Nakayama’s lemma is a powerful and useful tool in algebraic geometry. It sort of plays the role of the inverse function theorem in differential geometry.
In this post, we will consider various versions of Nakayama’s lemma and some applications.
Nakayama’s Lemma
We start with a standard technique due to Atiyah and Macdonald (Atiyah and Macdonald 1969).
Lemma 1 (Generalized Cayley–Hamilton Theorem) Let be a finitely generated -module, an ideal of and an element in such that . Then there exist elements such that
Proof. Let , ,, be a basis of . Then a endomorphism is represented by matrix , i.e. Since , we may assume that . Denote by the Kronecker delta. Then the above equality is equivalent to
By multiplying both sides the adjugate matrix, we find that
The desired equality follows by applying the Laplace expansion to the determinant.
Lemma 2 (Nakayama’s Lemma) Let be an ideal of a commutative ring and a finitely generated -module. If , then there exists such that for all .
Proof. Take in the Lemma Lemma 1, and let , we see that .
Corollary 1 Let be a commutative local ring, the maximal ideal and a finitely generated -module. If , then .
Proof. By Lemma Lemma 2, there is an element such that for all . Since is an unit in , multiplying the inverse , we find that that .
A Geometric Version of Nakayama’s Lemma
A good resource of various versions and application of Nakayama’s lemma is Mumford’s red book (Mumford 1999).
Corollary 2 Let be a commutative local ring, the maximal ideal and a finitely generated -module. Let , , be elements in such that , , generate . Then , , generate . In particular, generators of also generates .
Proof. Let be the submodule in . Then we have It then follows that (by the third isomorphism theorem or the snake lemma). Therefore,
Apply Nakayama’s Lemma to , we get and hence .
Let be scheme and be a -module. Then the map is an isomorphism. Indeed, if , then there is a unique homomorphism such that . This defines an inverse map.
Let , be a family of sections . We say that is generated by the family, if the corresponding homomorphism is surjective.
Equivalently, is generated by the family , if the family generates and for any closed point , generate the stalk as an -module.
The above corollary implies the following geometric version of Nakayama’s lemma.
Lemma 3 (Geometric Nakayama’s Lemma) Let be a Noetherian scheme, a closed point in , an open neighborhood of , and a coherent sheaf on . Let , , be sections in such that the germs , , in generate the sheaf . Then there exist an open neighborhood such that , , generate .
In particular, if , then there exists an open subset in such that =0.
Proof. Let be an affine open neighborhood of in . Then , , and , where is a finitely generated -module. Since , , generate , by Corollary @ref(cor:LiftingSection), we know that , , generate . Moreover, , , generates .
Those sections define a morphism such that the morphism at the stalk level is surjective.
By shrinking , we may assume that is surjective (see Lemma 17.9.4 in the Stack Project).
Therefore, , , generate .
In the lemma, the condition can be relax to any scheme and is quasicoherent sheaf of finite type (see for example Serre 1955, sec. II Proposition 1).
This geometric version has some useful corollaries.
Corollary 3 Let be a Noetherian scheme and a coherent sheaf on . Then is an upper semi-continuous function, i.e. the set is open for any .
Proof. It follows directly from Lemma 3.
Corollary 4 Let be a reduced Noetherian scheme and a coherent sheaf on . Then is a free -module in some neighborhood of if and only if is a constant near .
Proof. If is free in a neighborhood of , then is constant over that neighborhood.
Conversely, let be a neighborhood of such that is a constant for any . Then by shrinking , we may assume that is surjective. Let be the kernel of this morphism. If , then there is a nonzero element in . Since and is globally generated, we may view as a global section of . Since is reduced, since X is reduced, the section will be non-zero at some generic point . Localized at , we get a short exact sequence where is a field because is the generic point.
Because . There is a contraction by comparing the ranks of the -modules in the above short exact sequence.
Therefore, is an isomorphism.
Corollary 5 Let be a non-empty quasi-compact scheme and is a coherent sheaf on . If is globally generated at all closed points, then is globally generated at all points.
Proof. By Lemma Lemma 3, it suffices to show that there is a closed point in the closure of every point of . Indeed, let be an open neighborhood of such that is generated by it globally sections. Since , we see . Otherwise, if , then which implies a contradiction.
The fact that every point in has a closed point in its closure follows from the quasi-compactness of .
Since is non-empty quasi-compact, it admits an irredundant finite open cover . Restrict to , we get an irredundant finite open cover . Because a prime ideal is always contained in a maximal ideal. In each affine open set, there is a closed point. Let be a close point in . If it is a closed point of , then we are done. Otherwise, let such that and , where the closure is taken in the closed subset but not in . With loss of generality, we may assume that . Repeating this procedures, as the open cover is finite, we know that there must be a closed point in which is also closed in .
Note that without quasi-compactness, a closed subset of a scheme may not have a closed point (see Vakil 2017 Exercise 5.1.E).
The Theorem of the Square
Let be a morphism between schemes and be a coherent sheaf on . We denote .
Theorem 1 (Theorem of the Square) Let be a complete varieties and a reduced Noetherian Scheme. Let and be two line bundles on . If for all closed points , we have there exists a line bundle on such that , where is the projection onto .
Proof. From the statement, we are expect that . Indeed, this is true. First, because is complete and is trivial, then for any . By Grauert’s Theorem, we know that is locally free and moreover of rank 1.
We now show that the natural morphism is an isomorphism. Let be the restriction of on the fiber . Then by diagram chasing and the assumption, we have an isomorphism along the fiber . Consequently, for each closed point , the morphism is an isomorphism.
Therefore, by Nakayama lemma, is surjective. Because is of the rank 1, then the surjective morphism must be an isomorphism, i.e. .
Corollary 6 (See-Saw Principle) Suppose that, in addition to the hypotheses of Theorem of the Square, for some . Then .
Proof. Let be the line bundle on such that . Over , we have . Therefore, is trivial, which implies that is trivial and .
The see-saw principle is very useful to prove the Theorem of cube which says that an invertible sheaf on the product of three complete varieties is trivial if it is trivial along fibers to the three projections . Interested reader may find a proof, for example, in (Milne 2008).