SMF

Conducteur, descente et pincement

Conductor, Descent and Pinching

Daniel Ferrand
Conducteur, descente et pincement
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  • Année : 2003
  • Fascicule : 4
  • Tome : 131
  • Format : Électronique
  • Langue de l'ouvrage :
    Français
  • Class. Math. : 13Axx, 13Bxx, 14A15
  • Pages : 553-585
  • DOI : 10.24033/bsmf.2455
Une somme amalgamée de schémas est décrite localement par un produit fibré d'anneaux. Ce texte donne un résultat global d'existence (§ 5.4) de schémas définis comme certaines sommes amalgamées et un procédé algébrique (§ 2.2) pour décrire les modules sur produits fibrés d'anneaux correspondants.
This paper investigates some fiber products of rings, and dually some pushouts of schemes. The algebraic side is centered on the use of the conductor to solve some descent problems, problems which can be better reached by comparing suitable categories of modules on the fiber product of rings, with the fiber product of the similar categories for each of the factors. The main algebraic result (§2.2) asserts that these categories are very close to be equivalent, and that they are indeed equivalent as long as one restricts to flat modules. The geometric side is concerned with the existence of schemes defined by pinching : starting with a scheme $X'$, a closed subscheme $Y'$ and a finite morphism $Y' \rightarrow Y$, the “pinching construction” is intended to produce a scheme $X$ which is the pushout of $X'$ along the morphism $Y' \rightarrow Y$ ; such a scheme is proved to exist (§5.4) under the mild asumption that any finite set of points in $X'$ (resp. in $Y$) is contained in an open affine subset of $X'$ (resp. of $Y$).
Produit fibré, somme amalgamée, descente finie, pincement
Fiber product, pushout, finite descent, pinching