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Some results on ampleness and divisorial schemes

1967; Mathematical Sciences Publishers; Volume: 23; Issue: 2 Linguagem: Inglês

10.2140/pjm.1967.23.217

1945-5844

Mario Borelli,

Advanced Algebra and Geometry

The purpose of this note is twofold.Part I consists of an example of an algebraic scheme which is the union of two closed, quasi-projective subscheme, but which is not itself quasiprojective.The main result of Part II is a structure theorem for coherent sheaves over divisorial schemes and, as an application, the proof that Theorem 2 of Borel-Serre's paper "Le Theorem de Riemann-Roch", which is stated only for quasiprojective, nonsingular schemes, can be extended to arbitrary nonsingular schemes.(See the Remark on page 108 of the mentioned paper.)The example given in Part I shows furthermore that, if Sf is an invertible sheaf over a noncomplete scheme X, which induces ample sheaves over the irreducible components of X, £/f need not be ample.That Sf is ample if X is complete is shown by Grothendieck in Theorem 2.6.2., Chapter III of "Elements de Geometrie Algebrique".The example we give consists of the union of two quasi-affine closed subschemes (whence their respective sheaves of local rings are ample).Since the union itself is not quasi-projective, its sheaf of local rings is not ample.The result obtained in Part II is but a first step towards Riemann-Roch-type theorems for arbitrary nonsingular schemes.To the author's knowledge, no suitable definition of a ring structure for equivalence classes of sheaves (i.e. a satisfactory intersection theory for equivalence classes of cycles) has been found as yet over an arbitrary nonsingular scheme.(See [3] and the remark on page 143 of [41 "On ne peut pas . ..".)The essential part of the proof of Theorem 3.3.in Part II was communicated to the author by Steven Kleiman, to whom the author is indebted for this and other conversations.The notation and terminology we use are, unless otherwise specifically stated, those of [7] and [5].We consider only algebraic schemes, with an arbitrary, algebraically closed ground field.For the sake of convenience we drop the adjective "algebraic" and speak simply of schemes.Also, all rings we consider are understood to be commutative and with unity, and all ring homomorphisms to be such that 1-+1.When we refer to, say, Lemma 2.3 without further identification we mean Lemma 2.3 of the present work, to be found as the third Lemma of §2.