Sheaf cohomology / 06

Cech cohomology

Cech cochains are local data on intersections; their coboundary measures whether the data glue, split, or obstruct.

Bridge from forms: Cech cocycles are failed gluing records

The de Rham example of a closed non-exact form already contains a Cech 1-cocycle. Choose local primitives \(f_i\) for a closed 1-form \(\omega\) on a cover with connected overlaps. On overlaps, the locally constant differences

\[c_{ij}=f_j-f_i\]

satisfy \(c_{ij}+c_{jk}+c_{ki}=0\) on triple overlaps. Changing the primitives changes \(c_{ij}\) by a coboundary. Thus the Cech class records exactly the failure of local primitives to glue.

Example: the circle

On \(S^1\), local angle coordinates glue up to integer multiples of \(2\pi\). Those locally constant jumps form a nontrivial Cech class, the same class detected by integrating the invariant angular form, usually written \(d\theta\), around the circle.

Example: exact forms give coboundaries

If \(\omega=df\) globally, take \(f_i=f|_{U_i}\). Then \(c_{ij}=0\). If different local primitives are chosen, the resulting cocycle is a coboundary, so its cohomology class still vanishes.

Chapter 8 of the PDF develops Cech cohomology. It is the most computational form of sheaf cohomology because every cochain has a visible support: open sets, double overlaps, triple overlaps, and so on.

Cochains for an open cover

Let \(\mathfrak U=\{U_i\}_{i\in I}\) be an open cover of \(X\) and let \(\mathcal F\) be a sheaf of abelian groups. A \(q\)-cochain assigns a section to each nonempty \((q+1)\)-fold intersection:

\[C^q(\mathfrak U,\mathcal F)= \prod_{i_0<\cdots<i_q} \mathcal F(U_{i_0}\cap\cdots\cap U_{i_q}).\]

The coboundary \(\delta:C^q\to C^{q+1}\) is

\[(\delta c)_{i_0\cdots i_{q+1}} =\sum_{r=0}^{q+1}(-1)^r c_{i_0\cdots\widehat{i_r}\cdots i_{q+1}} \big|_{U_{i_0}\cap\cdots\cap U_{i_{q+1}}}.\]

Then \(\delta^2=0\), so one defines

\[\check H^q(\mathfrak U,\mathcal F)= \ker(\delta:C^q\to C^{q+1})/ \operatorname{im}(\delta:C^{q-1}\to C^q).\]

Proof sketch of \(\delta^2=0\)

Each term in \(\delta(\delta c)\) omits two indices. Omitting \(i_r\) and then \(i_s\) occurs with the opposite sign from omitting \(i_s\) and then \(i_r\). The restrictions land in the same intersection, so the two terms cancel.

Degree zero

A 0-cochain is a family \(s_i\in\mathcal F(U_i)\). It is a 0-cocycle when

\[s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}.\]

By the sheaf gluing axiom, this is precisely a global section. Therefore

\[\check H^0(\mathfrak U,\mathcal F)\cong\Gamma(X,\mathcal F).\]

Degree one

A 1-cochain is a family \(a_{ij}\in\mathcal F(U_i\cap U_j)\). It is a cocycle when

\[a_{jk}-a_{ik}+a_{ij}=0\]

on triple intersections, using additive notation. It is a coboundary if there are \(b_i\in\mathcal F(U_i)\) such that

\[a_{ij}=b_j-b_i.\]

Thus \(\check H^1\) measures compatible overlap data that cannot be obtained as differences of local data.

Example 1: line bundles as multiplicative 1-cocycles

For \(\mathcal O_X^*\), Cech addition is multiplication. A line bundle is represented by functions \(g_{ij}\in\mathcal O^*(U_i\cap U_j)\) satisfying

\[g_{ij}g_{jk}g_{ki}=1.\]

A change of local frame multiplies the cocycle by a coboundary factor. With the additive sign convention above, the multiplicative coboundary of local units \(h_i\) is \(h_jh_i^{-1}\); with the frame convention used in Article 01 the transition functions are multiplied by its inverse \(h_i h_j^{-1}\). These generate the same subgroup of coboundaries, but the inverse matters when comparing formulas term by term. Therefore line bundles on \(X\) are classes in \(\check H^1(\mathfrak U,\mathcal O^*)\), and after refinement in \(H^1(X,\mathcal O^*)\).

Example 2: \(S^1\) and the constant sheaf \(\mathbb Z\)

Cover \(S^1\) by two arcs \(U,V\) whose intersection has two connected components. A 1-cochain for \(\underline{\mathbb Z}\) assigns one integer to each component of \(U\cap V\). Coboundaries can shift both components by the same difference of local constants, so the difference between the two integers remains. Hence

\[\check H^1(S^1,\underline{\mathbb Z})\cong\mathbb Z.\]

This integer is the winding obstruction.

Refinements and direct limits

Cohomology computed from one cover can miss information if the cover is too coarse or intersections are not simple. A refinement \(\mathfrak V\to\mathfrak U\) induces maps

\[\check H^q(\mathfrak U,\mathcal F)\to \check H^q(\mathfrak V,\mathcal F).\]

The Cech cohomology of \(X\) is obtained by taking the direct limit over covers:

\[\check H^q(X,\mathcal F)=\varinjlim_{\mathfrak U} \check H^q(\mathfrak U,\mathcal F).\]

For good covers and many geometric sheaves, carefully chosen covers already compute the correct sheaf cohomology.

Example 3: the two-chart cover of \(\mathbb{CP}^1\)

Let \(U_0=\mathbb C\) and \(U_\infty=\mathbb C\) with overlap \(\mathbb C^*\). In the convention of Article 01, a multiplicative Cech 1-cocycle \(z^{-n}\) represents \(\mathcal O(n)\); the reciprocal convention uses \(z^n\). The integer \(n\) is recovered, up to that convention sign, by winding around the overlap annulus. Refining the cover does not remove this class because it records global degree.

Example 4: additive splitting on Stein intersections

For the sheaf \(\mathcal O\) on \(\mathbb{CP}^1\) with the standard two-chart cover, a 1-cocycle is a Laurent series on \(\mathbb C^*\). Terms with nonnegative powers extend to \(U_0\); terms with negative powers extend to \(U_\infty\). The cocycle splits, giving \(H^1(\mathbb{CP}^1,\mathcal O)=0\). For \(\mathcal O(-2)\), a residue-like term survives, giving a one-dimensional \(H^1\).

Interpretation

The Cech complex is a bookkeeping device for local-to-global failure. Degree zero asks whether local sections glue. Degree one asks whether overlap corrections are induced by local choices. Higher degrees repeat this question on higher intersections.