Cohomology of Sheaves

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.