Théorèmes de Bertini et Applications by J.P. Jouanolou

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.

1 - Ensembles constructibles. 2
2 - Morphismes de kind fini : théorèmes de Chevalley et platitude five
3 - Corps commutatifs : extensions séparables, primaires, 17
universellement intègres.
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
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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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.

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