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Local to global equivalence for homomorphisms. (English) Zbl 0899.13019

Let \(R\) be a commutative ring, \(A,B\) be finitely presented \(R\)-modules, \(f\in \operatorname{Hom}_R(A,B)\), \(G(f)\) be the set of equivalence classes of homomorphisms, which are locally equivalent to \(f\). A one-to-one correspondence between \(G(f)\) and the first cohomology group \(H^1 (X, {\mathcal E}^*)\) is established, where \({\mathcal E}^*\) is a suitable sheaf over \(X= \text{Spec} (R)\). Examples of calculation are presented, in particular, in a case where \(A\) and \(B\) are projective.

MSC:

13D45 Local cohomology and commutative rings
13B10 Morphisms of commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
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References:

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