Čech Cohomology 03: Degree Zero and Degree One
Čech cohomology / 03
Degree zero and degree one
The first two degrees contain the main local-to-global mechanism: sections glue in degree zero, and overlap errors survive in degree one, just as local potentials may fail to become one global potential.
Degree zero: compatible local sections
A \(0\)-cochain is a family
\[s_i\in\mathcal F(U_i).\]It is a cocycle when
\[s_i|_{U_{ij}}=s_j|_{U_{ij}}\]for every overlap. This is exactly the sheaf gluing condition. Hence
\[\check H^0(\mathfrak U,\mathcal F)\cong\mathcal F(X).\]This identity is the first sanity check in every computation. If your degree-zero answer is not the global sections, the cover, restrictions, or signs have been misread.
Degree one: overlap errors
A \(1\)-cochain is a family
\[a_{ij}\in\mathcal F(U_{ij}).\]It is a cocycle when, on triple overlaps,
\[a_{jk}-a_{ik}+a_{ij}=0.\]It is a coboundary if there are local sections \(b_i\in\mathcal F(U_i)\) such that
\[a_{ij}=b_j-b_i.\]Thus \(\check H^1\) measures overlap data that are locally consistent but cannot be expressed as differences of local corrections.
Patching interpretation
Suppose local objects are already chosen on each \(U_i\), but they disagree on overlaps by \(a_{ij}\). If \(a_{ij}=b_j-b_i\), changing the local choices by \(b_i\) removes the disagreement. If not, the class \([a_{ij}]\) is the obstruction.
Local potentials and closed forms
This is exactly the local-potential story from de Rham cohomology. Let \(\alpha\) be a closed \(1\)-form. On small enough open sets, the Poincare lemma gives functions \(f_i\) with
\[\alpha=df_i.\]On overlaps,
\[d(f_j-f_i)=df_j-df_i=\alpha-\alpha=0.\]Thus \(f_j-f_i\) is locally constant. The constants
\[c_{ij}=f_j-f_i\]form a Čech \(1\)-cocycle for the constant sheaf. Replacing \(f_i\) by \(f_i+b_i\) changes \(c_{ij}\) by the coboundary \(b_j-b_i\). Therefore the de Rham class of \(\alpha\) is seen in Čech language as the obstruction to patching local potentials.
Curl-free but not a global gradient
In vector calculus language, a curl-free field is locally a gradient. Around a puncture or hole, local scalar potentials may differ by constants after going around the loop. The surviving Čech \(1\)-class is the same obstruction detected by integrating the field around a closed loop.
Two-set covers
When \(X=U\cup V\), there are no triple intersections in an ordered two-set cover. Therefore every \(1\)-cochain is automatically a cocycle. The only question is whether it is a coboundary.
For a sheaf \(\mathcal F\),
\[\check H^1(\{U,V\},\mathcal F) = \frac{\mathcal F(U\cap V)} {\{s_V|_{U\cap V}-s_U|_{U\cap V}\}}.\]This formula is often the quickest way to compute examples.
Constant sheaf on a circle
Cover \(S^1\) by two arcs \(U,V\) whose intersection has two connected components \(W_1\sqcup W_2\). For the constant sheaf \(\underline{\mathbb Z}\),
\[C^1\cong\mathbb Z\oplus\mathbb Z.\]Coboundaries are diagonal pairs \((n,n)\), because a locally constant section on each arc restricts to the same integer on both components. Therefore
\[\check H^1(S^1,\underline{\mathbb Z})\cong (\mathbb Z\oplus\mathbb Z)/\Delta\mathbb Z \cong\mathbb Z.\]The surviving integer is the difference between the two overlap components.
Connected overlap gives vanishing
For the analogous two-set cover of \(S^2\) by north and south charts, \(U\cap V\) is connected. Then \(C^1\cong\mathbb Z\) and every element is a coboundary. This gives \(\check H^1(S^2,\underline{\mathbb Z})=0\) for that cover.
Practical test
To compute a degree-one class:
- Write the overlap section \(a_{ij}\).
- Check the triple-overlap cocycle equation.
- Try to solve \(a_{ij}=b_j-b_i\).
- If no solution exists, identify the invariant left after all possible choices of \(b_i\).