By V. I. Danilov (auth.), I. R. Shafarevich (eds.)

ISBN-10: 3642609252

ISBN-13: 9783642609251

ISBN-10: 3642646077

ISBN-13: 9783642646072

This EMS quantity contains components. the 1st half is dedicated to the exposition of the cohomology idea of algebraic forms. the second one half offers with algebraic surfaces. The authors, who're famous specialists within the box, have taken pains to give the cloth carefully and coherently. The publication includes quite a few examples and insights on quite a few themes. This booklet might be immensely necessary to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, advanced research and similar fields.

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**Extra info for Algebraic Geometry II: Cohomology of Algebraic Varieties. Algebraic Surfaces**

**Sample text**

The following theorem of H. Cartan allows us to establish the acyclicity of some coverings. Theorem. Let A be a class of open subsets of X that satisfies the following two conditions: a) A is closed under finite intersections, and b) A contains sufficiently small open subsets. We assume that for every U E A and an arbitrary A-covering U = (Ui) of U ( which means that every Ui E A), we get Hq (U, F) = 0 for q > 0. Then every A-covering is acyclic. ln particular, for every A-covering of X, we get an isomorphism H*(U, F) = H*(X, F).

By induction hypothesis, we get (p ~ 0, 0 < q < n) in the spectral sequence of our covering (see Sect. 4). This yields an exact sequence Since the image of a in Hn(Ui, F) is trivial for every i, the dass a lies in the group Hn((Ui), F), which is trivial by the assumption of the theorem. Chapter 2 Cohomology of Coherent Sheaves Since every algebraic variety is endowed with an algebraically defined Zariski topology, it makes sense to consider sheaves and their cohomology on algebraic varieties. However, it is not clear if we obtain something of interest.

So, by (b), one can find an A-covering (Ui) of U suchthat the image of aiUi in Hn(Ui, F) is trivial for every i. By induction hypothesis, we get (p ~ 0, 0 < q < n) in the spectral sequence of our covering (see Sect. 4). This yields an exact sequence Since the image of a in Hn(Ui, F) is trivial for every i, the dass a lies in the group Hn((Ui), F), which is trivial by the assumption of the theorem. Chapter 2 Cohomology of Coherent Sheaves Since every algebraic variety is endowed with an algebraically defined Zariski topology, it makes sense to consider sheaves and their cohomology on algebraic varieties.

### Algebraic Geometry II: Cohomology of Algebraic Varieties. Algebraic Surfaces by V. I. Danilov (auth.), I. R. Shafarevich (eds.)

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