Čech cohomology / 07

Exact sequences and connecting maps

Exact sequences turn one sheaf-cohomology calculation into another. Čech cochains make the connecting map concrete.

The recurring obstruction pattern is now familiar: solve locally, compare on overlaps, and keep what cannot be patched away. A connecting homomorphism is the same pattern one degree higher.

Let

\[0\to\mathcal F'\to\mathcal F\to\mathcal F''\to0\]

be a short exact sequence of sheaves. A section of \(\mathcal F''\) may lift locally to \(\mathcal F\) even when it does not lift globally.

The Čech recipe

Start with a Čech cocycle

\[c\in C^q(\mathfrak U,\mathcal F'')\]

representing a class in \(H^q(X,\mathcal F'')\). Choose local lifts

\[\tilde c\in C^q(\mathfrak U,\mathcal F).\]

Because \(c\) is a cocycle, \(\delta c=0\). Applying \(\delta\) to the lift gives

\[\delta\tilde c\in C^{q+1}(\mathfrak U,\mathcal F).\]

Its image in \(\mathcal F''\) is zero, so it actually lies in \(\mathcal F'\):

\[\delta\tilde c\in C^{q+1}(\mathfrak U,\mathcal F').\]

The class

\[[\delta\tilde c]\in H^{q+1}(X,\mathcal F')\]

is the connecting image of \([c]\).

Memory hook

The connecting map is not mysterious:

  1. lift locally;
  2. apply \(\delta\);
  3. the failure to remain compatible lands in the previous sheaf;
  4. take its cohomology class.

Degree-zero version

For a global section of \(\mathcal F''\), choose local lifts \(s_i\in\mathcal F(U_i)\). On overlaps,

\[s_j-s_i\]

maps to zero in \(\mathcal F''\), so it is a section of \(\mathcal F'\). The family \(\{s_j-s_i\}\) is a Čech \(1\)-cocycle. It vanishes in cohomology exactly when the local lifts can be corrected to one global lift.

Why this is computational

Many sheaf-cohomology computations are hard directly but easy through a sequence. Divisor sequences, quotient sheaves, logarithm sequences, and restriction sequences all ask the same practical question: can local lifts be chosen compatibly?

This article provides the operation; the next articles show it in two common settings: Laurent splitting on \(\mathbb{CP}^1\) and de Rham data converted into Čech cocycles.