{\partial }}_j\sigma }$

viewed as an element of the free abelian group spanned by the simplices of ${\displaystyle \mathcal{U}}$.

Cochain

A q-cochain of ${\displaystyle \mathcal{U}}$ with coefficients in ${\displaystyle \mathcal{F}}$ is a map which associates with each q-simplex σ an element of ${\displaystyle \mathcal{F}(|\sigma |)}$, and we denote the set of all q-cochains of ${\displaystyle \mathcal{U}}$ with coefficients in ${\displaystyle \mathcal{F}}$ by ${\displaystyle C^q(\mathcal{U},\mathcal{F})}$. ${\displaystyle C^q(\mathcal{U},\mathcal{F})}$ is an abelian group by pointwise addition.

Differential

The cochain groups can be made into a cochain complex ${\displaystyle (C^{\bullet }(\mathcal{U},\mathcal{F}),\delta )}$ by defining the coboundary operator ${\displaystyle {\delta }_q:C^q(\mathcal{U},\mathcal{F})\to C^{q+1}(\mathcal{U},\mathcal{F})}$ by:

${\displaystyle ({\delta }_{q}f)(\sigma ):=\sum _{j=0}^{q+1}(-1{)}^j{\mathrm{r}\mathrm{e}\mathrm{s}}_{|\sigma |}^{|{\mathrm{\partial }}_j\sigma |}f({\mathrm{\partial }}_j\sigma ),}$

where ${\displaystyle {\mathrm{r}\mathrm{e}\mathrm{s}}_{|\sigma |}^{|{\mathrm{\partial }}_j\sigma |}}$ is the restriction morphism from ${\displaystyle \mathcal{F}(|{\mathrm{\partial }}_j\sigma |)}$ to ${\displaystyle \mathcal{F}(|\sigma |).}$ (Notice that ∂_jσ ⊆ σ, but |σ| ⊆ |∂_jσ|.)

A calculation shows that ${\displaystyle {\delta }_{q+1}\circ {\delta }_q=0.}$

The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called the differential of the cochain complex.

Cocycle

A q-cochain is called a q-cocycle if it is in the kernel of ${\displaystyle \delta }$, hence ${\displaystyle Z^q(\mathcal{U},\mathcal{F}):=\ker ({\delta }_q)\subseteq C^q(\mathcal{U},\mathcal{F})}$ is the set of all q-cocycles.

Thus a (q−1)-cochain ${\displaystyle f}$ is a cocycle if for all q-simplices ${\displaystyle \sigma }$ the cocycle condition

${\displaystyle \sum _{j=0}^q(-1{)}^j{\mathrm{r}\mathrm{e}\mathrm{s}}_{|\sigma |}^{|{\mathrm{\partial }}_j\sigma |}f({\mathrm{\partial }}_j\sigma )=0}$

holds.

A 0-cocycle ${\displaystyle f}$ is a collection of local sections of ${\displaystyle \mathcal{F}}$ satisfying a compatibility relation on every intersecting ${\displaystyle A,B\in \mathcal{U}}$

${\displaystyle f(A){|}_{A\cap B}=f(B){|}_{A\cap B}}$

A 1-cocycle ${\displaystyle f}$ satisfies for every non-empty ${\displaystyle U=A\cap B\cap C}$ with ${\displaystyle A,B,C\in \mathcal{U}}$

${\displaystyle f(B\cap C){|}_U-f(A\cap C){|}_U+f(A\cap B){|}_U=0}$

Coboundary

A q-cochain is called a q-coboundary if it is in the image of ${\displaystyle \delta }$ and ${\displaystyle B^q(\mathcal{U},\mathcal{F}):=\mathrm{I}\mathrm{m}({\delta }_{q-1})\subseteq C^q(\mathcal{U},\mathcal{F})}$ is the set of all q-coboundaries.

For example, a 1-cochain ${\displaystyle f}$ is a 1-coboundary if there exists a 0-cochain ${\displaystyle h}$ such that for every intersecting ${\displaystyle A,B\in \mathcal{U}}$

${\displaystyle f(A\cap B)=h(A){|}_{A\cap B}-h(B){|}_{A\cap B}}$

Cohomology

The Čech cohomology of ${\displaystyle \mathcal{U}}$ with values in ${\displaystyle \mathcal{F}}$ is defined to be the cohomology of the cochain complex ${\displaystyle (C^{\bullet }(\mathcal{U},\mathcal{F}),\delta )}$. Thus the qth Čech cohomology is given by

${\displaystyle {\check{H}}^q(\mathcal{U},\mathcal{F}):=H^q((C^{\bullet }(\mathcal{U},\mathcal{F}),\delta ))=Z^q(\mathcal{U},\mathcal{F})/B^q(\mathcal{U},\mathcal{F})}$.

The Čech cohomology of X is defined by considering refinements of open covers. If ${\displaystyle \mathcal{V}}$ is a refinement of ${\displaystyle \mathcal{U}}$ then there is a map in cohomology ${\displaystyle {\check{H}}^{\ast }(\mathcal{U},\mathcal{F})\to {\check{H}}^{\ast }(\mathcal{V},\mathcal{F}).}$ The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in ${\displaystyle \mathcal{F}}$ is defined as the direct limit ${\displaystyle \check{H}(X,\mathcal{F}):=\underset{\mathcal{U}}{\underrightarrow{\lim }}\check{H}(\mathcal{U},\mathcal{F})}$ of this system.

The Čech cohomology of X with coefficients in a fixed abelian group A, denoted ${\displaystyle \check{H}(X;A)}$, is defined as ${\displaystyle \check{H}(X,{\mathcal{F}}_A)}$ where ${\displaystyle {\mathcal{F}}_A}$ is the constant sheaf on X determined by A.

A variant of Čech cohomology, called numerable Čech cohomology, is defined as above, except that all open covers considered are required to be numerable: that is, there is a partition of unity {ρ_i} such that each support ${\displaystyle \{x\mid {\rho }_i(x)>0\}}$ is contained in some element of the cover. If X is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.

Relation to other cohomology theories

If X is homotopy equivalent to a CW complex, then the Čech cohomology ${\displaystyle {\check{H}}^{\ast }(X;A)}$ is naturally isomorphic to the singular cohomology ${\displaystyle H^{\ast }(X;A)}$. If X is a differentiable manifold, then ${\displaystyle {\check{H}}^{\ast }(X;\mathbb{R})}$ is also naturally isomorphic to the de Rham cohomology; the article on de Rham cohomology provides a brief review of this isomorphism. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if X is the closed topologist's sine curve, then ${\displaystyle {\check{H}}^1(X;\mathbb{Z})=\mathbb{Z},}$ whereas ${\displaystyle H^1(X;\mathbb{Z})=0.}$

If X is a differentiable manifold and the cover ${\displaystyle \mathcal{U}}$ of X is a "good cover" (i.e. all the sets U_α are contractible to a point, and all finite intersections of sets in ${\displaystyle \mathcal{U}}$ are either empty or contractible to a point), then ${\displaystyle {\check{H}}^{\ast }(\mathcal{U};\mathbb{R})}$ is isomorphic to the de Rham cohomology.

If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology.

For a presheaf ${\displaystyle \mathcal{F}}$ on X, let ${\displaystyle {\mathcal{F}}^{+}}$ denote its sheafification. Then we have a natural comparison map

${\displaystyle \chi :{\check{H}}^{\ast }(X,\mathcal{F})\to H^{\ast }(X,{\mathcal{F}}^{+})}$

from Čech cohomology to sheaf cohomology. If X is paracompact Hausdorff, then $\chi$ is an isomorphism. More generally, $\chi$ is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.

In algebraic geometry

Čech cohomology can be defined more generally for objects in a site C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a scheme X. The Čech cohomology with values in some sheaf ${\displaystyle \mathcal{F}}$ is defined as

${\displaystyle {\check{H}}^n(X,\mathcal{F}):=\underset{\mathcal{U}}{\underrightarrow{\lim }}{\check{H}}^n(\mathcal{U},\mathcal{F}).}$

where the colimit runs over all coverings (with respect to the chosen topology) of X. Here ${\displaystyle {\check{H}}^n(\mathcal{U},\mathcal{F})}$ is defined as above, except that the r-fold intersections of open subsets inside the ambient topological space are replaced by the r-fold fiber product

${\displaystyle {\mathcal{U}}^{{\times }_X^r}:=\mathcal{U}{\times }_X\cdots {\times }_X\mathcal{U}.}$

As in the classical situation of topological spaces, there is always a map

${\displaystyle {\check{H}}^n(X,\mathcal{F})\to H^n(X,\mathcal{F})}$

from Čech cohomology to sheaf cohomology. It is always an isomorphism in degrees n = 0 and 1, but may fail to be so in general. For the Zariski topology on a Noetherian separated scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided that any finite set of points of X are contained in some open affine subscheme. This is satisfied, for example, if X is quasi-projective over an affine scheme.

The possible difference between Čech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Čech nerve

${\displaystyle N_X\mathcal{U}:\cdots \to \mathcal{U}{\times }_X\mathcal{U}{\times }_X\mathcal{U}\to \mathcal{U}{\times }_X\mathcal{U}\to \mathcal{U}.}$

A hypercovering K_∗ of X is a certain simplicial object in C, i.e., a collection of objects K_n together with boundary and degeneracy maps. Applying a sheaf ${\displaystyle \mathcal{F}}$ to K_∗ yields a simplicial abelian group $\mathcal{F}(K_{\ast })$ whose n-th cohomology group is denoted $H^n(\mathcal{F}(K_{\ast }))$. (This group is the same as ${\displaystyle {\check{H}}^n(\mathcal{U},\mathcal{F})}$ in case K_∗ equals ${\displaystyle N_X\mathcal{U}}$.) Then, it can be shown that there is a canonical isomorphism

${\displaystyle H^n(X,\mathcal{F})\cong \underset{K_{\ast }}{\underrightarrow{\lim }}{H}^n(\mathcal{F}(K_{\ast })),}$

where the colimit now runs over all hypercoverings.

Examples

The most basic example of Čech cohomology is given by the case where the presheaf ${\displaystyle \mathcal{F}}$ is a constant sheaf, e.g. ${\displaystyle \mathcal{F}=\mathbb{R}}$. In such cases, each ${\displaystyle q}$-cochain ${\displaystyle f}$ is simply a function which maps every ${\displaystyle q}$-simplex to ${\displaystyle \mathbb{R}}$. For example, we calculate the first Čech cohomology with values in ${\displaystyle \mathbb{R}}$ of the unit circle ${\displaystyle X=S^1}$. Dividing ${\displaystyle X}$ into three arcs and choosing sufficiently small open neighborhoods, we obtain an open cover ${\displaystyle \mathcal{U}=\{U_0,U_1,U_2\}}$ where ${\displaystyle U_i\cap U_j\ne \mathrm{\varnothing }}$ but ${\displaystyle U_0\cap U_1\cap U_2=\mathrm{\varnothing }}$.

Given any 1-cocycle ${\displaystyle f}$, ${\displaystyle \delta f}$ is a 2-cochain which takes inputs of the form ${\displaystyle (U_i,U_i,U_i),(U_i,U_i,U_j),(U_j,U_i,U_i),(U_i,U_j,U_i)}$ where ${\displaystyle i\ne j}$ (since ${\displaystyle U_0\cap U_1\cap U_2=\mathrm{\varnothing }}$ and hence ${\displaystyle (U_i,U_j,U_k)}$ is not a 2-simplex for any permutation ${\displaystyle \{i,j,k\}=\{1,2,3\}}$). The first three inputs give ${\displaystyle f(U_i,U_i)=0}$; the fourth gives

${\displaystyle \delta f(U_i,U_j,U_i)=f(U_j,U_i)-f(U_i,U_i)+f(U_i,U_j)=0⟹f(U_j,U_i)=-f(U_i,U_j).}$

Such a function is fully determined by the values of ${\displaystyle f(U_0,U_1),f(U_0,U_2),f(U_1,U_2)}$. Thus,

${\displaystyle Z^1(\mathcal{U},\mathbb{R})=\{f\in C^1(\mathcal{U},\mathbb{R}):f(U_i,U_i)=0,f(U_j,U_i)=-f(U_i,U_j)\}\cong {\mathbb{R}}^3.}$

On the other hand, given any 1-coboundary ${\displaystyle f=\delta g}$, we have

${\displaystyle \{\begin{array}{ll}f(U_i,U_i)=g(U_i)-g(U_i)=0& (i=0,1,2);\\ f(U_i,U_j)=g(U_j)-g(U_i)=-f(U_j,U_i)& (i\ne j)\end{array}}$

However, upon closer inspection we see that ${\displaystyle f(U_0,U_1)+f(U_1,U_2)=f(U_0,U_2)}$ and hence each 1-coboundary ${\displaystyle f}$ is uniquely determined by ${\displaystyle f(U_0,U_1)}$ and ${\displaystyle f(U_1,U_2)}$. This gives the set of 1-coboundaries:

${\displaystyle \begin{array}{cc}{B}^1(\mathcal{U},\mathbb{R})=\{f\in C^1(\mathcal{U},\mathbb{R}):& f(U_i,U_i)=0,f(U_j,U_i)=-f(U_i,U_j),\\ & f(U_0,U_2)=f(U_0,U_1)+f(U_1,U_2)\}\cong {\mathbb{R}}^2.\end{array}}$

Therefore, ${\displaystyle {\check{H}}^1(\mathcal{U},\mathbb{R})=Z^1(\mathcal{U},\mathbb{R})/B^1(\mathcal{U},\mathbb{R})\cong \mathbb{R}}$. Since ${\displaystyle \mathcal{U}}$ is a good cover of ${\displaystyle X}$, we have ${\displaystyle {\check{H}}^1(X,\mathbb{R})\cong \mathbb{R}}$ by Leray's theorem.

We may also compute the coherent sheaf cohomology of ${\displaystyle {\mathrm{\Omega }}^1}$ on the projective line ${\displaystyle {\mathbb{P}}_{\mathbb{C}}^1}$ using the Čech complex. Using the cover

${\displaystyle \mathcal{U}=\{U_1=\text{Spec}(\mathbb{C}[y]),U_2=\text{Spec}(\mathbb{C}[y^{-1}])\}}$

we have the following modules from the cotangent sheaf

${\displaystyle \begin{array}{cc}& {\mathrm{\Omega }}^1(U_1)=\mathbb{C}[y]dy\\ & {\mathrm{\Omega }}^1(U_2)=\mathbb{C}\left[y^{-1}\right]d{y}^{-1}\end{array}}$

If we take the conventions that ${\displaystyle d{y}^{-1}=-(1/y^2)dy}$ then we get the Čech complex

${\displaystyle 0\to \mathbb{C}[y]dy\oplus \mathbb{C}\left[y^{-1}\right]d{y}^{-1}\stackrel{d^0}{\to }\mathbb{C}\left[y,y^{-1}\right]dy\to 0}$

Since ${\displaystyle d^0}$ is injective and the only element not in the image of ${\displaystyle d^0}$ is ${\displaystyle y^{-1}dy}$ we get that

${\displaystyle \begin{array}{cc}& H^1({\mathbb{P}}_{\mathbb{C}}^1,{\mathrm{\Omega }}^1)\cong \mathbb{C}\\ & H^k({\mathbb{P}}_{\mathbb{C}}^1,{\mathrm{\Omega }}^1)\cong 0\text{ for }k\ne 1\end{array}}$