Inmathematics, specificallyalgebraic topology,Čech cohomology is acohomology theory based on the intersection properties ofopencovers of atopological space. It is named for the mathematicianEduard Čech.

LetX be a topological space, and let${\displaystyle \mathcal{U}}$ be an open cover ofX. Let${\displaystyle N(\mathcal{U})}$ denote thenerve of the covering. The idea of Čech cohomology is that, for an open cover${\displaystyle \mathcal{U}}$ consisting of sufficiently small open sets, the resulting simplicial complex${\displaystyle N(\mathcal{U})}$ should be a good combinatorial model for the spaceX. For such a cover, the Čech cohomology ofX is defined to be thesimplicialcohomology of the nerve. This idea can be formalized by the notion of agood cover. However, a more general approach is to take thedirect limit of the cohomology groups of the nerve over the system of all possible open covers ofX, ordered byrefinement. This is the approach adopted below.

LetX be atopological space, and let${\displaystyle \mathcal{F}}$ be apresheaf ofabelian groups onX. Let${\displaystyle \mathcal{U}}$ be anopen cover ofX.

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

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

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

Theboundary 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 thefree abelian group spanned by the simplices of${\displaystyle \mathcal{U}}$.

Aq-cochain of${\displaystyle \mathcal{U}}$ with coefficients in${\displaystyle \mathcal{F}}$ is a map which associates with eachq-simplex σ an element of${\displaystyle \mathcal{F}(|\sigma |)}$, and we denote the set of allq-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.

The cochain groups can be made into acochain complex${\displaystyle (C^{\bullet }(\mathcal{U},\mathcal{F}),\delta )}$ by defining thecoboundary 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 therestriction 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${\displaystyle {\delta }_{q+1}\circ {\delta }_q=0.}$

Thecoboundary operator is analogous to theexterior derivative ofDe Rham cohomology, so it sometimes called the differential of thecochain complex.

Aq-cochain is called aq-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 allq-cocycles.

Thus a (q−1)-cochain${\displaystyle f}$ is a cocycle if for allq-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${\displaystyle A,B,C\in \mathcal{U}}$

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

Aq-cochain is called aq-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 allq-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${\displaystyle A,B\in \mathcal{U}}$

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

TheČech 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 theqth 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 ofX is defined by consideringrefinements 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 ofX form adirected set under refinement, so the above map leads to adirect system of abelian groups. TheČech cohomology ofX with values in${\displaystyle \mathcal{F}}$ is defined as thedirect limit${\displaystyle \check{H}(X,\mathcal{F}):=\underset{\mathcal{U}}{\underrightarrow{\lim }}\check{H}(\mathcal{U},\mathcal{F})}$ of this system.

The Čech cohomology ofX with coefficients in a fixed abelian groupA, denoted${\displaystyle \check{H}(X;A)}$, is defined as${\displaystyle \check{H}(X,{\mathcal{F}}_A)}$ where${\displaystyle {\mathcal{F}}_A}$ is theconstant sheaf onX determined byA.

A variant of Čech cohomology, callednumerable Čech cohomology, is defined as above, except that all open covers considered are required to benumerable: that is, there is apartition of unity {ρ_i} such that each support${\displaystyle \{x\mid {\rho }_i(x)>0\}}$ is contained in some element of the cover. IfX isparacompact andHausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.

IfX ishomotopy equivalent to aCW complex, then the Čech cohomology${\displaystyle {\check{H}}^{\ast }(X;A)}$ isnaturally isomorphic to thesingular cohomology${\displaystyle H^{\ast }(X;A)}$. IfX is adifferentiable manifold, then${\displaystyle {\check{H}}^{\ast }(X;\mathbb{R})}$ is also naturally isomorphic to thede 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 ifX is theclosed topologist's sine curve, then${\displaystyle {\check{H}}^1(X;\mathbb{Z})=\mathbb{Z},}$ whereas${\displaystyle H^1(X;\mathbb{Z})=0.}$

IfX is a differentiable manifold and the cover${\displaystyle \mathcal{U}}$ ofX is a "good cover" (i.e. all the setsU_α arecontractible 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.

IfX is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic toAlexander-Spanier cohomology.

For a presheaf${\displaystyle \mathcal{F}}$ onX, let${\displaystyle {\mathcal{F}}^{+}}$ denote itssheafification. Then we have a natural comparison map

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

from Čech cohomology tosheaf cohomology. IfX is paracompact Hausdorff, then$\chi$ is an isomorphism. More generally,$\chi$ is an isomorphism whenever the Čech cohomology of all presheaves onX with zero sheafification vanishes.^[2]

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

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

where thecolimit runs over all coverings (with respect to the chosen topology) ofX. Here${\displaystyle {\check{H}}^n(\mathcal{U},\mathcal{F})}$ is defined as above, except that ther-fold intersections of open subsets inside the ambient topological space are replaced by ther-foldfiber 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 degreesn = 0 and 1, but may fail to be so in general. For theZariski topology on aNoetherianseparated scheme, Čech and sheaf cohomology agree for anyquasi-coherent sheaf. For theétale topology, the two cohomologies agree for any étale sheaf onX, provided that any finite set of points ofX are contained in some open affine subscheme. This is satisfied, for example, ifX isquasi-projective over anaffine scheme.^[3]

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

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

A hypercoveringK_∗ ofX is a certainsimplicial object inC, i.e., a collection of objectsK_n together with boundary and degeneracy maps. Applying a sheaf${\displaystyle \mathcal{F}}$ toK_∗ yields asimplicial abelian group$\mathcal{F}(K_{\ast })$ whosen-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 caseK_∗ 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.^[4]

The most basic example of Čech cohomology is given by the case where the presheaf${\displaystyle \mathcal{F}}$ is aconstant 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

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:

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 agood cover of${\displaystyle X}$, we have${\displaystyle {\check{H}}^1(X,\mathbb{R})\cong \mathbb{R}}$ byLeray'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

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

• Bott, Raoul; Loring Tu (1982).Differential Forms in Algebraic Topology. Springer.ISBN 0-387-90613-4.
• Hatcher, Allen (2002).Algebraic Topology(PDF). Cambridge University Press.ISBN 0-521-79540-0.
• Wells, Raymond (1980)."2. Sheaf Theory: Appendix A. Cech Cohomology with Coefficients in a Sheaf".Differential Analysis on Complex Manifolds. Springer. pp. 63–64.doi:10.1007/978-1-4757-3946-6_2.ISBN 978-3-540-90419-9.

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