By J.P. Jouanolou
Bertini theorems are one of the most valuable
statements in algebraic geometry. This
monograph fills the necessity for a latest, systematic
exposition. it's going to supply an advent to the
field for graduate scholars, in addition to a reference
for experts. integrated during this quantity are fresh
applications to the constitution of projective modules
of finite style and to connectivity theorems which
demonstrate the worth of those theorems.
TABLE DE MATIERES
I - PROPRIETES CONSTRUCTIBLES ET THEOREMES DE BERTINI.
1 - Ensembles constructibles. 2
2 - Morphismes de kind fini : théorèmes de Chevalley et platitude five
3 - Corps commutatifs : extensions séparables, primaires, 17
4 - Constructibilité de certaines propriétés géométriques. 29
5-- Corps commutatifs : dérivations et différentielles. forty seven
6 - Théorèmes de Bertini. sixty two
7 - software à des questions de connexité. ninety one
II - constitution DES MODULES PROJECTIFS.
1 — Rang libre d'un module. ninety eight
2 - Théorème de Serre. ninety nine
3 - Théorème de simplification de Bass. one zero five
4 - Théorème de simplification de Suslin, 109
5 - Un théorème de Bertini. 121
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Extra resources for Théorèmes de Bertini et Applications
4) on the level of complexes in degree -diMhgfi without any intervention of signs. 6) in place such define as D (X) by wX1 y [nJ 0 f L wxly[n]. 4) Recall that in we [RD] use the in order without the intervention of isomorphism (gf) translations, triple composites, and to -+ L = the definition is modified definition D (Y) f an with respect to use the modified definition -a signs. 7) instead of the one in [RD]. 6)). 4). ), to be where 7 : -4 D+(X) (X, 61x) -+ flat map of ringed spaces. For example, if X* is of quasi-coherent injective sheaves on Y, then the (Y, f* ex) is the canonical bounded below complex complex of quasi-coherent a f* Ox -modules Je om y (f* 61X, X *) can be viewed as a complex of quasi-coherent on X, and this represents the complex f (j*).
3. Let X be a locally noetherian category of quasi-coherent sheaves in the category of Cx-modules. in the PROOF. 18], there is an on X. scheme, J an Then _0 is an injection i : J -4 injective object iniective object f for some quasi- / which is injective as an OX-module. r as an ex-module on j, this injection splits. Thus, Y is a direct summand of and so is injective as an OX-module. coherent Ox-module By this lemma, the injective objects in the category of quasi-coherent 6IXon a locally noetherian scheme X are exactly the injective ex-modules which are quasi-coherent as sheaves.
Let 1` and I/ denote the respective canonical truncations in rows < n. By the theory qc of injective resolutions in abelian categories, we can choose a map of double complexes pi lqc over f 016 - fOWq, and a map of double complexes in the tion of P2 : _ Kq*,* -+ Iq*c* over f001q*,. canonical truncations in rows Le ,t < p', and p2 denote the induced maps on the n. qc and consist of f,,-acyclics, applying f, to Tot (p2) yields a Since X and TotB(p) 2 quasi-isomorphism. Beware that applying f, to Tbt ED isomorphism.