Darij Grinberg, A few facts on
integrality (version 5
September 2009).
PDF file.
Old version (version 20
August 2009) with slightly weaker Theorem 1 but better
proof (it's the same argument, but the generalization
made it harder to write up).
This is a four-part note about commutative algebra. Rings
mean commutative rings with unity.
Part 1 (Integrality over rings) is an
(over-formalized) writeup of proofs to some known and
less known results about integrality over rings. If A is
a subring of a ring B, and n is an integer, then an
element u of B is said to be n-integral over A
if there exists a monic polynomial P of degree n with
coefficients in A such that P(u) = 0. We show that:
- (Theorem 1) An element u of B is n-integral over A if
and only if there exists an n-generated (= generated by n
elements) A-submodule U of B such that uU is a subset of
U and such that v = 0 for every v in B satisfying vU = 0.
- (Theorem 1 as well) An element u of B is n-integral
over A if and only if there exists an n-generated (=
generated by n elements) A-submodule U of B such that 1
lies in U and uU is a subset of U.
- (Theorem 1 in the alternative
version) An element u of B is n-integral over A if
and only if there exists a faithful n-generated (=
generated by n elements) A-submodule U of some B-module
such that uU is a subset of U.
- (Theorem 2) If a0, a1, ..., an
are elements of A and v is an element of B such that
SUM_{i=0}^{n} aivi
= 0, then SUM_{i=0}^{n-k} ai+kvi
is n-integral over A for every 0 <= k <= n. (This
result, and its Corollary 3, generalize exercise
2-5 in J.
S. Milne's Algebraic Number Theory.)
- (Theorem 4) If some element v of B is m-integral over
A, and some element u of B is n-integral over A[v], then
u is nm-integral over A. (This is a known fact. I derive
it from Theorem 1, just as most people do. Maybe I will
also write up a different proof using resultants.)
- (Theorem 5) Any element of A is 1-integral over A. If
some element x of B is m-integral over A, and some
element y of B is n-integral over A, then x+y and xy are
nm-integral over A. (This is known again. I use Theorem 4
to prove this.)
- (Corollary 6) Let v be an element of B, and n and m two
positive integers. Let P be a polynomial of degree n-1
with coefficients in A, and let u = P(v). If vu is
m-integral over A, then u is nm-integral over A. (This
follows from Theorems 2 and 5 but may turn out useful,
though I don't expect much.)
Part 2 (Integrality over ideal semifiltrations)
gives a common generalization to integrality over rings
(as considered in Part 1) and integrality
over ideals (a less known, but still important
notion).
We define an ideal semifiltration of a ring A as
a sequence (Ii)i>=0. of ideals of A such
that I0 = A and IaIb
is a subset of Ia+b for
any a >= 0 and b >= 0. (This notion is weaker than
that of an ideal filtration, since we do not require that
In+1 is a subset of In for every n >= 0.)
If A is a subring of a ring B, if (Ii)i>=0 is an ideal
semifiltration of A, and if n is an integer, then an
element u of B is said to be n-integral over (A,(Ii)i>=0) if
there exists a monic polynomial P of degree n with
coefficients in A such that P(u) = 0 and the i-th
coefficient of P lies in Ideg P - i
for every i in {0, 1, ..., deg P}.
While this notion is much more general than integrality
over rings (which is its particular case when (Ii)i>=0
= (A)i>=0) and
integrality over ideals (which is its particular case
when B = A and (Ii)i>=0 = (Ii)i>=0 for some fixed ideal
I), it still can be reduced to basic integrality over
rings by a base change. Namely:
- (Theorem 7) The element u of B is n-integral over (A,(Ii)i>=0)
if and only if the element uY of the polynomial ring B[Y]
is n-integral over the Rees algebra A[(Ii)i>=0*Y]. (This Rees algebra
A[(Ii)i>=0*Y]
is defined as the subring I0Y0
+ I1Y1 + I2Y2 + ... of the
polynomial ring A[Y]. Not that I would particularly like
the notation A[(Ii)i>=0*Y], but I have not
seen a better one.)
(The idea underlying this theorem is not new, but I
haven't seen it stated in standard texts on integrality.)
Using this reduction, we can generalize Theorems 4 and 5:
- (Theorem 8, generalizing Theorem 5) An element of A is
1-integral over (A,(Ii)i>=0) if and only if it
lies in I1. If some
element x of B is m-integral over (A,(Ii)i>=0), and some element y
of B is n-integral over (A,(Ii)i>=0), then x+y is
nm-integral over (A,(Ii)i>=0). If some element x of
B is m-integral over (A,(Ii)i>=0), and some element y
of B is n-integral over A (not necessarily over (A,(Ii)i>=0)
!), then xy is nm-integral over (A,(Ii)i>=0).
- (Theorem 9, generalizing Theorem 4) If some element v
of B is m-integral over A, and some element u of B is
n-integral over (A[v], (IiA[v])i>=0), then u is
nm-integral over (A,(Ii)i>=0).
Note that Theorem 9 doesn't seem to yield Theorem 8 as
easily as Theorem 5 could be derived from Theorem 4 !
Part 3 (Generalizing to two ideal
semifiltrations) continues Part 2, generalizing
a part of it even further:
Let A be a subring of a ring B. Let (Ii)i>=0 and (Ji)i>=0 be two ideal
semifiltrations of A. Then, (IiJi)i>=0
is an ideal semifiltration of A, as well. Now, we can
give a "relative" version of Theorem 7:
- (Theorem 11) An element u of B is n-integral over (A,(IiJi)i>=0) if and only if the
element uY of the polynomial ring B[Y] is n-integral over
the (A[I], (JiA[I])i>=0), where A[I] is a shorthand for the
Rees algebra A[(Ii)i>=0*Y].
Using this, we can generalize the xy part of Theorem 8
even further:
- (Theorem 13) If some element x of B is m-integral over
(A,(Ii)i>=0),
and some element y of B is n-integral over (A,(Ji)i>=0),
then xy is nm-integral over (A,(IiJi)i>=0).
Part 4 (Accelerating ideal semifiltrations)
extends Theorem 7:
- (Theorem 16) We will work under the same conditions at
Let s >= 0 be an integer. The element u of B is
n-integral over (A,(Isi)i>=0) if and only if the
element uYs of the polynomial ring B[Y] is
n-integral over the Rees algebra A[(Ii)i>=0*Y].
Actually, this can be further generalized in the vein of
Theorem 11 (to Theorem 15).
As a consequence, Theorem 2 is generalized.
