Nakayama lemma

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

Lemma (Nakayama): Let R be a commutative ring with identity 1, 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.

Corollary 1: With conditions as above, if I is contained in the Jacobson radical of R, then necessarily M = 0.

Proof: I is in the Jacobson radical iff 1 + x is invertible for any x ∈ I, and r as above is such an element.

Corollary 2: If M ≅ N ⊕ IN′ for some ideal I in the Jacobson radical of A and N′ is finitely-generated, then M ≅ N.

Proof: Apply Corollary 1 to M/N.

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

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.

[edit] References

• Atiyah, M.F. and Macdonald, I.G (1969). Introduction to Commutative Algebra. Addison-Wesley, Reading, MA.
• Matsumura H., Commutative Algebra, 2nd ed. Benjamin/Cummings, 1980.