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If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

We investigate some properties of (geometric) fields in toposes of sheaves over Boolean spaces and establish the internal validity of a number of classical theorems from Algebraic Geometry and the theory of ordered fields. We then use our results to obtain, via sheaf representations, some know theorems about (von Neumann) regular rings as well as some new theorems for regular f-rings. By contrast with previous investigations in these last two subjects (Saracino and Weispfenning {39} and van den Dries {42}) a more natural approach, inspired by work of Macintyre {30}, Loullis {29}, Bunge-Reyes {7} and Bunge {4},{5} is employed here. In addition to sheaf theoretic methods we use a variety of logical methods from geometric logic, infinitary intuitionistic logic and model theory. We also prove some new theorems on the transfer of subobjects along certain morphisms and a "lifting theorem" taking truth from statements about global sections to their internal validity.

Thesis (Ph.D.)--McMaster University (Canada), 1991. / Source: Dissertation Abstracts International, Volume: 53-11, Section: B, page: 5753. Supervisor: G. Bruns.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves 170-171).

Thesis (doctoral)--Universidad de Santiago de Compostela, 1981. / Bibliography: p. 150-158.

No description available.

Este trabalho estuda a demonstração de existência de uma teoria de cohomologia em estruturas o-minimais arbitrárias, conforme o trabalho de Edmundo, Jones e Peatfield. / This work studies the proof of the existence of sheaf cohomology theory in arbitrary o-minimal structures, following the work of Edmundo, Jones and Peatfield.

Este trabalho estuda a demonstração de existência de uma teoria de cohomologia em estruturas o-minimais arbitrárias, conforme o trabalho de Edmundo, Jones e Peatfield. / This work studies the proof of the existence of sheaf cohomology theory in arbitrary o-minimal structures, following the work of Edmundo, Jones and Peatfield.