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Ai are surjective. Set ai = ker 'l/Ji cAl" so that = a Ai, and finally set a = Ai 1 (ai). The morphism ip makes Y into a scheme over A, that is, the sheaf Oy has an A-module structure. But since a C Ai 1 (ai), under the action of A on Oy(ip-l(Ui )), the ideal a acts trivially. In other words, Y is a scheme over Aja.

Schemes Ua Figure 24. The Glueing Conditions restriction homomorphisms p~, for W' eWe Va are defined in the obvious way. Thus the presheaf Ox is not defined on all open subsets of X, but the W on which it is defined form a basis for the open sets of the topology. The situation is as for the definition of the structure sheaf of Spec A. In the same way as there, we can extend the definition of Ox (U) to all open sets U C X as the projective limit limOx(W), taken over open sets W C U where Ox(W) +-is already defined.

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