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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 $$\operatorname{Spec} B' \rightarrow \operatorname{Spec} B$$.

##### MSC:
 14A15 Schemes and morphisms 13C99 Theory of modules and ideals in commutative rings
##### Keywords:
ringed space; fiber product
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