Čech cohomology / 09

De Rham to Čech computation

To convert de Rham data into Čech data, choose local potentials and record their overlap differences.

This article returns to the starting intuition and turns it into a concrete calculation. A closed form gives local solutions. The Čech cocycle records the failure of those local solutions to patch.

Closed one-forms

Let \(\alpha\) be a closed \(1\)-form. Choose a good cover \(\{U_i\}\). By local exactness, pick functions \(f_i\) with

\[\alpha=df_i\]

on each \(U_i\). On overlaps,

\[d(f_j-f_i)=0.\]

Therefore

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

is locally constant. The family \(c=\{c_{ij}\}\) is a Čech \(1\)-cocycle for the constant sheaf.

Exactness as a coboundary test

If constants \(b_i\) exist with

\[c_{ij}=b_j-b_i,\]

then the corrected local potentials \(f_i-b_i\) glue to a global function \(f\) with \(df=\alpha\). If no such constants exist, \(\alpha\) is closed but not exact, and \([c_{ij}]\) is the Čech representative of the obstruction.

Vector calculus translation

For \(H^1\), read this as:

\[\nabla\times F=0 \quad\Rightarrow\quad F=\nabla f_i\text{ locally}.\]

The failure of the scalar potentials \(f_i\) to patch is the same obstruction detected by integrating \(F\) around closed loops.

Closed two-forms and divergence-free fields

In three-dimensional vector calculus, a closed \(2\)-form corresponds to a divergence-free field \(B\). Locally one may write

\[B=\nabla\times A_i.\]

On overlaps,

\[\nabla\times(A_j-A_i)=0,\]

so \(A_j-A_i\) is locally a gradient. The next layer of local scalar potentials produces higher overlap data. The same ladder appears again: local solve, overlap difference, obstruction.

Practical workflow

To compute a de Rham class by Čech data:

  1. choose a cover where local primitives are easy;
  2. solve the differential equation locally;
  3. subtract local solutions on overlaps;
  4. check the Čech cocycle condition;
  5. quotient by changes of local primitives;
  6. interpret the surviving class as periods, flux, or a sheaf-cohomology class.

Punctured-plane pattern

The angular form is locally the differential of an angle function. Different branches of the angle differ by constants on overlaps. Going around the puncture changes the branch by a period. Čech records the constant jumps.