Sheaf Cohomology 09: Long Exact Sequences

Sheaf cohomology / 09

Long exact sequences

A short exact sequence of sheaves is local; its long exact cohomology sequence records the global cost of lifting sections.

Bridge from Stokes: obstruction propagation

Stokes’ theorem explains how an interior derivative produces a boundary term. A long exact sequence explains how a local lifting problem produces a cohomology obstruction one degree higher. In both cases, the point is not the formal symbol but the bookkeeping of what remains after local cancellations.

Example: integration by parts

The formula

\[\int_a^b u\,v'\,dx=[uv]_a^b-\int_a^b v\,u'\,dx\]

moves a derivative and creates a boundary term. The connecting homomorphism in a long exact sequence moves a lifting problem and creates an obstruction class.

Example: local lifts

If a section of \(\mathcal F''\) lifts locally to \(\mathcal F\), the differences of those local lifts on overlaps land in \(\mathcal F'\). Their Cech class is the boundary term of the lifting problem.

Chapter 11 of the PDF develops the long exact sequence in cohomology. This is the main computational tool of sheaf cohomology: once a sheaf is related to simpler sheaves, its cohomology can often be read from the exact sequence.

The theorem

Given a short exact sequence of sheaves

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

there is a natural long exact sequence

\[\begin{aligned} 0&\to H^0(X,\mathcal F')\to H^0(X,\mathcal F)\to H^0(X,\mathcal F'')\\ &\xrightarrow{\delta}H^1(X,\mathcal F')\to H^1(X,\mathcal F)\to H^1(X,\mathcal F'')\\ &\xrightarrow{\delta}H^2(X,\mathcal F')\to\cdots . \end{aligned}\]

The connecting homomorphism \(\delta\) sends an object that is locally liftable to the obstruction to lifting it globally.

Cech construction of the connecting map

Let \(s''\in H^0(X,\mathcal F'')\). Choose a cover \(\{U_i\}\) and local lifts \(s_i\in\mathcal F(U_i)\). On overlaps,

\[s_j-s_i\]

maps to zero in \(\mathcal F^{\prime\prime}\), so it comes from a section \(a_{ij}\in\mathcal F^{\prime}(U_i\cap U_j)\). The family \(a_{ij}\) is a Cech 1-cocycle. Its class is

\[\delta(s'')\in H^1(X,\mathcal F').\]

Changing the local lifts changes \(a_{ij}\) by a coboundary, so the cohomology class is well-defined.

Example 1: exponential sequence

For

\[0\to\mathbb Z\to\mathcal O_X\to\mathcal O_X^*\to0,\]

the connecting map sends a nowhere-zero holomorphic function to its logarithm monodromy, and sends a line bundle class to its first Chern class one degree later. The obstruction to choosing logarithms globally becomes an integral cohomology class.

Example 2: evaluating at a divisor

For a line bundle \(L\) and an effective divisor \(D\) there is an exact sequence

\[0\to L(-D)\to L\to L|_D\to0.\]

The connecting map

\[H^0(D,L|_D)\to H^1(X,L(-D))\]

measures whether prescribed jets along \(D\) extend to global sections of \(L\). This is the cohomological form of interpolation with constraints.

Naturality

A morphism between short exact sequences induces a morphism between their long exact cohomology sequences, and the squares commute. This matters because many computations compare an unknown sheaf with a model sheaf by restriction, tensoring, or pullback. The connecting maps are compatible with those comparisons.

Proof sketch

Using injective resolutions, sheaf cohomology is a derived functor, and derived functors of a left exact functor produce long exact sequences from short exact sequences. In the Cech model, the construction above gives the connecting map explicitly; exactness follows by checking that a cocycle vanishes precisely when the preceding lifting problem can be solved.

Mayer-Vietoris as an exact sequence

If \(X=U\cup V\) and \(\mathcal F\) is computed by an acyclic resolution \(\mathcal A^\bullet\) whose restriction-difference maps are surjective, then one has a short exact sequence of complexes

\[0\to \mathcal A^\bullet(X)\to \mathcal A^\bullet(U)\oplus\mathcal A^\bullet(V) \to\mathcal A^\bullet(U\cap V)\to0.\]

For smooth forms, and more generally for fine resolutions used in differential geometry, this surjectivity comes from partitions of unity. The associated long exact sequence is the Mayer-Vietoris sequence.

Example 3: computing \(H^1(S^1,\mathbb R)\)

Cover \(S^1\) by two arcs \(U,V\) with two overlap components. Since arcs are contractible, their first cohomology vanishes. The Mayer-Vietoris boundary identifies the one-dimensional difference between the two overlap components, giving

\(H^1(S^1,\mathbb R)\cong\mathbb R.\)

Example 4: vanishing on \(\mathbb{CP}^1\) from a divisor sequence

The sequence

\[0\to\mathcal O(n-1)\to\mathcal O(n)\to\mathcal O(n)|_p\to0\]

relates sections of consecutive line bundles. Starting from \(H^1(\mathbb{CP}^1,\mathcal O(-1))=0\) and using induction, one obtains the familiar dimensions of \(H^0(\mathcal O(n))\) for \(n\ge0\).

Exactness as a diagnostic

When a computation seems wrong, locate the term that should be the kernel of the next map and the image of the previous map. The long exact sequence is not only a theorem; it is a consistency check for dimensions, maps, and obstruction classes.