Čech cohomology

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 ${\displaystyle \sigma =(U_{i})_{i\in \{0,\ldots ,q\}}}$ 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:

${\displaystyle \partial _{j}\sigma$ :=(U_{i})_{i\in \{0,\ldots ,q\}\setminus \{j\}}.}

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

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

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

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.

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:

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

where ${\displaystyle \mathrm {res} _{|\sigma |}^{|\partial _{j}\sigma |}}$ is the restriction morphism from ${\displaystyle {\mathcal {F}}(|\partial _{j}\sigma |)}$ 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.

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

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

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 ${\displaystyle A,B,C\in {\mathcal {U}}}$

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

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

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

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 ${\displaystyle {\check {H}}^{*}({\mathcal {U}},{\mathcal {F}})\to {\check {H}}^{*}({\mathcal {V}},{\mathcal {F}}).}$ 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 ${\displaystyle {\check {H}}(X,{\mathcal {F}}):=\varinjlim _{\mathcal {U}}{\check {H}}({\mathcal {U}},{\mathcal {F}})}$ 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.

For example, we can compute the coherent sheaf cohomology of ${\displaystyle \Omega ^{1}}$ 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

${\displaystyle {\begin{aligned}&\Omega ^{1}(U_{1})=\mathbb {C} [y]dy\\&\Omega ^{1}(U_{2})=\mathbb {C} \left[y^{-1}\right]dy^{-1}\end{aligned}}}$

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

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

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

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

• 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.