Sheaf (mathematics)
Computing sheaf cohomology

Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). For example, a partition of unity argument shows that the sheaf of smooth functions on a manifold is soft. The higher cohomology groups H i (U, F) {\displaystyle H^{i}(U,{\mathcal {F}})} for i > 0 {\displaystyle i>0} vanish for soft sheaves, which gives a way of computing cohomology of other sheaves. For example, the de Rham complex is a resolution of the constant sheaf R _ {\displaystyle {\underline {\mathbf {R} }}} on any smooth manifold, so the sheaf cohomology of R _ {\displaystyle {\underline {\mathbf {R} }}} is equal to its de Rham cohomology.

A different approach is by Čech cohomology. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations, such as computing the coherent sheaf cohomology of complex projective space P n {\displaystyle \mathbb {P} ^{n}}. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct H 1 {\displaystyle H^{1}} but incorrect higher cohomology groups. To get around this, Jean-Louis Verdier developed hypercoverings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of Pierre Deligne 's mixed Hodge structures.

Many other coherent sheaf cohomology groups are found using an embedding i: X ↪ Y {\displaystyle i:X\hookrightarrow Y} of a space X {\displaystyle X} into a space with known cohomology, such as P n {\displaystyle \mathbb {P} ^{n}}, or some weighted projective space. In this way, the known sheaf cohomology groups on these ambient spaces can be related to the sheaves i ∗ F {\displaystyle i_{*}{\mathcal {F}}}, giving H i (Y, i ∗ F) ≅ H i (X, F) {\displaystyle H^{i}(Y,i_{*}{\mathcal {F}})\cong H^{i}(X,{\mathcal {F}})}. For example, computing the coherent sheaf cohomology of projective plane curves is easily found. One big theorem in this space is the Hodge decomposition found using a spectral sequence associated to sheaf cohomology groups, proved by Deligne. Essentially, the E 1 {\displaystyle E_{1}} -page with terms

E 1 p, q = H p (X, Ω X q) {\displaystyle E_{1}^{p,q}=H^{p}(X,\Omega _{X}^{q})}

the sheaf cohomology of a smooth projective variety X {\displaystyle X}, degenerates, meaning E 1 = E ∞ {\displaystyle E_{1}=E_{\infty }}. This gives the canonical Hodge structure on the cohomology groups H k (X, C) {\displaystyle H^{k}(X,\mathbb {C})}. It was later found these cohomology groups can be easily explicitly computed using Griffiths residues. See Jacobian ideal. These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, the decomposition theorem, paving the path for Mixed Hodge modules.

Another clean approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of Lie groups. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds.

In many cases there is a duality theory for sheaves that generalizes Poincaré duality. See Grothendieck duality and Verdier duality.
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