Integral element explained
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.[1]
If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).
The case of greatest interest in number theory is that of complex numbers integral over Z (e.g.,
or
); in this context, the integral elements are usually called
algebraic integers. The algebraic integers in a finite
extension field k of the
rationals Q form a subring of
k, called the
ring of integers of
k, a central object of study in
algebraic number theory.
In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
Definition
Let
be a ring and let
be a subring of
An element
of
is said to be
integral over
if for some
there exists
in
such that
The set of elements of
that are integral over
is called the
integral closure of
in
The integral closure of any subring
in
is, itself, a subring of
and contains
If every element of
is integral over
then we say that
is
integral over
, or equivalently
is an
integral extension of
Examples
Integral closure in algebraic number theory
(or
).
Integral closure of integers in rationals
Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.
Quadratic extensions
The Gaussian integers are the complex numbers of the form
, and are integral over
Z.
is then the integral closure of
Z in
. Typically this ring is denoted
.
The integral closure of Z in
is the ring
} \right]This example and the previous one are examples of
quadratic integers. The integral closure of a quadratic extension
can be found by constructing the
minimal polynomial of an arbitrary element
and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the quadratic extensions article.
Roots of unity
Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z[ζ]. This can be found by using the minimal polynomial and using Eisenstein's criterion.
Ring of algebraic integers
The integral closure of Z in the field of complex numbers C, or the algebraic closure
} is called the
ring of algebraic integers.
Other
The roots of unity, nilpotent elements and idempotent elements in any ring are integral over Z.
Integral closure in algebraic geometry
In geometry, integral closure is closely related with normalization and normal schemes. It is the first step in resolution of singularities since it gives a process for resolving singularities of codimension 1.
- For example, the integral closure of
is the ring
since geometrically, the first ring corresponds to the
-plane unioned with the
-plane. They have a codimension 1 singularity along the
-axis where they intersect.
- u−1 is integral over R if and only if u−1 ∈ R[''u''].
is integral over
R.
- The integral closure of the homogeneous coordinate ring of a normal projective variety X is the ring of sections
oplusn\operatorname{H}0(X,l{O}X(n)).
Integrality in algebra
is an
algebraic closure of a field
k, then
is integral over
- The integral closure of C[[x]] in a finite extension of C((x)) is of the form
(cf.
Puiseux series)
Equivalent definitions
Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent:
(i) b is integral over A;
(ii) the subring A[''b''] of B generated by A and b is a finitely generated A-module;
(iii) there exists a subring C of B containing A[''b''] and which is a finitely generated A-module;
(iv) there exists a faithful A[''b'']-module M such that M is finitely generated as an A-module.
The usual proof of this uses the following variant of the Cayley–Hamilton theorem on determinants:
Theorem Let u be an endomorphism of an A-module M generated by n elements and I an ideal of A such that
. Then there is a relation:
un+a1un-1+ … +an-1u+an=0,ai\inIi.
This theorem (with
I =
A and
u multiplication by
b) gives (iv) ⇒ (i) and the rest is easy. Coincidentally,
Nakayama's lemma is also an immediate consequence of this theorem.
Elementary properties
Integral closure forms a ring
It follows from the above four equivalent statements that the set of elements of
that are integral over
forms a subring of
containing
. (Proof: If
x,
y are elements of
that are integral over
, then
are integral over
since they stabilize
, which is a finitely generated module over
and is annihilated only by zero.)
[2] This ring is called the
integral closure of
in
.
Transitivity of integrality
Another consequence of the above equivalence is that "integrality" is transitive, in the following sense. Let
be a ring containing
and
. If
is integral over
and
integral over
, then
is integral over
. In particular, if
is itself integral over
and
is integral over
, then
is also integral over
.
Integral closed in fraction field
If
happens to be the integral closure of
in
, then
A is said to be
integrally closed in
. If
is the
total ring of fractions of
, (e.g., the
field of fractions when
is an
integral domain), then one sometimes drops the qualification "in
and simply says "integral closure of
" and "
is
integrally closed."
[3] For example, the ring of integers
is integrally closed in the field
.
Transitivity of integral closure with integrally closed domains
Let A be an integral domain with the field of fractions K and A' the integral closure of A in an algebraic field extension L of K. Then the field of fractions of A' is L. In particular, A' is an integrally closed domain.
Transitivity in algebraic number theory
This situation is applicable in algebraic number theory when relating the ring of integers and a field extension. In particular, given a field extension
the integral closure of
in
is the ring of integers
.
Remarks
Note that transitivity of integrality above implies that if
is integral over
, then
is a union (equivalently an
inductive limit) of subrings that are finitely generated
-modules.
If
is
noetherian, transitivity of integrality can be weakened to the statement:
There exists a finitely generated
-submodule of
that contains
.
Relation with finiteness conditions
Finally, the assumption that
be a subring of
can be modified a bit. If
is a
ring homomorphism, then one says
is
integral if
is integral over
. In the same way one says
is
finite (
finitely generated
-module) or of
finite type (
finitely generated
-algebra). In this viewpoint, one has that
is finite if and only if
is integral and of finite type.
Or more explicitly,
is a finitely generated
-module if and only if
is generated as an
-algebra by a finite number of elements integral over
.
Integral extensions
Cohen-Seidenberg theorems
An integral extension A ⊆ B has the going-up property, the lying over property, and the incomparability property (Cohen–Seidenberg theorems). Explicitly, given a chain of prime ideals
ak{p}1\subset … \subsetak{p}n
in
A there exists a
ak{p}'1\subset … \subsetak{p}'n
in
B with
(going-up and lying over) and two distinct prime ideals with inclusion relation cannot contract to the same prime ideal (incomparability). In particular, the
Krull dimensions of
A and
B are the same. Furthermore, if
A is an integrally closed domain, then the going-down holds (see below).
In general, the going-up implies the lying-over. Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over".
When A, B are domains such that B is integral over A, A is a field if and only if B is a field. As a corollary, one has: given a prime ideal
of
B,
is a
maximal ideal of
B if and only if
is a maximal ideal of
A. Another corollary: if
L/
K is an algebraic extension, then any subring of
L containing
K is a field.
Applications
Let B be a ring that is integral over a subring A and k an algebraically closed field. If
is a homomorphism, then
f extends to a homomorphism
B →
k. This follows from the going-up.
Geometric interpretation of going-up
Let
be an integral extension of rings. Then the induced map
\begin{cases}f\#:\operatorname{Spec}B\to\operatorname{Spec}A\ p\mapstof-1(p)\end{cases}
is a closed map; in fact,
for any ideal
I and
is
surjective if
f is
injective. This is a geometric interpretation of the going-up.
Geometric interpretation of integral extensions
Let B be a ring and A a subring that is a noetherian integrally closed domain (i.e.,
is a
normal scheme.) If
B is integral over
A, then
\operatorname{Spec}B\to\operatorname{Spec}A
is submersive; i.e., the
topology of
is the
quotient topology. The proof uses the notion of
constructible sets. (See also:
Torsor (algebraic geometry).)
Integrality, base-change, universally-closed, and geometry
If
is integral over
, then
is integral over
R for any
A-algebra
R. In particular,
\operatorname{Spec}(B ⊗ AR)\to\operatorname{Spec}R
is closed; i.e., the integral extension induces a "
universally closed" map. This leads to a
geometric characterization of integral extension. Namely, let
B be a ring with only finitely many
minimal prime ideals (e.g., integral domain or noetherian ring). Then
B is integral over a (subring)
A if and only if
\operatorname{Spec}(B ⊗ AR)\to\operatorname{Spec}R
is closed for any
A-algebra
R. In particular, every
proper map is universally closed.
[4] Galois actions on integral extensions of integrally closed domains
G=\operatorname{Gal}(L/K)
acts transitively on each fiber of
\operatorname{Spec}B\to\operatorname{Spec}A
.
Proof. Suppose
for any
in
G. Then, by
prime avoidance, there is an element
x in
such that
for any
.
G fixes the element
y=\prod\nolimits\sigma\sigma(x)
and thus
y is
purely inseparable over
K. Then some power
belongs to
K; since
A is integrally closed we have:
Thus, we found
is in
but not in
; i.e.,
ak{p}1\capA\neak{p}2\capA
.
Application to algebraic number theory
The Galois group
then acts on all of the prime ideals
ak{q}1,\ldots,ak{q}k\inSpec(l{O}L)
lying over a fixed prime ideal
.
[5] That is, if
then there is a Galois action on the set
Sak{p}=\{ak{q}1,\ldots,ak{q}k\}
. This is called the
Splitting of prime ideals in Galois extensions.
Remarks
The same idea in the proof shows that if
is a purely inseparable extension (need not be normal), then
\operatorname{Spec}B\to\operatorname{Spec}A
is
bijective.
Let A, K, etc. as before but assume L is only a finite field extension of K. Then
(i)
\operatorname{Spec}B\to\operatorname{Spec}A
has finite fibers.
(ii) the going-down holds between A and B: given
ak{p}1\subset … \subsetak{p}n=ak{p}'n\capA
, there exists
ak{p}'1\subset … \subsetak{p}'n
that contracts to it.Indeed, in both statements, by enlarging
L, we can assume
L is a normal extension. Then (i) is immediate. As for (ii), by the going-up, we can find a chain
that contracts to
. By transitivity, there is
such that
and then
are the desired chain.
Integral closure
See also: Integral closure of an ideal. Let A ⊂ B be rings and A' the integral closure of A in B. (See above for the definition.)
Integral closures behave nicely under various constructions. Specifically, for a multiplicatively closed subset S of A, the localization S−1A' is the integral closure of S−1A in S−1B, and
is the integral closure of
in
.
[6] If
are subrings of rings
, then the integral closure of
in
is
where
are the integral closures of
in
.
The integral closure of a local ring A in, say, B, need not be local. (If this is the case, the ring is called unibranch.) This is the case for example when A is Henselian and B is a field extension of the field of fractions of A.
If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation rings of K containing A.
Let A be an
-graded subring of an
-
graded ring B. Then the integral closure of
A in
B is an
-graded subring of
B.
[7] There is also a concept of the integral closure of an ideal. The integral closure of an ideal
, usually denoted by
, is the set of all elements
such that there exists a monic polynomial
with
with
as a root.
[8] [9] The
radical of an ideal is integrally closed.
[10] [11] For noetherian rings, there are alternate definitions as well.
if there exists a
not contained in any minimal prime, such that
for all
.
if in the normalized blow-up of
I, the pull back of
r is contained in the inverse image of
I. The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings.
The notion of integral closure of an ideal is used in some proofs of the going-down theorem.
Conductor
See main article: Conductor (ring theory).
Let B be a ring and A a subring of B such that B is integral over A. Then the annihilator of the A-module B/A is called the conductor of A in B. Because the notion has origin in algebraic number theory, the conductor is denoted by
. Explicitly,
consists of elements
a in
A such that
. (cf.
idealizer in abstract algebra.) It is the largest
ideal of
A that is also an ideal of
B.
[12] If
S is a multiplicatively closed subset of
A, then
S-1ak{f}(B/A)=ak{f}(S-1B/S-1A)
.If
B is a subring of the
total ring of fractions of
A, then we may identify
ak{f}(B/A)=\operatorname{Hom}A(B,A)
.
Example: Let k be a field and let
(i.e.,
A is the coordinate ring of the
affine curve
.)
B is the integral closure of
A in
. The conductor of
A in
B is the ideal
. More generally, the conductor of
,
a,
b relatively prime, is
with
.
Suppose B is the integral closure of an integral domain A in the field of fractions of A such that the A-module
is finitely generated. Then the conductor
of
A is an ideal defining the
support of
; thus,
A coincides with
B in the complement of
in
. In particular, the set
\{ak{p}\in\operatorname{Spec}A\midAak{p}isintegrallyclosed\}
, the complement of
, is an
open set.
Finiteness of integral closure
An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results.
The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring. This is a consequence of the Krull–Akizuki theorem. In general, the integral closure of a noetherian domain of dimension at most 2 is noetherian; Nagata gave an example of dimension 3 noetherian domain whose integral closure is not noetherian. A nicer statement is this: the integral closure of a noetherian domain is a Krull domain (Mori–Nagata theorem). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain.
Let A be a noetherian integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure
of
A in
L is a finitely generated
A-module. This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form.)
Let A be a finitely generated algebra over a field k that is an integral domain with field of fractions K. If L is a finite extension of K, then the integral closure
of
A in
L is a finitely generated
A-module and is also a finitely generated
k-algebra. The result is due to Noether and can be shown using the
Noether normalization lemma as follows. It is clear that it is enough to show the assertion when
L/
K is either separable or purely inseparable. The separable case is noted above, so assume
L/
K is purely inseparable. By the normalization lemma,
A is integral over the
polynomial ring
. Since
L/
K is a finite purely inseparable extension, there is a power
q of a
prime number such that every element of
L is a
q-th root of an element in
K. Let
be a finite extension of
k containing all
q-th roots of coefficients of finitely many rational functions that generate
L. Then we have:
The ring on the right is the field of fractions of
, which is the integral closure of
S; thus, contains
. Hence,
is finite over
S; a fortiori, over
A. The result remains true if we replace
k by
Z.
The integral closure of a complete local noetherian domain A in a finite extension of the field of fractions of A is finite over A. More precisely, for a local noetherian ring R, we have the following chains of implications:
(i) A complete
A is a
Nagata ring(ii) A is a Nagata domain
A analytically unramified
the integral closure of the completion
is finite over
the integral closure of
A is finite over A.
Noether's normalization lemma
See main article: Noether normalization lemma.
Noether's normalisation lemma is a theorem in commutative algebra. Given a field K and a finitely generated K-algebra A, the theorem says it is possible to find elements y1, y2, ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[''y''<sub>1</sub>,..., ''y''<sub>''m''</sub>]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendental extension and B ⊂ A is finite.[13]
Integral morphisms
In algebraic geometry, a morphism
of
schemes is
integral if it is affine and if for some (equivalently, every) affine open cover
of
Y, every map
is of the form
\operatorname{Spec}(A)\to\operatorname{Spec}(B)
where
A is an integral
B-algebra. The class of integral morphisms is more general than the class of
finite morphisms because there are integral extensions that are not finite, such as, in many cases, the algebraic closure of a field over the field.
Absolute integral closure
Let A be an integral domain and L (some) algebraic closure of the field of fractions of A. Then the integral closure
of
A in
L is called the
absolute integral closure of
A.
[14] It is unique up to a non-canonical
isomorphism. The
ring of all algebraic integers is an example (and thus
is typically not noetherian).
See also
References
- Book: Atiyah . Michael Francis . Michael Atiyah . Macdonald . Ian G. . Ian G. Macdonald . Introduction to commutative algebra . 1994 . 1969 . Addison–Wesley . 0-201-40751-5.
- Book: Bourbaki . Nicolas . Nicolas Bourbaki . Algèbre commutative . 2006 . Springer . Berlin . 978-3-540-33937-3.
- Book: Kaplansky, Irving . Commutative Rings . Lectures in Mathematics . September 1974 . . 0-226-42454-5 . registration .
- H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.
- Web site: James Milne (mathematician) . J. S. . Milne . Algebraic number theory . 19 July 2020.
- M. Reid, Undergraduate Commutative Algebra, London Mathematical Society, 29, Cambridge University Press, 1995.
Further reading
for a regular sequence
?]
Notes and References
- The above equation is sometimes called an integral equation and b is said to be integrally dependent on A (as opposed to algebraic dependent.)
- This proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.)
- Chapter 2 of
- Web site: Section 32.14 (05JW): Universally closed morphisms—The Stacks project. stacks.math.columbia.edu. 2020-05-11.
- Book: Stein. Computational Introduction to Algebraic Number Theory. 101.
- An exercise in
- Proof: Let
be a ring homomorphism such that
if
is homogeneous of degree n. The integral closure of
in
is
, where
is the integral closure of A in B. If b in B is integral over A, then
is integral over
; i.e., it is in
. That is, each coefficient
in the polynomial
is in A.
- Exercise 4.14 in
- Definition 1.1.1 in
- Exercise 4.15 in
- Remark 1.1.3 in
- Chapter 12 of
- Chapter 4 of Reid.
- [Melvin Hochster]