Nakayama lemma

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In mathematics, Nakayama's lemma is an important technical lemma in commutative algebra and algebraic geometry, named after Tadashi Nakayama. 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.

There is also a graded version of Nakayama's lemma. Let R be a graded ring (over the integers), and let R ₊ the ideal generated by positively graded elements. Then if M is a graded module over R for which M_i = 0 for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative degree) such that R ₊ M = M, then M = 0. Of particular importance is the case that R is a polynomial ring with the standard grading, and M is a finitely generated module.

The proof is much easier than in the ungraded case: taking i to be the least integer such that , we see that M_i does not appear in R ₊ M, so either , or such an i does not exist, i.e., M = 0.

[edit] References

• Atiyah, M.F. and Macdonald, I.G, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA (1969).
• Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics 52 Springer-Verlag (1977).