Nakayama lemma

In mathematics, Nakayama's lemma is an important technical lemma in commutative algebra and algebraic geometry. It is a consequence of the Cayley-Hamilton theorem. One of its many equivalent statements is as follows:

Related Topics:
Lemma - Commutative algebra - Algebraic geometry - Cayley-Hamilton theorem

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Let R be a commutative ring with identity 1, let I an ideal in R, and M a finitely-generated module over R. If IM = M, then there exists an r ∈ R with r ≡ 1 (mod I), such that rM = 0. Furthermore, if I is contained in the Jacobson radical of R, then necessarily M = 0.

Related Topics:
Commutative ring - Ideal - Finitely-generated module - Jacobson radical

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In the language of coherent sheaves, the Nakayama lemma can be stated as follows:

Related Topics:
Coherent - Sheaves

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Let F be a coherent sheaf. Then the stalk at x, denoted by F_x, is zero if and only if F|_U =0 for some neighborhood U of x.

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