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 ...
, the Artin–Rees lemma is a basic result about
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
, along with results such as the
Hilbert basis theorem. It was proved in the 1950s in independent works by the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
s
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
and
David Rees David or Dai Rees may refer to:
Entertainment
* David Rees (author) (1936–1993), British children's author
* Dave Rees (born 1969), American drummer for SNFU and Wheat Chiefs
* David Rees (cartoonist) (born 1972), American cartoonist and televis ...
; a special case was known to
Oscar Zariski
, birth_date =
, birth_place = Kobrin, Russian Empire
, death_date =
, death_place = Brookline, Massachusetts, United States
, nationality = American
, field = Mathematics
, work_institutions = ...
prior to their work.
One consequence of the lemma is the
Krull intersection theorem 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 algebraic num ...
. The result is also used to prove the exactness property of
completion. The lemma also plays a key role in the study of
â„“-adic sheaves.
Statement
Let ''I'' 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 considere ...
in a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
''R''; let ''M'' be a
finitely generated ''R''-module and let ''N'' a submodule of ''M''. Then there exists an integer ''k'' ≥ 1 so that, for ''n'' ≥ ''k'',
:
Proof
The lemma immediately follows from the fact that ''R'' is Noetherian once necessary notions and notations are set up.
For any ring ''R'' and an ideal ''I'' in ''R'', we set
(''B'' for blow-up.) We say a decreasing sequence of submodules
is an ''I''-filtration if
; moreover, it is stable if
for sufficiently large ''n''. If ''M'' is given an ''I''-filtration, we set
; it is a
graded module
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
over
.
Now, let ''M'' be a ''R''-module with the ''I''-filtration
by finitely generated ''R''-modules. We make an observation
:
is a finitely generated module over
if and only if the filtration is ''I''-stable.
Indeed, if the filtration is ''I''-stable, then
is generated by the first
terms
and those terms are finitely generated; thus,
is finitely generated. Conversely, if it is finitely generated, say, by some homogeneous elements in
, then, for
, each ''f'' in
can be written as
with the generators
in
. That is,
.
We can now prove the lemma, assuming ''R'' is Noetherian. Let
. Then
are an ''I''-stable filtration. Thus, by the observation,
is finitely generated over
. But