Nakayama's lemma

Summary

In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem[1] — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring.

The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in Nakayama (1951), although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya (1951).[2] In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem, an observation made by Michael Atiyah (1969). The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson (1945), and so the noncommutative Nakayama lemma is sometimes known as the Jacobson–Azumaya theorem.[1] The latter has various applications in the theory of Jacobson radicals.[3]

StatementEdit

Let   be a commutative ring with identity 1. The following is Nakayama's lemma, as stated in Matsumura (1989):

Statement 1: Let   be an ideal in  , and   a finitely-generated module over  . If  , then there exists an   with  , such that  .

This is proven below.

The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.[4]

Statement 2: If   is a finitely-generated module over  ,   is the Jacobson radical of  , and  , then  .

Proof:   (with   as above) is in the Jacobson radical so   is invertible.

More generally, one has that   is a superfluous submodule of   when   is finitely-generated.

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.[5]

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.[6] Here separatedness means that the I-adic topology satisfies the T1 separation axiom, and is equivalent to  

ConsequencesEdit

Local ringsEdit

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.

Geometric interpretationEdit

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.[7] 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  .

Module epimorphismsEdit

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.[8]

Over a local ring, one can say more about module epimorphisms:[9]

  • 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.

Homological versionsEdit

Nakayama's lemma also has several versions in homological algebra. The above statement about epimorphisms can be used to show:[9]

  • 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.

More generally, one has[10]

  • 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
 
Here   is the residue field of   and   is the tor functor.

ProofEdit

A standard proof of the Nakayama lemma uses the following technique due to Atiyah & Macdonald (1969).[11]

  • 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.

Noncommutative caseEdit

A version of the lemma holds for right modules over non-commutative unital rings R. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem.[12]

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.[13] 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.[14] 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.[15] 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.[16]

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.[16]

ProofEdit

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

 .

Therefore,

 .

Thus   is a linear combination of the elements from  . This contradicts the minimality of   and establishes the result.[17]

Graded versionEdit

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.,  .

See alsoEdit

NotesEdit

  1. ^ a b Nagata 1962, §A.2
  2. ^ Nagata 1962, §A.2; Matsumura 1989, p. 8
  3. ^ Isaacs 1993, Corollary 13.13, p. 184
  4. ^ Eisenbud 1995, Corollary 4.8; Atiyah & Macdonald (1969, Proposition 2.6)
  5. ^ Eisenbud 1995, Corollary 4.8(b)
  6. ^ Eisenbud 1995, Exercise 7.2
  7. ^ Eisenbud 1993, §4.4
  8. ^ Matsumura 1989, Theorem 2.4
  9. ^ a b Griffiths & Harris 1994, p. 681
  10. ^ Eisenbud 1993, Corollary 19.5
  11. ^ Matsumura 1989, p. 7: "A standard technique applicable to finite A-modules is the 'determinant trick'..." See also the proof contained in Eisenbud (1995, §4.1).
  12. ^ Nagata 1962, §A2
  13. ^ Isaacs 1993, p. 182
  14. ^ Isaacs 1993, p. 183
  15. ^ Isaacs 1993, Theorem 12.19, p. 172
  16. ^ a b Isaacs 1993, Theorem 13.11, p. 183
  17. ^ Isaacs 1993, Theorem 13.11, p. 183; Isaacs 1993, Corollary 13.12, p. 183

ReferencesEdit

  • Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to Commutative Algebra, Reading, MA: Addison-Wesley.
  • Azumaya, Gorô (1951), "On maximally central algebras", Nagoya Mathematical Journal, 2: 119–150, doi:10.1017/s0027763000010114, ISSN 0027-7630, MR 0040287.
  • Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
  • Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, doi:10.1002/9781118032527, ISBN 978-0-471-05059-9, MR 1288523
  • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag.
  • Isaacs, I. Martin (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-534-19002-2
  • Jacobson, Nathan (1945), "The radical and semi-simplicity for arbitrary rings", American Journal of Mathematics, 67 (2): 300–320, doi:10.2307/2371731, ISSN 0002-9327, JSTOR 2371731, MR 0012271.
  • Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461.
  • Nagata, Masayoshi (1975), Local rings, Robert E. Krieger Publishing Co., Huntington, N.Y., ISBN 978-0-88275-228-0, MR 0460307.
  • Nakayama, Tadasi (1951), "A remark on finitely generated modules", Nagoya Mathematical Journal, 3: 139–140, doi:10.1017/s0027763000012265, ISSN 0027-7630, MR 0043770.

LinksEdit

  • How to understand Nakayama's Lemma and its Corollaries