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descent of affine schemes

In algebraic geometry the most basic and often considered case of descent theory is the descent of quasicoherent sheaves. However, SGA I also considers the descent of affine schemes over a base scheme.

  • SGA I.8.2: Descente des préschémas affines sur un autre

The basic theorem is that the fibered category \mathcal{F} of affine morphisms f:XSf: X\to S over the site of base schemes SS with fpqc topology is a Grothendieck 1-stack (i.e. every fpqc morphism is of effective \mathcal{F}-descent).

This theorem uses more specific reasoning than the theory of descent for quasicoherent sheaves and unlike the former, it does not generalize to the noncommutative case.

Created on August 6, 2011 17:54:55 by Zoran Škoda (212.15.171.95)