Sheaf cohomology / 04

Sheaves, presheaves, and locality

A sheaf is a rule for assigning objects to open sets, together with an exact test for when local data assemble into global data.

Bridge from differential forms: local primitives as sheaf data

The sheaf axioms formalize a question already present in differential forms: if an object exists locally and agrees on overlaps, does it exist globally? For a closed 1-form $\omega$, the Poincare lemma gives local functions $f_i$ with $df_i=\omega$ on sufficiently small open sets. On overlaps,

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

so the differences $f_i-f_j$ are locally constant. The obstruction to choosing the $f_i$ so that they glue is the prototype for sheaf cohomology.

Example: exact form

If $\omega=df$ globally, then each local primitive can be chosen as $f_i=f|_{U_i}$, and the overlap differences vanish. Gluing succeeds.

Example: punctured plane

For the global angular form $\alpha=(-y\,dx+x\,dy)/(x^2+y^2)$ on $\mathbb C^*$, local angle functions $\theta_i$ satisfy $d\theta_i=\alpha$. After going once around the origin the angle changes by $2\pi$. The local data are valid; the global primitive is obstructed.

Chapter 7 of the PDF begins the abstract language needed for cohomology. The point of sheaves is not abstraction for its own sake: the same two axioms simultaneously describe holomorphic functions, smooth forms, local sections of a bundle, locally constant functions, and divisor data.

Presheaves

A presheaf $\mathcal F$ of abelian groups on a topological space $X$ assigns:

to each open set $U\subset X$ an abelian group $\mathcal F(U)$;
to each inclusion $V\subset U$ a restriction map

\[\rho^U_V:\mathcal F(U)\to\mathcal F(V)\]

such that

\[\rho^U_U=\operatorname{id},\qquad \rho^V_W\circ\rho^U_V=\rho^U_W \quad(W\subset V\subset U).\]

Elements of $\mathcal F(U)$ are called sections over $U$.

Sheaf axioms

A presheaf is a sheaf if for every open cover $U=\bigcup_i U_i$:

Locality. If $s,t\in\mathcal F(U)$ and $$s

_{U_i}=t

_{U_i}$for all$i$, then$s=t$$.

Gluing. If $s_i\in\mathcal F(U_i)$ satisfy

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

for all $i,j$, then there is a unique $s\in\mathcal F(U)$ with $$s

_{U_i}=s_i$$.

The first axiom says global sections are determined locally. The second says compatible local sections actually come from a global section.

Example 1: holomorphic functions

Let $X$ be a complex manifold and set $\mathcal O_X(U)$ equal to the ring of holomorphic functions on $U$. If two holomorphic functions agree on an open cover, they agree everywhere. If holomorphic functions on the pieces agree on overlaps, they glue to a holomorphic function because holomorphicity is local. Thus $\mathcal O_X$ is a sheaf of rings.

Example 2: smooth differential forms

The assignment

\[U\mapsto \mathcal A^k_X(U)\]

of smooth $k$-forms is a sheaf. A form is a collection of smooth coefficient functions in local coordinates, and smoothness can be tested locally. This sheaf becomes central when fine resolutions compute de Rham cohomology.

What fails for a presheaf

Some natural assignments have restrictions but fail gluing.

Non-example 1: bounded holomorphic functions

Let $\mathcal B(U)$ be bounded holomorphic functions on $U$. Restrictions exist. However, take $U=\mathbb C$ and cover it by disks $U_n=\{|z|<n\}$. The function $f_n(z)=z$ is bounded on each $U_n$, and these local sections agree on overlaps. They glue to $f(z)=z$ on $\mathbb C$, but $z$ is not bounded on $\mathbb C$. Therefore $\mathcal B$ is not a sheaf.

Warning: local logarithms are a cohomological obstruction, not a sheaf-axiom failure

On $\mathbb C^*$, the function $z$ has holomorphic logarithms on simply connected sectors, but no global holomorphic logarithm. This does not mean that $\mathcal O$ fails the sheaf axiom: if local branches of $\log z$ agreed on overlaps, they would glue. The obstruction is that the branches differ by locally constant multiples of $2\pi i$ around the origin, so no compatible family of local logarithms exists.

Sheaves of sections

Every vector bundle $E\to X$ determines a sheaf $\mathcal E$:

\[\mathcal E(U)=\{\text{holomorphic sections }U\to E\}.\]

For a line bundle $L$ represented by transition functions $g_{ij}$, a section over $U$ is a family of holomorphic functions $s_i$ on $U\cap U_i$ satisfying the induced compatibility relation from the transition functions. This is why line bundles and sheaves are equivalent languages for locally free rank-one objects.

Example 3: sections of $\mathcal O(1)$

For $\mathbb{CP}^1=U_0\cup U_\infty$, a section of $\mathcal O(1)$ is a pair

\[s_0(z),\qquad s_\infty(w)\]

with overlap relation $s_\infty(w)=z^{-1}s_0(z)=w\,s_0(z)$ in the section convention of Article 01, or the reciprocal relation if the frames are reversed. The sheaf condition ensures that compatible local expressions define one global section.

Example 4: locally constant functions

Let $\underline{\mathbb Z}$ be the sheaf of locally constant integer-valued functions. On a connected open set, its sections are constant integers. On a disconnected open set, a section may take different integers on different components. This example is small but important: it remembers topology rather than analytic regularity.

Why sheaves enter cohomology

The sheaf axioms make degree zero local-to-global behavior exact. Higher cohomology measures the failure of local compatible data of higher degree to be globally trivial. Thus sheaf cohomology is not an auxiliary formalism; it is the systematic measurement of gluing obstructions.