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.
The following result manifests Nakayama's lemma in terms of generators.
-adic topology satisfies the T1 separation axiom, and is equivalent to
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.
, and if we take a basis of the vector bundle at a point in the scheme
[7] It asserts: 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: Over a local ring, one can say more about module epimorphisms:[9] Nakayama's lemma also has several versions in homological algebra.
The above statement about epimorphisms can be used to show:[9] A geometrical and global counterpart to this is the Serre–Swan theorem, relating projective modules and coherent sheaves.
More generally, one has[10] Nakayama's lemma is used to prove a version of the inverse function theorem in algebraic geometry: A standard proof of the Nakayama lemma uses the following technique due to Atiyah & Macdonald (1969).
[12] This assertion is precisely a generalized version of the Cayley–Hamilton theorem, and the proof proceeds along the same lines.
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.
[13] 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.
So U·J(R) is necessarily a subset of V, by the definition of J(R) and the fact that U/V is simple.
However, this need not hold for arbitrary modules U over R, for U need not contain any maximal submodules.
[16] 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.
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.
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