Sheaf Cohomology 07: Sheaf Cohomology and Acyclic Resolutions

Sheaf cohomology / 07

Sheaf cohomology and acyclic resolutions

Sheaf cohomology is the derived obstruction to taking global sections; acyclic resolutions make it computable.

Bridge from de Rham: derived functors measure failed exactness

In positive-degree de Rham cohomology, a closed form is locally exact but may not be globally exact. In sheaf language, the functor of global sections preserves kernels but not necessarily cokernels. Sheaf cohomology is the systematic derived measurement of that failure.

Example: closed forms

The sequence

\[\mathcal A^0\xrightarrow{d}\mathcal A^1\xrightarrow{d}\mathcal A^2\]

is locally exact at \(\mathcal A^1\) by the Poincare lemma, but global exactness can fail. The quotient of global closed 1-forms by global exact 1-forms is \(H^1_{\mathrm{dR}}(X)\).

Example: global sections as the fragile step

Local equations can often be solved after shrinking the open set. Passing to all of \(X\) at once is the non-exact operation. Derived functors isolate exactly what is lost at that global step.

Chapter 9 of the PDF explains how Cech cohomology connects to sheaf cohomology. The conceptual definition uses derived functors, while practical computations use resolutions whose higher cohomology vanishes.

Global sections are left exact

The global-section functor

\[\Gamma(X,-):\mathcal F\mapsto\mathcal F(X)\]

is left exact. For a short exact sequence

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

one obtains

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

but the final map may not be surjective. The right derived functors of \(\Gamma(X,-)\) are sheaf cohomology:

\[R^q\Gamma(X,\mathcal F)=H^q(X,\mathcal F).\]

Injective resolutions

Choose an injective resolution

\[0\to\mathcal F\to\mathcal I^0\to\mathcal I^1\to\mathcal I^2\to\cdots.\]

Apply global sections:

\[0\to\Gamma(X,\mathcal I^0)\to \Gamma(X,\mathcal I^1)\to \Gamma(X,\mathcal I^2)\to\cdots.\]

Then \(H^q(X,\mathcal F)\) is the cohomology of this complex. This definition is canonical but usually not computational, because injective sheaves are large and abstract.

Why derived functors are needed

If global sections were exact, all gluing problems would be solved by local solutions. The exponential sequence and the logarithm on \(\mathbb C^*\) already show this is false. Derived functors systematically record the successive failures of exactness.

Acyclic resolutions

A sheaf \(\mathcal A\) is \(\Gamma\)-acyclic if

\[H^q(X,\mathcal A)=0\qquad(q>0).\]

If \(\mathcal F\) has an exact resolution by acyclic sheaves,

\[0\to\mathcal F\to\mathcal A^0\to\mathcal A^1\to\cdots,\]

then \(H^q(X,\mathcal F)\) is computed by the global-section complex

\[0\to\Gamma(X,\mathcal A^0)\to\Gamma(X,\mathcal A^1)\to\cdots.\]

This is the engine behind de Rham and Dolbeault cohomology.

Example 1: de Rham resolution

On a smooth manifold,

\[0\to\underline{\mathbb R}\to \mathcal A^0\xrightarrow{d}\mathcal A^1\xrightarrow{d} \mathcal A^2\xrightarrow{d}\cdots\]

is exact by the Poincare lemma, and the sheaves \(\mathcal A^k\) are fine, hence acyclic. Therefore

\[H^q(X,\underline{\mathbb R}) \cong H^q(\Gamma(X,\mathcal A^\bullet),d),\]

which is de Rham cohomology.

Example 2: Dolbeault resolution

On a complex manifold, the sheaf of holomorphic functions fits into

\[0\to\mathcal O_X\to \mathcal A^{0,0}\xrightarrow{\bar\partial} \mathcal A^{0,1}\xrightarrow{\bar\partial} \mathcal A^{0,2}\to\cdots.\]

The local exactness is the \(\bar\partial\)-Poincare lemma. The sheaves of smooth forms are fine, so

\(H^q(X,\mathcal O_X) \cong H^{0,q}_{\bar\partial}(X).\)

Leray covers and Cech comparison

An open cover \(\mathfrak U\) is acyclic for \(\mathcal F\) if every finite intersection \(U_{i_0}\cap\cdots\cap U_{i_q}\) has

\[H^p(U_{i_0}\cap\cdots\cap U_{i_q},\mathcal F)=0 \qquad(p>0).\]

For such a cover,

\[\check H^q(\mathfrak U,\mathcal F)\cong H^q(X,\mathcal F).\]

This is often called Leray’s theorem. It justifies doing sheaf cohomology by overlap calculations when the cover is chosen well.

Example 3: coherent sheaves on Stein covers

For a coherent analytic sheaf on a complex manifold, choose a Stein cover whose finite intersections are Stein, for instance sufficiently small coordinate polydiscs in a good cover. Cartan acyclicity on those intersections lets this cover compute sheaf cohomology by Cech cochains. This is why local analytic coordinates can compute global holomorphic invariants.

Example 4: a good cover of a circle

For the constant sheaf \(\underline{\mathbb Z}\) on \(S^1\), a cover by two arcs with disconnected intersection is not a good cover. Refining to three arcs with contractible intersections gives the usual cellular cohomology. Both approaches can compute \(H^1(S^1,\mathbb Z)\), but the good-cover method matches the abstract theorem directly.

Fine, soft, and flabby sheaves

The PDF lists three common acyclicity conditions:

Type	Useful property	Typical example
fine	partitions of unity act on sections	smooth functions, smooth forms
soft	sections over closed sets extend	smooth functions on paracompact manifolds
flabby	all restriction maps are surjective	sheaves of discontinuous functions

Fine sheaves are central in geometry because partitions of unity let one patch local solutions while controlling support.