Nakayama's lemma

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 + IM for some ideal I in the Jacobson radical of R and M is finitely-generated, then M = N.

Proof: Apply Corollary 1 to M/N.

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

Let F be a coherent sheaf over an arbitrary scheme X; then the fibre of F at x, F(x) = F_x/m_xF_x (where F_x is the stalk at x), is zero if and only if F_x = 0 or, equivalently, if $F|_{U}=0$ for some neighborhood U of x.

References

• Atiyah, M.F. and Macdonald, I.G (1969). Introduction to Commutative Algebra. Addison-Wesley, Reading, MA.