A short exact sequence \(0 \to \mcf' \to \mcf \to \mcf'' \to 0\) of sheaves of abelian groups gives a long exact sequence \[ 0 \to H^0(\mcf') \to H^0(\mcf) \to H^0(\mcf'') \mapsvia{\delta} H^1(\mcf') \to \cdots , \] natural in morphisms of short exact sequences.
(Long exact sequence of a short exact sequence of complexes)
Let \(0 \to C'^\bullet \xrightarrow{u} C^\bullet \xrightarrow{v} C''^\bullet \to 0\) be a short exact sequence of complexes in an abelian category. Then there are connecting morphisms \(\delta^i \colon h^i(C''^\bullet) \to h^{i+1}(C'^\bullet)\), natural in the sequence, making \[\cdots \to h^i(C'^\bullet) \to h^i(C^\bullet) \to h^i(C''^\bullet) \xrightarrow{\delta^i} h^{i+1}(C'^\bullet) \to \cdots\] exact.
For derived-functor cohomology, the proposition follows by applying the lemma to \(0 \to \Gamma(X, I'^\bullet) \to \Gamma(X, I^\bullet) \to \Gamma(X, I''^\bullet) \to 0\) for compatible injective resolutions, which remains exact because \(I'^\bullet\) is injective in each degree. For Čech cohomology of quasicoherent sheaves on a quasicompact separated scheme with a finite affine cover \(\mathfrak{U}\), the sequence \(0 \to C^\bullet(\mathfrak{U}, \mcf') \to C^\bullet(\mathfrak{U}, \mcf) \to C^\bullet(\mathfrak{U}, \mcf'') \to 0\) is exact because every finite intersection of the cover is affine and sections of quasicoherent sheaves over affines form exact sequences, and the lemma applies directly.
(What the connecting map is)
\(\delta\) sends a global section of \(\mcf''\) to the obstruction to lifting it: lift locally on a cover, take the differences of the lifts on overlaps, and read the result as a class in \(H^1(\mcf')\). So \(H^1(\mcf') = 0\) is precisely the statement that every global section of the quotient lifts.
(Standard computations)
For \(Y \subseteq \PP^n\) closed with ideal sheaf \(\mci_Y\), twist the ideal sequence: \[ 0 \to \mci_Y(d) \to \OO_{\PP^n}(d) \to \OO_Y(d) \to 0 . \] Known cohomology of the middle term plus known cohomology of the left term determines the right, which is the unknown. For a smooth hypersurface of degree \(e\), \(\mci_Y = \OO(-e)\) and everything is a binomial coefficient; this is how one gets \(h^0(\OO_Y(d))\), the Hilbert polynomial of \(Y\), and the genus of a plane curve as \(\binom{e-1}{2}\).
The other constant use is the skyscraper sequence \(0 \to \OO(D-p) \to \OO(D) \to k(p) \to 0\) on a curve, which is how \(\ell(D)\) moves by at most one when a point is added, and hence how Riemann–Roch is proved by induction.
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T-COHFINFiniteness of cohomology for coherent sheaves on a projective schemeuses