Sheaf Cohomology 05: Morphisms, Stalks, and Exactness

Sheaf cohomology / 05

Morphisms, stalks, and exactness

Sheaf exactness is tested on germs, not on global sections. This distinction is the source of cohomology.

Sections 7.3-7.5 of the PDF introduce morphisms, exact sequences, stalks, and examples. The main lesson is that sheaves are local objects, while global sections are not an exact functor.

Morphisms

A morphism of sheaves \(\varphi:\mathcal F\to\mathcal G\) is a compatible family of maps

\[\varphi_U:\mathcal F(U)\to\mathcal G(U)\]

for all open \(U\subset X\), satisfying

\[\rho^U_V\circ\varphi_U=\varphi_V\circ\rho^U_V \quad(V\subset U).\]

Kernels are computed sectionwise:

\[(\ker\varphi)(U)=\ker(\varphi_U).\]

Images require care: the sectionwise image presheaf may fail to be a sheaf, so the image sheaf is the sheafification of that presheaf.

Stalks and germs

The stalk of \(\mathcal F\) at \(p\in X\) is the direct limit

\[\mathcal F_p=\varinjlim_{p\in U}\mathcal F(U).\]

An element of \(\mathcal F_p\) is a germ: two local sections represent the same germ if they agree on some smaller neighborhood of \(p\).

Example 1: holomorphic function germs

The stalk \(\mathcal O_{\mathbb C,0}\) consists of convergent power series

\[a_0+a_1z+a_2z^2+\cdots\]

near \(0\). A holomorphic function on a disk and its restriction to a smaller disk define the same germ. The maximal ideal consists of germs vanishing at \(0\).

Example 2: skyscraper sheaf

Fix \(p\in X\) and an abelian group \(A\). The skyscraper sheaf \(A_p\) is defined by

\[A_p(U)= \begin{cases} A,& p\in U,\\ 0,& p\notin U. \end{cases}\]

Its stalk at \(p\) is \(A\), while every other stalk is \(0\). It models data concentrated at a single point, such as the quotient \(\mathcal O_p/\mathfrak m_p\).

Exactness is stalkwise

A sequence of sheaves

\[\mathcal F'\xrightarrow{\alpha}\mathcal F\xrightarrow{\beta}\mathcal F''\]

is exact if for every point \(p\),

\[\operatorname{im}(\alpha_p)=\ker(\beta_p)\]

inside the stalk \(\mathcal F_p\). This is equivalent to exactness after restricting sufficiently near every point.

Why stalks are the correct test

Suppose a local section \(s\in\mathcal F(U)\) maps to zero in \(\mathcal F''(U)\). Exactness should mean that near each point \(p\in U\), the germ \(s_p\) is locally the image of a section of \(\mathcal F'\). There may be no single global preimage on all of \(U\). Stalks encode exactly this local lifting condition.

Global sections are only left exact

The functor \(\Gamma(X,-)\) sends a sheaf to its group of global sections. From a short exact sequence

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

one always gets an exact sequence

\[0\to\Gamma(X,\mathcal F')\to\Gamma(X,\mathcal F)\to\Gamma(X,\mathcal F'').\]

Surjectivity on global sections can fail. Sheaf cohomology measures this failure.

Example 3: the exponential map on \(\mathbb C^*\)

The morphism

\[\mathcal O_{\mathbb C^*}\xrightarrow{\exp(2\pi i\,\cdot)}\mathcal O_{\mathbb C^*}^*\]

is locally surjective: every nowhere-zero holomorphic function has a local normalized logarithm. But the global section \(z\in\mathcal O^*(\mathbb C^*)\) has no global normalized logarithm, equivalently no global preimage under \(f\mapsto\exp(2\pi i f)\). Thus local surjectivity does not imply global surjectivity.

Example 4: an effective divisor exact sequence

For a point \(p\) on a Riemann surface there is a short exact sequence

\[0\to\mathcal O(-p)\to\mathcal O\to\mathbb C_p\to0,\]

where the last map evaluates a germ at \(p\). On global sections over a compact connected Riemann surface, constants map onto \(\mathbb C\), so this example is globally exact. But replacing \(\mathcal O\) by a more restrictive line bundle can make the evaluation map fail to be surjective, and the obstruction appears in \(H^1\).

Constant sheaf and constant presheaf

The constant sheaf \(\underline A\) consists of locally constant \(A\)-valued functions. It is not the same as the presheaf assigning \(A\) to every nonempty open set. On a disconnected open set \(U=U_1\sqcup U_2\), a locally constant section can choose independent values on the two components, so

\[\underline A(U)\cong A\times A.\]

This distinction is a basic example of sheafification: one repairs a presheaf by adding the local gluing behavior it lacks.