Nakayama’s lemma is a simple but fundamental result of commutative algebra frequently used to lift information from the fiber of a sheaf over a point (as for example a coherent sheaf over a scheme) to give information on the stalk at that point.
Nakayama’s lemma is frequently stated in a general but slightly unilluminating form. We begin with an easier and more intuitive form. In this article, all rings are assumed to be commutative.
Here is a sample application. Suppose is an -module map, giving rise to an exact sequence
Tensoring with is a right exact functor, so we have an exact sequence
Nakayama’s lemma says that if , then . Equivalently, that if is epic, then is epic. In particular, to check whether a finite set of elements generates , it suffices to check whether the residue classes generate the vector space , which is a linear algebra calculation.
Suppose is a Noetherian local ring. A typical example is the stalk at a point of a Noetherian scheme as locally ringed space, and we will write as if we were in that situation. Being Noetherian, its maximal ideal is finitely generated. Suppose – the cotangent space – is a vector space of dimension . We would like to know whether a collection of functions that vanish at form a local coordinate system.
For this, it suffices to check whether the differentials at , belonging to the cotangent space , are linearly independent. (For then they span the cotangent space, and one concludes from Nakayama that the generate as an -module, thereby forming a local coordinate system at .) In this way, Nakayama’s lemma operates as a kind of “inverse function theorem”.
To cement this further, the following statement is offered in Harris as a corollary of Nakayama’s lemma (corollary 14.10, page 179):
(Inverse Function Theorem) A map between complex projective varieties of dimension which is a bijection and has injective derivative at every point is an isomorphism.
We turn now to a general statement of Nakayama’s lemma.
(To be continued)