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sheaf cohomology

/Type /Page {\displaystyle X} {\displaystyle {\mathcal {F}}}

>> endobj : is always finitely presented as a module over itself, but it is not always coherent.

{\displaystyle U\mapsto \operatorname {Hom} (F|_{U},G|_{U})} .

U , but now gives a way to distinguish different complex manifolds. This terminology and notation is by analogy with sections of fiber bundles or sections of the étalé space of a sheaf; see below.

.

For example, the de Rham complex is a resolution of the constant sheaf _ on any smooth manifold, so the sheaf cohomology of _ is equal to its de Rham cohomology.

They also find use in constructions such as Godement resolutions. M ) {\displaystyle f\in {\mathcal {H}}(U)} _ If we have two sheaves on different spaces, then their data is indexed differently. ) Given a complex submanifold

which shows that π represents the functor Γ. E is constructed so that the projection map π is a covering map. which takes an open subset U /MediaBox [0 0 595.276 841.89]

. f

This shows that some of the facets of sheaf theory can also be traced back as far as Leibniz. .

U

Speci cally, the sheaf coho-mology groups of a sheaf F on a topological space X, denoted Hn(X,F), are de ned to be the right derived functors of the global section functor. . /Length 8

1 M The set Heuristically speaking, a morphism of sheaves is analogous to a function between them.

n

) )

can be embedded inside some

, are isomorphic. (not necessarily surjective), the kernel of φ is of finite type. 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. ~ X << /Length 5 0 R /Filter /FlateDecode >> ( Homology, Cohomology, and Sheaf Cohomology Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania U ⊕ A sheaf of groups under pointwise multiplication. F

This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space. {\displaystyle \varphi :G\to F} /Length 919 are continuous, and aF is the sheaf of continuous sections of φ for this topology. {\displaystyle C^{\infty }(M)} ���q�92k�j���-�V�n¨9��Y?D{\�ܿ��c�w�x\�=�ꆐ�ǖu���)��=��Bi�<8����8��1��_����M2]�~LL���9&�� �W�u��ޏ�,`&0���D��g; �x���4��\4��ȨST�����T��,`O2���#߸J-���-�h�0h�^!��PM%8j�����i��HG�U{�B�fw"�珉$��gF��ٷ��A��=&�͑Q�76���@� #��"dkV�Q�b*Y����[�`{qˈ�#�̂EJ�SQ��5����HH��3��&BpU�qCZ�T�܈g 5��������O-�M�m�����o��w���b���F��r��t���!��.+㲒T�6*B�p2tT�&�|���:1�m���.�6�$��2�{�52u�Ͽ-x%6f���8�a�� �>�&��xu.$�C���X�%�\1ZB&j%��%��I��DԠ+��?�?�?#\�� ��?��&6�n�7[ z��跗:��_��yD��(b�?�_�.

O → -module and for every inclusion of open sets V ⊆ U, the restriction map . n X /Parent 6 0 R The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf

The category of sheaves of abelian groups on a topological space X is an abelian category, and so it makes sense to ask when a morphism f: B → C of sheaves is injective or surjective. O O ( {\displaystyle U\mapsto \operatorname {Hom} (F,G)} is the exceptional inverse image functor.)

− !

. . See Grothendieck duality and Verdier duality. /Resources 1 0 R X | X The kernel of this map is not finitely generated, so 7 0 obj << U

∗ ∈ X

) {\displaystyle X} and Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. {\displaystyle {\mathcal {O}}_{X}} {\displaystyle U=\bigcup _{i\in I}U_{i}}, f If F is a sheaf over X, then the étalé space of F is a topological space E together with a local homeomorphism π : E → X such that the sheaf of sections of π is F. The space E is usually very strange, and even if the sheaf F arises from a natural topological situation, E may not have any clear topological interpretation.

X

on any smooth manifold, so the sheaf cohomology of

as a topological space with charts where the transition functions are biholomorphic, we could consider the underlying topological space of In fact, such a viewpoint is encapsulated in the theory of (locally) ringed spaces.

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