In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
and
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, Nakayama's lemma — also known as the Krull–Azumaya theorem
— governs the interaction between the
Jacobson radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
of a
ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
(typically a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
) 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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s over a
field. It is an important tool in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, because it allows local data on
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, in the form of modules over
local ring
In mathematics, more specifically 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 algebraic varieties or manifolds, or of ...
s, to be studied pointwise as vector spaces over the
residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...
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
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
and then in general by
Goro Azumaya (
1951
Events
January
* January 4 – Korean War: Third Battle of Seoul – Chinese and North Korean forces capture Seoul for the second time (having lost the Second Battle of Seoul in September 1950).
* January 9 – The Government of the Uni ...
). 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
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...
(
1969). 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. Perhaps most familiar as a p ...
version of the lemma for right ideals appears in
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
Biography
Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awa ...
(
1945
1945 marked the end of World War II, the fall of Nazi Germany, and the Empire of Japan. It is also the year concentration camps were liberated and the only year in which atomic weapons have been used in combat.
Events
World War II will be ...
), 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
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
s.
Statement
Let
be a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
with identity 1. The following is Nakayama's lemma, as stated in :
Statement 1: Let
be an
ideal in
, and
a
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts i ...
over
. If
, then there exists
with
such that
.
This is proven
below
Below may refer to:
*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 Belo ...
. A useful mnemonic for Nakayama's lemma is "
". This summarizes the following alternative formulation:
Statement 2: Let
be an
ideal in
, and
a
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts i ...
over
. If
, then there exists an
such that
for all
.
:''Proof'': Take
in Statement 1.
The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears.
Statement 3: If ''
'' is a finitely generated module over ''
'',
is the
Jacobson radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
of
, and
, then
.
:''Proof'':
(with
as in Statement 1) is in the Jacobson radical so
is invertible.
More generally, one has that
is a
superfluous submodule of
when
is finitely generated.
Statement 4: If ''
'' is a finitely generated module over ''
'', ''
'' is a submodule of
, and ''
'' = ''
'', then ''
'' = ''
''.
:''Proof'': Apply Statement 3 to ''
''.
The following result manifests Nakayama's lemma in terms of generators.
Statement 5: If ''
'' is a finitely generated module over ''
'' and the images of elements ''
''
1,...,''
''
'''' of ''
'' in ''
'' generate ''
'' as an ''
''-module, then ''
''
1,...,''
''
'''' also generate ''
'' as an ''
''-module.
:''Proof'': Apply Statement 4 to
.
If one assumes instead that ''
'' is
complete and ''
'' is separated with respect to the ''
''-adic topology for an ideal ''
'' in ''
'', this last statement holds with ''
'' in place of ''
'' and without assuming in advance that ''
'' is finitely generated. Here separatedness means that the ''
''-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 mathematics, more specifically 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 algebraic varieties or manifolds, or of ...
with
maximal ideal
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 ...
, the quotient
is a vector space over the field
. Statement 5 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
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by a ...
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
germ
Germ or germs may refer to:
Science
* Germ (microorganism), an informal word for a pathogen
* Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually
* Germ layer, a primary layer of cells that forms during embry ...
s of functions at a point. Finitely generated modules over local rings arise quite often as germs of
sections 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 eve ...
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 refer ...
. 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
-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 subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''.
If ''A'', ''B'' are fields, then the notions of "integral over" and ...
of commutative rings, and
a
prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
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 (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. 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 group homomorp ...
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
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizatio ...
of
over
is equal to the length of every minimal
free resolution
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to de ...
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 const ...
.
Inverse function theorem
Nakayama's lemma is used to prove a version of the
inverse function theorem
In mathematics, the inverse function theorem is a theorem that asserts that, if a real function ''f'' has a continuous derivative near a point where its derivative is nonzero, then, near this point, ''f'' has an inverse function. The inverse fu ...
in algebraic geometry:
* Let
be a
projective morphism
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometr ...
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 non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
::
: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 of ...
. 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:
and
.
Noncommutative case
A version of the lemma holds for right modules over
non-commutative
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. Perhaps most familiar as a pr ...
unital ring
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in ...
s ''R''. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem.
Let J(''R'') be the
Jacobson radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
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 ...
of ''U'', then ''U''/''V'' is
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by John ...
. 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 In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
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
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...
*
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
Theorems in ring theory
Algebraic geometry
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