Download Brauer Groups in Ring Theory and Algebraic Geometry: by F. Oystaeyen PDF

By F. Oystaeyen

The workshop ''Brauer teams in Geometry and Ring Theory'', prepared on the college of Antwerp, U.I.A. in August 1981, has been financially supported by means of the Belgian starting place for medical study N.F.W.O. and by means of U.I.A. We thank either associations for his or her non-stop aid.

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Extra resources for Brauer Groups in Ring Theory and Algebraic Geometry: Proceedings, Univ. of Antwerp Wilrijk, Belgium, Aug 17-28, 1981

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Then D is ample if and only if it satisfies either of the following properties: (i). For every irreducible subvariety V ⊆ X of positive dimension, there is a positive integer m = m(V ), together with a non-zero section 0 = s = sV ∈ H 0 V, OV (mD) , such that s vanishes at some point of V . (ii). For every irreducible subvariety V ⊆ X of positive dimension, χ V, OV (mD) → ∞ as m → ∞. (It is enough to prove this when X is reduced and irreducible. Suppose that (i) holds. Taking first V = X and replacing D by a multiple, we can assume that D is effective.

224, Chapter 1, §2]). f. ∩ H 1,1 X; C . Finally we say a word about functoriality. Let f : Y −→ X be a morphism of complete varieties or projective schemes. If α ∈ Pic(X) is a class mapping to zero in N 1 (X), then it follows from the projection formula that f ∗ (α) is numerically trivial on Y . 5 a functorial induced homomorphism f ∗ : N 1 (X) −→ N 1 (Y ). 22. (Non-projective schemes). A, the integrality and projectivity hypotheses in the previous paragraph arise only in order to use the functorial properties of line bundles to discuss divisors.

Dk · V 6 , V D1 · . . g. Chern classes of vector bundles — in Part Two of this book. Working topologically allows one to bypass potential complications involved in specifying groups to receive the classes in question. It then seemed natural to use topologically-based intersection theory throughout. 16 Chapter 1. Ample and Nef Line Bundles (or a small variant thereof) for the intersection product in question. By linearity one can replace V by an arbitrary k-cycle, and evidently this product depends only on the linear equivalence class of the Di .

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