General Preface (to both Volumes, I, II) |
Preface (to Volume I) |
Acknowledgements |
Contents of Volume II |
Vector Sheaves. General Theory / Part 1: |
Sheaf Theory / Chapter I.: |
Preliminaries. Definitions / 1.: |
Sections of a sheaf / 2.: |
A sheaf is its sections. Defining families of sections / 3.: |
Examples. Germs and sheaves / 4.: |
The sheaf of germs of locally constant functions. The constant sheaf / 4.(a).: |
The sheaf of germs of continuous functions / 4.(b).: |
Sheaves of germs of C[superscript infinity]-functions and of holomorphic ones / 4.(c).: |
Presheaves. Basic definitions / 5.: |
Morphisms of presheaves / 6.: |
Stalks of a presheaf. Sheafification, or sheaf generated by a presheaf / 7.: |
The sheafification functor / 8.: |
Presheaf of sections of a sheaf. The section functor / 9.: |
Further examples of presheaves / 10.: |
The constant presheaf / 10.(a).: |
Presheaves of functions / 10.(b).: |
Complete presheaves. Examples / 11.: |
Morphisms of presheaves (contn'd) / 12.: |
The section functor and its (left) adjoint: The sheafification functor (contn'd) / 13.: |
Change of base space / 14.: |
The direct image (or push-out) functor / 14.(a).: |
Inverse image (or pull-back) functor / 14.(b).: |
Sheaves and Presheaves with Algebraic Structure / Chapter II.: |
Preliminaries. Basic definitions |
Sheaves with algebraic structures. A-modules / 1.(a).: |
Presheaves with algebraic structure. A-presheaves / 1.(b).: |
Section-wise definition of algebraic structure in a given sheaf (of sets) / 1.(c).: |
A-morphisms. Exact sequences |
Morphisms of A-presheaves / 2.(a).: |
Quotients of A-modules / 2.(b).: |
The (left) exactness of the section functor / 2.(c).: |
Change of base space. The direct, resp. inverse, image functors for A-modules and A-presheaves / 2.(d).: |
Direct (or Whitney) sum of A-modules. Free A-modules |
Locally free A-modules. Vector sheaves |
Tensor products of A-modules |
The functor Hom[subscript A] |
Sheaf Cohomology / 6.(a).: |
Preliminaries. Fundamental concepts |
Derived functors |
Derived functor cohomology |
Cech cohomology |
The directed set of (proper) open coverings of a given topological space |
Cech cochain complex |
Passage to the limit |
Leray's Theorem / 4.(d).: |
Lifting of cocycles |
Passage to the limit (of liftable cocycles) / 5.(a).: |
Long exact sequences in Cech cohomology (contn'd) / 5.(b).: |
Cech cohomolgy with coefficients in a presheaf |
Long exact (Cech) cohomology sequence for presheaves |
Cech cohomology for presheaves on paracompact spaces / 6.(b).: |
Cohomology of double complexes. Hypercohomology |
Cech hypercohomology / 7.(a).: |
Cohomology theories in action. (Interrelations) |
Fine resolutions. De Rham cohomology / 8.(a).: |
The functors Ext[subscript A] and Ext[subscript A] |
The functor Ext[subscript A] / 9.(a).: |
A-extensions / 9.(b).: |
Ext[superscript 1 subscript A] and A-extensions |
The 1st cohomology set |
0-dimensional (non-abelian) cohomology / 11.(a).: |
Linear and Multilinear Algebra of Vector Sheaves / Chapter IV.: |
Classical group sheaves |
A (full) matrix algebra sheaf |
The general linear group sheaf |
An example |
Free A-modules (contn'd) |
Matrix representations of A-morphisms / 3.(a).: |
The determinant morphism |
Dual A-modules |
Certain basic formulae relating the functors Hom[subscript A] and Hom[subscript A] |
Exterior algebra of A-modules |
Determinant line sheaf |
Riemannian A-modules |
Positive partitions of unity |
Hermitian A-modules |
Gram-Schmidt orthogonalization |
Volume element |
The Hodge (or * -) operator |
Vector Sheaves and Cohomology / Chapter V.: |
Vector sheaves and 1-cocycles / 1: |
Section-wise definition of being a vector sheaf locally free |
Cohomological classification of vector sheaves |
Geometrical meaning of the 1st cohomology set (a la Grothendieck) |
Examples |
Behaviour of vector sheaves in cohomology |
Line sheaves. The Picard group |
The determinant morphism (contn'd) |
Flat vector sheaves |
Line sheaves and Chern classes |
The functors Ext[subscript A] and Ext[subscript A] (contn'd) |
Splittings of A-extensions (contn'd) |
Dual A-extensions |
Classical group sheaves (contn'd) |
The orthogonal group sheaf |
The unitary group sheaf |
Category jargon / Appendix.: |
Bibliography |
Notational Index |
Subject Index |
General Preface (to both Volumes, I, II) |
Preface (to Volume I) |
Acknowledgements |