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Conductor, descent and pinching. (Conducteur, descente et pincement.) (French) Zbl 1058.14003
From the point of view of algebra, this paper studies injective homomorphisms of rings \(f: A \rightarrow A'\) which have a non-zero conductor \(I\) (the annihilator of \(A'/A\) as an \(A\)-module). In this case the ring \(A\) is isomorphic to the fiber product \(A'\times _{A'/I} A/I\); the ”descente” problem is to relate properties of \(A\) to properties of \(A'\) and \(A'/I\). It is proved here that to give a flat \(A\) module \(P\) is equivalent to giving a flat \(A'\)-module \(P'\), a flat \(A/I\)-module \(Q\) and an \(A'/I\) isomorphism \(A'\otimes_A Q \rightarrow P'/IP\). From the geometric point of view, in the situation above, let \(B=A/I\) and \(B'=A'/I\), hence \(A\cong A'\times_{B'}B\). Then \(\operatorname{Spec} A\) can be identified with the ringed space given by \(\operatorname{Spec} A' \sqcup _{\operatorname{Spec} B'} \operatorname{Spec} B\) (Theorem 5.1), i.e. \(\operatorname{Spec} A\) comes from “pinching” \(\operatorname{Spec} A'\) along the closed set \(\operatorname{Spec} B'\) via the morphism \(\operatorname{Spec} B' \rightarrow \operatorname{Spec} B\).

14A15 Schemes and morphisms
13C99 Theory of modules and ideals in commutative rings
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