Cohomology of sheaves.

*(English)*Zbl 0559.55001
Lecture Notes Series. Aarhus University 55. Aarhus: University of Aarhus, Department of Mathematics. 237 p. (1984).

This text contains a systematically exposition of the basic properties and some applications of the cohomology of sheaves. From different ways of introducing cohomology, the author takes as his basic definition the derived functors of the global section functor.

The text begins with a chapter on homological algebra, including all the usual basic concepts. The second chapter is an introduction in sheaf theory and in the study of cohomology of sheaves. To obtain essential results the class of locally compact spaces is considered and for this class of spaces the cohomology with compact support is studied using soft sheaves (Chapter III). As an example of a soft sheaf, the sheaf of smooth functions on a smooth manifold is considered and the De Rham theorem is proved.

Chapter IV is entitled ”Poincaré duality”. In first form it is a duality between cohomology and cohomology with compact support. In Chapters V, VI a more general Poincaré duality theory for a continuous map between locally compact spaces is developed. In the framework of local cohomology the special case of closed subspace of a topological space is considered (Chapter VI). This theory is used for the construction of characteristic classes: Stiefel-Whitney, Chern and Pontryagin (Chapter VII). In Chapter VIII a homology theory for locally compact spaces and proper maps is developed. This allows to express Poincaré duality as an isomorphism between homology and cohomology. Some applications to the classical theory of topological manifolds are given: submanifolds, degree, trace maps, the diagonal class, etc.

In Chapter IX, devoted to applications in algebraic geometry, the homology theory is used for the study of algebraic varieties. A detailed introduction to (co)homology classes of algebraic cycles, including a topological definition of the local intersection symbol, is given. Finally, in Chapter X sheaves on paracompact spaces are considered. If a locally compact space is paracompact then cohomology with compact support can be computed by means of resolutions of soft sheaves. This principle allows to obtain some applications to differential geometry, complex analysis, distributions and hyperfunctions.

The book offers in a self-contained manner important and many new results on cohomology of sheaves.

The text begins with a chapter on homological algebra, including all the usual basic concepts. The second chapter is an introduction in sheaf theory and in the study of cohomology of sheaves. To obtain essential results the class of locally compact spaces is considered and for this class of spaces the cohomology with compact support is studied using soft sheaves (Chapter III). As an example of a soft sheaf, the sheaf of smooth functions on a smooth manifold is considered and the De Rham theorem is proved.

Chapter IV is entitled ”Poincaré duality”. In first form it is a duality between cohomology and cohomology with compact support. In Chapters V, VI a more general Poincaré duality theory for a continuous map between locally compact spaces is developed. In the framework of local cohomology the special case of closed subspace of a topological space is considered (Chapter VI). This theory is used for the construction of characteristic classes: Stiefel-Whitney, Chern and Pontryagin (Chapter VII). In Chapter VIII a homology theory for locally compact spaces and proper maps is developed. This allows to express Poincaré duality as an isomorphism between homology and cohomology. Some applications to the classical theory of topological manifolds are given: submanifolds, degree, trace maps, the diagonal class, etc.

In Chapter IX, devoted to applications in algebraic geometry, the homology theory is used for the study of algebraic varieties. A detailed introduction to (co)homology classes of algebraic cycles, including a topological definition of the local intersection symbol, is given. Finally, in Chapter X sheaves on paracompact spaces are considered. If a locally compact space is paracompact then cohomology with compact support can be computed by means of resolutions of soft sheaves. This principle allows to obtain some applications to differential geometry, complex analysis, distributions and hyperfunctions.

The book offers in a self-contained manner important and many new results on cohomology of sheaves.

Reviewer: I.Pop (Iasi)

##### MSC:

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |

55N30 | Sheaf cohomology in algebraic topology |

55M05 | Duality in algebraic topology |

58A12 | de Rham theory in global analysis |

58A30 | Vector distributions (subbundles of the tangent bundles) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

57N65 | Algebraic topology of manifolds |

32L05 | Holomorphic bundles and generalizations |

18G10 | Resolutions; derived functors (category-theoretic aspects) |