Sheaf cohomology

Coarse cohomology with twisted coefficients

Mathematica Slovaca ◽

10.1515/ms-2017-0440 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1413-1444

Author(s):

Elisa Hartmann

Keyword(s):

Sheaf Cohomology ◽

Grothendieck Topology ◽

Coarse Structure ◽

Relative Cohomology ◽

Number Of Ends ◽

Constant Coefficients ◽

Coarse Space ◽

Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

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Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

Georgian Mathematical Journal ◽

10.1515/gmj.2003.37 ◽

2003 ◽

Vol 10 (1) ◽

pp. 37-43

Author(s):

E. Ballico

Keyword(s):

Projective Space ◽

Projective Spaces ◽

Free Resolution ◽

Complete Intersections ◽

Vanishing Theorem ◽

Sheaf Cohomology ◽

Minimal Free Resolution ◽

Branched Coverings ◽

Infinite Dimensional ◽

Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

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