Editing Sheaf (mathematics) (section)

== Sheaf cohomology ==
{{Main|Sheaf cohomology}}

In contexts where the open set <math>U</math> is fixed, and the sheaf is regarded as a variable, the set <math>F(U)</math> is also often denoted <math>\Gamma(U, F).</math>
As was noted above, this functor does not preserve epimorphisms. Instead, an epimorphism of sheaves <math>\mathcal F \to \mathcal G</math> is a map with the following property: for any section <math>g \in \mathcal G(U)</math> there is a covering <math>\mathcal{U} = \{U_i\}_{i \in I}</math> where <blockquote><math>U = \bigcup_{i \in I} U_i</math> </blockquote>of open subsets, such that the restriction <math>g|_{U_i}</math> are in the image of <math>\mathcal F(U_i)</math>. However, <math>g</math> itself need not be in the image of <math>\mathcal F(U)</math>. A concrete example of this phenomenon is the exponential map 
:<math>\mathcal O \stackrel{\exp} \to \mathcal O^\times</math>
between the sheaf of [[holomorphic function]]s and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function <math>g</math> (on some open subset in <math>\C</math>, say), admits a [[complex logarithm]] ''locally'', i.e., after restricting <math>g</math> to appropriate open subsets. However, <math>g</math> need not have a logarithm globally.

Sheaf cohomology captures this phenomenon. More precisely, for an [[exact sequence]] of sheaves of abelian groups
:<math>0 \to \mathcal F_1 \to \mathcal F_2 \to \mathcal F_3 \to 0,</math>
(i.e., an epimorphism <math>\mathcal F_2 \to \mathcal F_3</math> whose kernel is <math>\mathcal F_1</math>), there is a long exact sequence<math display="block">0 \to \Gamma(U, \mathcal F_1) \to \Gamma(U, \mathcal F_2) \to \Gamma(U, \mathcal F_3) \to H^1(U, \mathcal F_1) \to H^1(U, \mathcal F_2) \to H^1(U, \mathcal F_3) \to H^2(U, \mathcal F_1) \to \dots</math>By means of this sequence, the first cohomology group <math>H^1(U, \mathcal F_1)</math> is a measure for the non-surjectivity of the map between sections of <math>\mathcal F_2</math> and <math>\mathcal F_3</math>.

There are several different ways of constructing sheaf cohomology. {{harvtxt|Grothendieck|1957}} introduced them by defining sheaf cohomology as the [[derived functor]] of <math>\Gamma</math>. This method is theoretically satisfactory, but, being based on [[injective resolution]]s, of little use in concrete computations. [[Godement resolution]]s are another general, but practically inaccessible approach.

=== Computing sheaf cohomology ===
Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by [[soft sheaf|soft sheaves]], [[fine sheaf|fine sheaves]], and [[flabby sheaf|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 <math>H^i(U, \mathcal F)</math> for <math>i > 0</math> 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 <math>\underline{\mathbf{R}}</math> on any smooth manifold, so the sheaf cohomology of <math>\underline{\mathbf{R}}</math> 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 <math>\mathbb{P}^n</math>.<ref>Hartshorne (1977), Theorem III.5.1.</ref> 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 <math>H^1</math> but incorrect higher cohomology groups. To get around this, [[Jean-Louis Verdier]] developed [[hypercover]]ings. 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 structure]]s.

Many other coherent sheaf cohomology groups are found using an embedding <math>i:X \hookrightarrow Y</math> of a space <math>X</math> into a space with known cohomology, such as <math>\mathbb{P}^n</math>, or some [[weighted projective space]]. In this way, the known sheaf cohomology groups on these ambient spaces can be related to the sheaves <math>i_*\mathcal{F}</math>, giving <math>H^i(Y,i_*\mathcal{F}) \cong H^i(X,\mathcal{F})</math>. For example, computing the [[Coherent sheaf cohomology#Sheaf cohomology of plane-curves|coherent sheaf cohomology of projective plane curves]] is easily found. One big theorem in this space is the [[Hodge structure|Hodge decomposition]] found using a [[Leray spectral sequence|spectral sequence associated to sheaf cohomology groups]], proved by Deligne.<ref>{{Cite journal|last=Deligne|first=Pierre|date=1971|title=Théorie de Hodge : II|url=http://www.numdam.org/item/?id=PMIHES_1971__40__5_0|journal=[[Publications Mathématiques de l'IHÉS]]|language=en|volume=40|pages=5–57|doi=10.1007/BF02684692 |s2cid=118967613 }}</ref><ref>{{Cite journal|last=Deligne|first=Pierre|date=1974|title=Théorie de Hodge : III|url=http://www.numdam.org/item/PMIHES_1974__44__5_0/|journal=Publications Mathématiques de l'IHÉS|language=en|volume=44|pages=5–77|doi=10.1007/BF02685881 |s2cid=189777706 }}</ref> Essentially, the <math>E_1</math>-page with terms<blockquote><math>E_1^{p,q} = H^p(X,\Omega^q_X)</math></blockquote>the sheaf cohomology of a [[Smooth variety|smooth]] [[projective variety]] <math>X</math>, degenerates, meaning <math>E_1 = E_\infty</math>. This gives the canonical Hodge structure on the cohomology groups <math>H^k(X,\mathbb{C})</math>. It was later found these cohomology groups can be easily explicitly computed using [[Poincaré residue|Griffiths residues]]. See [[Jacobian ideal]]. These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, [[Decomposition theorem|the decomposition theorem]], paving the path for [[Mixed Hodge module]]s.

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

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

=== Derived categories of sheaves ===

The [[derived category]] of the category of sheaves of, say, abelian groups on some space ''X'', denoted here as <math>D(X)</math>, is the conceptual haven for sheaf cohomology, by virtue of the following relation: 
:<math>H^n(X, \mathcal F) = \operatorname{Hom}_{D(X)}(\mathbf Z, \mathcal F[n]).</math>
The adjunction between <math>f^{-1}</math>, which is the left adjoint of <math>f_*</math> (already on the level of sheaves of abelian groups) gives rise to an adjunction 
:<math>f^{-1} : D(Y) \rightleftarrows D(X) : R f_*</math> (for <math>f: X \to Y</math>),
where <math>Rf_*</math> is the derived functor. This latter functor encompasses the notion of sheaf cohomology since <math>H^n(X, \mathcal F) = R^n f_* \mathcal F</math> for <math>f: X \to \{*\}</math>.

{{Images of sheaves}}
Like <math>f_*</math>, the direct image with compact support <math>f_!</math> can also be derived. By virtue of the following isomorphism <math>R f_! \mathcal{F}</math> parametrizes the [[cohomology with compact support]] of the [[fiber (mathematics)|fibers]] of <math>f</math>:
:<math>(R^i f_! \mathcal F)_y = H^i_c(f^{-1}(y), \mathcal F).</math><ref>{{harvtxt|Iversen|1986|loc=Chapter VII, Theorem 1.4}}</ref>
This isomorphism is an example of a [[base change theorems|base change theorem]]. There is another adjunction
:<math>Rf_! : D(X) \rightleftarrows D(Y) : f^!.</math>
Unlike all the functors considered above, the twisted (or exceptional) inverse image functor <math>f^!</math> is in general only defined on the level of [[derived category|derived categories]], i.e., the functor is not obtained as the derived functor of some functor between [[Abelian category|abelian categories]]. If <math>f: X \to \{*\}</math> and ''X'' is a smooth [[orientable manifold]] of dimension ''n'', then
:<math>f^! \underline \mathbf R \cong \underline \mathbf R [n].</math><ref>{{harvtxt|Kashiwara|Schapira|1994|loc=Chapter III, §3.1}}</ref>
This computation, and the compatibility of the functors with duality (see [[Verdier duality]]) can be used to obtain a high-brow explanation of [[Poincaré duality]]. In the context of quasi-coherent sheaves on schemes, there is a similar duality known as [[coherent duality]].

[[Perverse sheaf|Perverse sheaves]] are certain objects in <math>D(X)</math>, i.e., complexes of sheaves (but not in general sheaves proper). They are an important tool to study the geometry of [[singularity (mathematics)|singularities]].<ref>{{harvtxt|de Cataldo|Migliorini|2010}}</ref>

==== Derived categories of coherent sheaves and the Grothendieck group ====
Another important application of derived categories of sheaves is with the derived category of [[Coherent sheaf|coherent sheaves]] on a scheme <math>X</math> denoted <math>D_{Coh}(X)</math>. This was used by Grothendieck in his development of [[intersection theory]]<ref>{{cite web|last=Grothendieck|title=Formalisme des intersections sur les schema algebriques propres|url=http://library.msri.org/books/sga/sga/6/6t_519.html}}</ref> using [[derived categories]] and [[K-theory]], that the intersection product of subschemes <math>Y_1, Y_2</math> is represented in [[Grothendieck group|K-theory]] as<blockquote><math>[Y_1]\cdot[Y_2] = [\mathcal{O}_{Y_1}\otimes_{\mathcal{O}_X}^{\mathbf{L}}\mathcal{O}_{Y_2}] \in K(\text{Coh(X)})</math></blockquote>where <math>\mathcal{O}_{Y_i}</math> are [[coherent sheaves]] defined by the <math>\mathcal{O}_X</math>-modules given by their [[Structure sheaf|structure sheaves]].