Statement 3: If is a finitely-generated module over R, N is a submodule of , and M = N + J(R)M, then M = N.
Proof: Apply Statement 2 to M/N.
The following result manifests Nakayama's lemma in terms of generators.
Statement 4: If M is a finitely-generated module over R and the images of elements m1,...,mn of M in M / J(R)M generate M / J(R)M as an R-module, then m1,...,mn also generate M as an R-module.
Proof: Apply Statement 3 to N = ΣiRmi.
If one assumes instead that R is complete and M is separated with respect to the I-adic topology for an ideal I in R, this last statement holds with I in place of J(R) and without assuming in advance that M is finitely generated. Here separatedness means that the I-adic topology satisfies the T1 separation axiom, and is equivalent to
In the special case of a finitely generated module over a local ring with maximal ideal, the quotient is a vector space over the field . Statement 4 then implies that a basis of lifts to a minimal set of generators of . Conversely, every minimal set of generators of is obtained in this way, and any two such sets of generators are related by an invertible matrix with entries in the ring.
In this form, Nakayama's lemma takes on concrete geometrical significance. Local rings arise in geometry as the germs of functions at a point. Finitely generated modules over local rings arise quite often as germs of sections of vector bundles. Working at the level of germs rather than points, the notion of finite-dimensional vector bundle gives way to that of a coherent sheaf. Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle in some sense. More precisely, let be a coherent sheaf of -modules over an arbitrary scheme. The stalk of at a point , denoted by , is a module over the local ring and the fiber of at is the vector space . Nakayama's lemma implies that a basis of the fiber lifts to a minimal set of generators of . That is:
Any basis of the fiber of a coherent sheaf at a point comes from a minimal basis of local sections.
Reformulating this geometrically, if is a locally free -modules representing a vector bundle , and if we take a basis of the vector bundle at a point in the scheme , this basis can be lifted to a basis of sections of the vector bundle in some neighborhood of the point. We can organize this data diagrammatically
where is an n-dimensional vector space, to say a basis in (which is a basis of sections of the bundle ) can be lifted to a basis of sections for some neighborhood of .
Going up and going downEdit
The going up theorem is essentially a corollary of Nakayama's lemma. It asserts:
Let be an integral extension of commutative rings, and a prime ideal of . Then there is a prime ideal in such that . Moreover, can be chosen to contain any prime of such that .
Nakayama's lemma makes precise one sense in which finitely generated modules over a commutative ring are like vector spaces over a field. The following consequence of Nakayama's lemma gives another way in which this is true:
If is a finitely generated -module and is a surjective endomorphism, then is an isomorphism.
Over a local ring, one can say more about module epimorphisms:
Suppose that is a local ring with maximal ideal , and are finitely generated -modules. If is an -linear map such that the quotient is surjective, then is surjective.
Nakayama's lemma also has several versions in homological algebra. The above statement about epimorphisms can be used to show:
Let be a finitely generated module over a local ring. Then is projective if and only if it is free. This can be used to compute the Grothendieck group of any local ring as .
A geometrical and global counterpart to this is the Serre–Swan theorem, relating projective modules and coherent sheaves.
Let be a local ring and a finitely generated module over . Then the projective dimension of over is equal to the length of every minimal free resolution of . Moreover, the projective dimension is equal to the global dimension of , which is by definition the smallest integer such that
Let M be an R-module generated by n elements, and φ: M → M an R-linear map. If there is an ideal I of R such that φ(M) ⊂ IM, then there is a monic polynomial
with pk ∈ Ik, such that
as an endomorphism of M.
This assertion is precisely a generalized version of the Cayley–Hamilton theorem, and the proof proceeds along the same lines. On the generators xi of M, one has a relation of the form
where aij ∈ I. Thus
The required result follows by multiplying by the adjugate of the matrix (φδij − aij) and invoking Cramer's rule. One finds then det(φδij − aij) = 0, so the required polynomial is
To prove Nakayama's lemma from the Cayley–Hamilton theorem, assume that IM = M and take φ to be the identity on M. Then define a polynomial p(x) as above. Then
has the required property.
A version of the lemma holds for right modules over non-commutativeunital ringsR. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem.
Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U.
If V is a maximal submodule of U, then U/V is simple. So U·J(R) is necessarily a subset of V, by the definition of J(R) and the fact that U/V is simple. Thus, if U contains at least one (proper) maximal submodule, U·J(R) is a proper submodule of U. However, this need not hold for arbitrary modules U over R, for U need not contain any maximal submodules. Naturally, if U is a Noetherian module, this holds. If R is Noetherian, and U is finitely generated, then U is a Noetherian module over R, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that U is finitely generated as an R-module (and no finiteness assumption on R), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma.
Precisely, one has:
Nakayama's lemma: Let U be a finitely generated right module over a (unital) ring R. If U is a non-zero module, then U·J(R) is a proper submodule of U.
Let be a finite subset of , minimal with respect to the property that it generates . Since is non-zero, this set is nonempty. Denote every element of by for . Since generates ,.
Suppose , to obtain a contradiction. Then every element can be expressed as a finite combination for some .
Each can be further decomposed as for some . Therefore, we have
Since is a (two-sided) ideal in , we have for every , and thus this becomes
for some , .
Putting and applying distributivity, we obtain
Choose some . If the right ideal were proper, then it would be contained in a maximal right ideal and both and would belong to , leading to a contradiction (note that by the definition of the Jacobson radical). Thus and has a right inverse in . We have
Thus is a linear combination of the elements from . This contradicts the minimality of and establishes the result.
There is also a graded version of Nakayama's lemma. Let R be a ring that is graded by the ordered semigroup of non-negative integers, and let denote the ideal generated by positively graded elements. Then if M is a graded module over R for which for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative degree) such that , then . 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 does not appear in , so either , or such an i does not exist, i.e., .