In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
and
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, Nakayama's lemma — also known as the Krull–Azumaya theorem
— governs the interaction between the
Jacobson radical of a
ring (typically a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
) and its
finitely generated modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
. Informally, the
lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), ...
immediately gives a precise sense in which finitely generated modules over a commutative ring behave like
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over a
field. It is an important tool in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, because it allows local data on
algebraic varieties, in the form of modules over
local ring In abstract algebra, more specifically 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 algebrai ...
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
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
Indiana University (IU) is a system of public ...
(
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
This year is notable for Apollo 11's first landing on the moon.
Events January
* January 4 – The Government of Spain hands over Ifni to Morocco.
* January 5
**Ariana Afghan Airlines Flight 701 crashes into a house on its approach to ...
). The special case of the
noncommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
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 radicals.
Statement
Let
be a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
with identity 1. The following is Nakayama's lemma, as stated in :
Statement 1: Let
be an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
in
, and
a
finitely-generated module over
. If
, then there exists an
with
, such that
.
This is proven
below
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*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.
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.
Statement 4: If ''M'' is a finitely-generated module over ''R'' and the images of elements ''m''
1,...,''m''
''n'' of ''M'' in ''M'' / ''J''(''R'')''M'' generate ''M'' / ''J''(''R'')''M'' as an ''R''-module, then ''m''
1,...,''m''
''n'' also generate ''M'' as an ''R''-module.
:''Proof'': Apply Statement 3 to ''N'' = Σ
''i''''Rm''
''i''.
If one assumes instead that ''R'' is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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
''T''1 separation axiom, and is equivalent to
Consequences
Local rings
In the special case of a finitely generated module
over a
local ring In abstract algebra, more specifically 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 algebrai ...
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 interpretation
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
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* Section (music), a complete, but not independent, musical idea
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** Section sig ...
of
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
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
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
. 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 A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
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* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
. 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
-module 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 down
The going up theorem is essentially a corollary of Nakayama's lemma. It asserts:
* Let
be an
integral extension In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' i ...
of commutative rings, and
a
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
of
. Then there is a prime ideal
in
such that
. Moreover,
can be chosen to contain any prime
of
such that
.
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
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.
Homological versions
Nakayama's lemma also has several versions in
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...
. 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
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
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
* 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
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to co ...
.
Inverse function theorem
Nakayama's lemma is used to prove a version of the
inverse function theorem in algebraic geometry:
* Let
be a
projective morphism between
quasi-projective varieties. Then
is an isomorphism if and only if it is a bijection and the
differential is injective for all
.
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
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\ ...
::
:with ''p''
''k'' ∈ ''I''
''k'', 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 ''x''
''i'' of ''M'', one has a relation of the form
:
where ''a''
''ij'' ∈ ''I''. Thus
:
The required result follows by multiplying by the
adjugate of the matrix (φδ
''ij'' − ''a''
''ij'') and invoking
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
. One finds then det(φδ
''ij'' − ''a''
''ij'') = 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 unital rings ''R''. 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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals co ...
of ''U'', then ''U''/''V'' is
simple
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Arts and entertainment
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* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
. 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''.
Proof
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.
[; ]
Graded version
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 also
*
Module theory
*
Serre–Swan theorem
Notes
References
*.
*.
*
*
*.
*
*.
*.
*.
*{{Citation , last1=Nakayama , first1=Tadasi , title=A remark on finitely generated modules , mr=0043770 , year=1951 , journal=Nagoya Mathematical Journal , issn=0027-7630 , volume=3 , pages=139–140 , doi=10.1017/s0027763000012265, doi-access=free .
Links
How to understand Nakayama's Lemma and its Corollaries
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