Nakayama lemma
Encyclopedia
In mathematics
, more specifically modern algebra and commutative algebra
, Nakayama's lemma also known as the Krull–Azumaya theorem 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 space
s over a field
. It is a significant tool in algebraic geometry
, because it allows local data on algebraic varieties
, in the form of modules over local ring
s, 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 , 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). 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. The latter has various applications in the theory of Jacobson radical
s.
with identity 1. The following is Nakayama's lemma, as stated in :
Statement 1: Let I be an ideal
in R, and M a finitely-generated module
over R. If IM = M, then there exists an r ∈ R with r ≡ 1 (mod I), such that rM = 0.
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.
Statement 2: With conditions as above, if I is contained in the Jacobson radical
of R, then necessarily M = 0.
More generally, one has
Statement 3: If M = N + IM for some ideal I in the Jacobson radical of R and M is finitely-generated, then M = N.
The following result manifests Nakayama's lemma in terms of generators
Statement 4: Let I be an ideal in the Jacobson radical of R, and suppose that M is finitely-generated. If m1,...,mn have images in M/IM that generate it as an R-module, then m1,...,mn also generate M as an R-module.
This conclusion of the last corollary holds without assuming in advance that M is finitely generated, provided that M is assumed to be a complete
and separated module with respect to the I-adic topology. Here separatedness means that the I-adic topology satisfies the T1
separation axiom, and is equivalent to
R with maximal ideal
m, the quotient M/mM is a vector space over the field R/m. Statement 4 then implies that a basis of M/mM lifts to a minimal set of generators of M. Conversely, every minimal set of generators of M 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 germ
s of functions at a point. Finitely generated modules over local rings arise quite often as germs of sections
of vector bundle
s. 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 F be a coherent sheaf of OX-modules over an arbitrary scheme
X. The stalk of F at a point p ∈ X, denoted by Fp, is a module over the local ring Op. The fibre of F at p is the vector space F(p) = Fp/mpFp where mp is the maximal ideal of Op. Nakayama's lemma implies that a basis of the fibre F(p) lifts to a minimal set of generators of Fp. That is:
To give geometrical context for this result, integral extensions correspond to proper map
s of algebraic varieties. For varieties over the complex field, proper simply means that the inverse image of a compact set in the usual topology is again compact. Going up then implies that the image of an algebraic subvariety under a proper map is again an algebraic subvariety.
Over a local ring, one can say more about module epimorphisms:
. The above statement about epimorphisms can be used to show:
A geometrical and global counterpart to this is the Serre–Swan theorem, relating projective modules and coherent sheaves.
More generally, one has
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.
Let J(R) be the Jacobson radical
of R. If U is a right module over a ring, R, and I is an 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 (right) 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:
. Since X generates U,
Suppose,
, to obtain a contradiction. Since, 
, we conclude,
, for 
, and 
By associativity,
Since J(R) is a (two-sided) ideal in R, we have
for every i, and thus this becomes
, for 
Applying distributivity,
Since
, it is quasiregular
and thus,
, for all i, where U(R) denotes the group of units
in R. Choose some j and write,
Therefore,
Thus xj is a linear combination of the elements of X distinct from xj. This contradicts the minimality of X and establishes the result.
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., 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically modern algebra and commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, Nakayama's lemma also known as the Krull–Azumaya theorem governs the interaction between the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
(typically a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
) and its finitely generated
Finitely generated
In mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...
modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. It is a significant tool in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, because it allows local data on algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
, in the form of modules over local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
s, to be studied pointwise as vector spaces over the residue field of the ring.
The lemma is named after the Japanese mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Tadashi Nakayama
Tadashi Nakayama (mathematician)
was a mathematician who made important contributions to representation theory. He received his degrees from Tokyo University and Osaka University and held permanent positions at Osaka University and Nagoya University. He had visiting positions at Princeton University, Illinois University, and...
and introduced in its present form in , although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull
Wolfgang Krull
Wolfgang Krull was a German mathematician working in the field of commutative algebra.He was born in Baden-Baden, Imperial Germany and died in Bonn, West Germany.- See also :* Krull dimension* Krull topology...
and then in general by Goro Azumaya
Goro Azumaya
was a Japanese mathematician who introduced the notion of Azumaya algebra in 1951. His advisor was Shokichi Iyanaga. At the time of his death he was an emeritus professor at Indiana University.-External links:...
(1951). In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
, an observation made by Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
(1969). The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson
Nathan Jacobson
Nathan Jacobson was an American mathematician....
(1945), and so the noncommutative Nakayama lemma is sometimes known as the Jacobson–Azumaya theorem. The latter has various applications in the theory of Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
s.
Statement
Let R be a commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
with identity 1. The following is Nakayama's lemma, as stated in :
Statement 1: Let I be an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in R, and M a finitely-generated module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over R. If IM = M, then there exists an r ∈ R with r ≡ 1 (mod I), such that rM = 0.
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.
Statement 2: With conditions as above, if I is contained in the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
of R, then necessarily M = 0.
- Proof: r−1 (with r as above) is in the Jacobson radical so r is invertible.
More generally, one has
Statement 3: If M = N + IM for some ideal I in the Jacobson radical of R and M is finitely-generated, then M = N.
- Proof: Apply Statement 2 to M/N.
The following result manifests Nakayama's lemma in terms of generators
Statement 4: Let I be an ideal in the Jacobson radical of R, and suppose that M is finitely-generated. If m1,...,mn have images in M/IM that generate it as an R-module, then m1,...,mn also generate M as an R-module.
- Proof: Apply Statement 2 to N = M/ΣiRmi.
This conclusion of the last corollary holds without assuming in advance that M is finitely generated, provided that M is assumed to be a complete
Completion (ring theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
and separated module with respect to the I-adic topology. Here separatedness means that the I-adic topology satisfies the T1
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
separation axiom, and is equivalent to
Local rings
In the special case of a finitely generated module M over a local ringLocal ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
R with maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
m, the quotient M/mM is a vector space over the field R/m. Statement 4 then implies that a basis of M/mM lifts to a minimal set of generators of M. Conversely, every minimal set of generators of M 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 germ
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
s of functions at a point. Finitely generated modules over local rings arise quite often as germs of sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s. Working at the level of germs rather than points, the notion of finite dimensional vector bundle gives way to that of a coherent sheaf
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
. 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 F be a coherent sheaf of OX-modules over an arbitrary scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
X. The stalk of F at a point p ∈ X, denoted by Fp, is a module over the local ring Op. The fibre of F at p is the vector space F(p) = Fp/mpFp where mp is the maximal ideal of Op. Nakayama's lemma implies that a basis of the fibre F(p) lifts to a minimal set of generators of Fp. That is:
- Any basis of the fiber of a coherent sheaf F at a point comes from a minimal basis of local sections.
Going up and going down
The going up theorem is essentially a corollary of Nakayama's lemma. It asserts:- Let R ⊂ S be an integral extension of commutative rings, and P a prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
of R. Then there is a prime ideal Q in S such that Q ∩ R = P. Moreover, Q can be chosen to contain any prime Q1 of S such that Q1 ∩ R ⊂ P.
To give geometrical context for this result, integral extensions correspond to proper map
Proper map
In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...
s of algebraic varieties. For varieties over the complex field, proper simply means that the inverse image of a compact set in the usual topology is again compact. Going up then implies that the image of an algebraic subvariety under a proper map is again an algebraic subvariety.
Module epimorphisms
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 M is a finitely generated R-module and ƒ : M → M is a surjective endomorphism, then ƒ is an isomorphism.
Over a local ring, one can say more about module epimorphisms:
- Suppose that R is a local ring with maximal ideal m, and M, N are finitely generated R-modules. If φ : M → N is an R-linear map such that the quotient φm : M/mM → N/mN is surjective, then φ is surjective.
Homological versions
Nakayama's lemma also has several versions in homological algebraHomological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
. The above statement about epimorphisms can be used to show:
- Let M be a finitely generated module over a local ring. Then M is projectiveProjective moduleIn mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
if and only if it is freeFree moduleIn mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
.
A geometrical and global counterpart to this is the Serre–Swan theorem, relating projective modules and coherent sheaves.
More generally, one has
- Let R be a local ring and M a finitely generated module over R. Then the projective dimension of M over R is equal to the length of every minimal free resolution of M. Moreover, the projective dimension is equal to the global dimension of M, which is by definition the smallest integer i ≥ 0 such that
-
- Here k is the residue field of R and Tor is the tor functorTor functorIn homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....
.
Proof
A standard proof of the Nakayama lemma uses the following technique due to .- 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
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
, 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
Cramer's rule
In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...
. 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 case
A version of the lemma holds for right modules over non-commutative unitary rings R. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem.Let J(R) be the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
of R. If U is a right module over a ring, R, and I is an 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 (right) submodule of U, then U/V is simple
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
. 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
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
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 ring R. If U is a non-zero module, then U·J(R) is a proper submodule of U.
Proof
Let X be a finite subset of U, minimal with respect to the property that it generates U. Since U is non-zero, this set X is nonempty. Denote every element of X by xi, for. Since X generates U,Suppose,
, to obtain a contradiction. Since, , we conclude,, for , and By associativity,
Since J(R) is a (two-sided) ideal in R, we have
for every i, and thus this becomes, for Applying distributivity,
Since
, it is quasiregularQuasiregular element
In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. Intuitively, quasiregularity captures what it means for an element of a ring to be "bad"; that is, have undesirable properties...
and thus,
, for all i, where U(R) denotes the group of unitsUnit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
in R. Choose some j and write,
Therefore,
Thus xj is a linear combination of the elements of X distinct from xj. This contradicts the minimality of X and establishes the result.
Graded version
There is also a graded version of Nakayama's lemma. Let R be a graded ring (over the 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., .