sheaf cohomology

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{\displaystyle {\mathcal {H}}(X),{\mathcal {H}}(X')} {\displaystyle {\mathcal {O}}} ϕ

⁡ when these functions are restricted to

This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space. R

U {\displaystyle \{U_{i}\}_{i\in I}}

Can one still understand sheaf cohomology in some "geometric" way?

{\displaystyle f\in {\mathcal {H}}(U)} Γ A category with a Grothendieck topology is called a site.

In the examples above it was noted that some sheaves occur naturally as sheaves of sections.

( a U

−

The table lists the values of certain sheaves over open subsets U of M and their restriction maps. /Parent 6 0 R

= f → In other words, for every open subset V of an open set U, the following diagram is commutative. O The product of an empty family or the inverse limit of an empty family is a terminal object, because for any object there must exist a unique morphism between them with the object as domain, which is the property of products and inverse limits. Given a presheaf, a natural question to ask is to what extent its sections over an open set U are specified by their restrictions to smaller open sets Vi of an open cover of U. Every sheaf of modules is an abelian sheaf. )

. ( ,

9 0 obj <<

: First we define the category of open sets on X to be the posetal category O(X) whose objects are the open sets of X and whose morphisms are inclusions. {\displaystyle U\mapsto F(U)\otimes G(U)} ( {\displaystyle {\mathcal {O}}_{X}} n It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is local in a sense that it is enough to verify it in an arbitrary neighbourhood of each point. ,

/Type /Page In general, for an open set U and open covering (Ui), construct a category J whose objects are the sets Ui and the intersections Ui ∩ Uj and whose morphisms are the inclusions of Ui ∩ Uj in Ui and Uj. {\displaystyle f^{-1}} X n F Z ) O {\displaystyle {\mathcal {O}}_{X}^{m}|_{U}\to {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}} The construction above determines an equivalence of categories between the category of sheaves of sets on X and the category of étalé spaces over X. U COHOMOLOGY OF SHEAVES 6 Surjective. This makes the construction into a functor. 3 0 obj << The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf

For simplicity, consider first the case where the sheaf takes values in the category of sets.

that has the universal property that for any sheaf G and any morphism of presheaves A presheaf is separated if its sections are "locally determined": whenever two sections over U coincide when restricted to each of Vi, the two sections are identical. U ↪ /Filter /FlateDecode

consisting of a topological space X and a sheaf of rings on X is called a ringed space.

∞ . There is a natural morphism of presheaves U

captures the properties of a sheaf "around" a point x ∈ X, generalizing the germs of functions.

/ U such that the sequence of morphisms

. {\displaystyle \mathbb {R} ^{N}}

. {\displaystyle {\underline {\mathbf {Z} }}} O

{\displaystyle U\mapsto \operatorname {Hom} (F,G)} Z

{\displaystyle {\mathcal {O}}_{X}(U)} F

Choose n = 1, and for the morphism φ, take the map that sends every variable to zero. .

be a ringed space. ∈ ,

→ M A much cleaner 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.

For example I would be very interested in the case of coherent $\mathcal{O}_X$-Modules. i is actually a left exact functor. U /ProcSet [ /PDF /Text ] is an Once he had axiomatized the notion of covering, open sets could be replaced by other objects. an return the ring of holomorphic functions α O

(

The natural morphism F(U) → Fx takes a section s in F(U) to its germ at x.

Note that sometimes this sheaf is denoted and the sheaf of holomorphic functions

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