Cohomology of Sheaves

Häftad, Engelska, 1986

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The most satis factory general class is that of locally compact spaces and it is the study of such spaces which occupies the central part of this text. The fundamental concepts in the study of locally compact spaces is cohomology with compact support and a particular class of sheaves,the so-called soft sheaves.

Produktinformation

Utgivningsdatum1986-04-01
Mått170 x 244 x 30 mm
Vikt818 g
FormatHäftad
SpråkEngelska
SerieUniversitext
Antal sidor464
FörlagSpringer-Verlag Berlin and Heidelberg GmbH & Co. KG
ISBN9783540163893

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Topologi inom Naturvetenskap och teknik

I. Homological Algebra.- 1. Exact categories.- 2. Homology of complexes.- 3. Additive categories.- 4. Homotopy theory of complexes.- 5. Abelian categories.- 6. Injective resolutions.- 7. Right derived functors.- 8. Composition products.- 9. Resume of the projective case.- 10. Complexes of free abelian groups.- 11. Sign rules.- II. Sheaf Theory.- 0. Direct limits of abelian groups.- 1. Presheaves and sheaves.- 2. Localization.- 3. Cohomology of sheaves.- 4. Direct and inverse image of sheaves. f*,f*.- 5. Continuous maps and cohomology!,.- 6. Locally closed subspaces, h!h.- 7. Cup products.- 8. Tensor product of sheaves.- 9. Local cohomology.- 10. Cross products.- 11. Flat sheaves.- 12. Hom(E,F).- III. Cohomology with Compact Support.- 1. Locally compact spaces.- 2. Soft sheaves.- 3. Soft sheaves on $$\mathbb {R}$$n.- 4. The exponential sequence.- 5. Cohomology of direct limits.- 6. Proper base change and proper homotopy.- 7. Locally closed subspaces.- 8. Cohomology of the n-sphere.- 9. Dimension of locally compact spaces.- 10. Wilder’s finiteness theorem.- IV. Cohomology and Analysis.- 1. Homotopy invariance of sheaf cohomology.- 2. Locally compact spaces, countable at infinity.- 3. Complex logarithms.- 4. Complex curve integrals. The monodromy theorem.- 5. The inhomogenous Cauchy-Riemann equations.- 6. Existence theorems for analytic functions.- 7. De Rham theorem.- 8. Relative cohomology.- 9. Classification of locally constant sheaves.- V. Duality with Coefficient in a Field.- 1. Sheaves of linear forms.- 2. Verdier duality.- 3. Orientation of topological manifolds.- 4. Submanifolds of $$\mathbb {R}$$n of codimension 1.- 5. Duality for a subspace.- 6. Alexander duality.- 7. Residue theorem for n-1 forms on $$\mathbb {R}$$n.- VI. Poincare Duality with GeneralCoefficients.- 1. Verdier duality.- 2. The dualizing complex D.- 3. Lefschetz duality.- 4. Algebraic duality.- 5. Universal coefficients.- 6. Alexander duality.- VII. Direct Image with Proper Support.- 1. The functor f!.- 2. The Künneth formula.- 3. Global form of Verdier duality.- 4. Covering spaces.- 5. Local form of Verdier duality.- VIII. Characteristic Classes.- 1. Local duality.- 2. Thom class.- 3. Oriented microbundles.- 4. Cohomology of real projective space.- 5. Stiefel-Whitney classes.- 6. Chern classes.- 7. Pontrjagin classes.- IX. Borel Moore Homology.- 1. Proper homotopy invariance.- 2. Restriction maps.- 3. Cap products.- 4. Poincare duality.- 5. Cross products and the Künneth formula.- 6. Diagonal class of an oriented manifold.- 7. Gysin maps.- 8. Lefschetz fixed point formula.- 9. Wu’s formula.- 10. Preservation of numbers.- 11. Trace maps in homology.- X. Application to Algebraic Geometry.- 1. Dimension of algebraic varieties.- 2. The cohomology class of a subvariety.- 3. Homology class of a subvariety.- 4. Intersection theory.- 5. Algebraic families of cycles.- 6. Algebraic cycles and Chern classes.- XI. Derived Categories.- 1. Categories of fractions.- 2. The derived category D (A).- 3. Triangles associated to an exact sequence.- 4. Yoneda extensions.- 5. Octahedra.- 6. Localization.