Nakayama lemma

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

Let R be a commutative ring with identity 1 , let I an Ideal (ring theory) in R , and M a Module (mathematics) 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 .

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 .

=References=

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