# LOCALIZATIONS OVER THE STEENROD ALGEBRA
LOCALIZATIONS OVER THE STEENROD ALGEBRA
LOCALIZATIONS OVER THE STEENROD ALGEBRA
LOCALIZATIONS OVER THE STEENROD ALGEBRA
LOCALIZATIONS OVER THE STEENROD ALGEBRA
THE LOST CHAPTER
THE LOST CHAPTER
THE LOST CHAPTER
THE LOST CHAPTER
THE LOST CHAPTER
MARA D. NEUSEL
MARA D. NEUSEL
MARA D. NEUSEL
MARA D. NEUSEL
MARA D. NEUSEL
EMAIL:MDN@SUNRISE.UNIMATH.GWDG.DE
Wyoming
Summer 1998
AMS CODE: 55S10 Steenrod Algebra, 13BXX Ring Extensions and Related Top
ics, 55XX Algebraic Topology, 13XX Commutative Rings and Algebras
KEYWORDS: Steenrod Algebra, Unstable Algebras over the Steenrod Alge
bras, Unstable Part, Localizations, Noetherianess, Integral Closure, Dickson
Algebra
Typeset by LS T E X
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : Let H * be an unstable algebra over the Steenrod algebra, and let
S # H * be a multiplicatively closed subset. The localization at S, i.e. S -1 H * ,
inherits an action of the Steenrod algebra from H * , which is, however, in general
no longer unstable. In this note we consider the following three statements.
(1) H * is Noetherian,
(2) the integral closure, H * S -1 H * , of H * in the localization with respect to
S is Noetherian,
(3) H * S -1 H * = Un(S -1 H * ), where Un(-) denotes the unstable part.
If the set S contains only (nonzero) non zero divisors and the algebras are
reduced then
(1) Û (2).
If S contains zero divisors, then only (1) Þ (2) remains true, to show the con
verse is false we construct a counter example.
The implication (2) Þ (3) is always true, while its converse (3) Þ (2) needs a
weird bunch of technical assumptions to remain true. However, none of them
can be removed: we illustrate this also with examples.
Finally, as a technical tool, we characterize Dfinite algebras.
Let H * be a non negatively graded connected commutative algebra over a Galois
field IF with q elements and characteristic p. We assume H * to be provided
with an unstable action of the Steenrod algebra 1 P P P P P * .
Let S # H * be a multiplicatively closed subset, and form the ring of fractions
S -1 H * := h
s #h ÎH * , s ÎS .
There is a unique extension of the action of the Steenrod algebra to S -1 H * given
by requiring
P(n)( h
s )P(n)(s) = P(n)(h)
for any h
s ÎS -1 H * , compare [15] Proposition 1.1 (a), where P(n) denotes the giant
Steenrod operation. By construction, this action satisfies the Cartan formulae,
but not the unstability condition.
Recall from [3], Definition 2.2, the notion of the unstable part:
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION : Let M be a graded module over the Steenrod algebra. For
a multi index I = (i 1 , . . . , i s ) denote by P I the product P i 1 · · · P i s . Then the
unstable part of M in degree k is the IFvector subspace defined by
Un(M) (k) := n m ÎM (k) #P r P I (m) = 0, r > k + deg(P I ), " multi indices I o .
Consider the following statements:
(1) H * is Noetherian.
(2) The integral closure (in its total ring 2 of fractions RF(H * )), H * is
Noetherian.
(3) The unstable part of the total ring of fractions is Un(RF(H * )) =H * . The
algebra H * is reduced, has only finitely many minimal prime ideals p
and the transcendence degree trzdeg(FF(H * / p) / IF) < #.
Starting with Emmy Noether's paper, [13], in the twenties, integral dependence
has been among the most powerful and useful tools in ring and ideal theory.
E.g. the simple question whether the integral closure of a Noetherian ring is
1 For an algebraic introduction of the Steenrod algebra over arbitrary Galois fields see Chapter 10
and 11 in [14] or the introduction of [11].
2 See Section 19 in [17] for the classical definition, or Section 4.2 in [11] for the graded case.
MARA D. NEUSEL
again Noetherian has led to an enormous amount of new concepts such as the
notions of Krull rings, Nagata rings and excellent rings.
The Steenrod algebra and algebras over it have been proven to be decisive
tools in algebraic topology (where they were born) for over 50 years and find
applications in cohomology theory, polynomial invariant theory of finite groups,
and most recently in algebraic geometry. In particular the Un(-) functor was
used (and studied) in, e.g., [3] and [4].
In this paper we combine both points of view and show in Section 2 that
(2) Þ (3).
If the involved algebras are reduced, i.e. the nil radical Nil is trivial, then 3 we
prove in Section 3 that
(1) Û (2).
In Section 5 we show that
(2) Ü (3).
Moreover, if we take an arbitrary multiplicatively closed subset S # H * and
look at the analogous statements:
(1') H * is Noetherian.
(2') The integral closure H * S -1 H * of H * in S -1 H * is Noetherian.
(3') The unstable part of the localization is Un(S -1 H * ) = H * S -1 H * . The al
gebra H * is reduced, has only finitely minimal prime ideals p and the
transcendence degree trzdeg(FF(H * / p) / IF) < #.
If S does not contain zero nor any zerodivisor, then the equivalence
(1 # ) Û (2 # )
remains true (for reduced rings). The implication (2 # ) §2
Þ (3 # ) is again always
true. In contrast
(3 # ) §5
Þ (2 # )
fails even for integral domains H * : we illustrate this with an example.
In Section 6 we consider the case where S does contain zerodivisors. The
implication (1 # ) Þ (2 # ) remains true for (reduced rings), while its converse fails:
we construct a counter example. (2 # ) Þ (3 # ) is again o.k., but not its converse.
3 Note that this says that the integral closure, H * , behaves like the P P P P P *
inseparable closure, q H * ,
in the sense that
H * Noetherian and reduced Þ q H * Noetherian
by Theorem 6.3.1 in [11] and
H * Noetherian Ü q H * Noetherian
by Proposition 4.2.3 in [11].
2
THE LOST CHAPTER
Finally we establish some technical results in Section 4 (and 5). In particular
we show that integral domains H * are Dfinite in the sense of Section 1.2 in
[11] if and only if trzdeg(FF(H * ) / IF) < # if and only if there exists an integral
ring extension D * (n) q l
# Un(FF(H * )) (where D * (n) q l
denotes a fractal of the
Dickson algebra) if and only if Un(FF(H * )) is Noetherian, see Theorem 4.7 and
Proposition 5.1. This is as far as possible generalized to rings with zerodivisors
(which is the source of the weird assumptions on H * mentioned earlier).
Acknowledgement
The main ideas of this paper were developed during a journey through a piece
of paradise called Wyoming in June 1998.
§1. Preliminaries and Recollections
Let A * be a non negatively graded commutative algebra 4 over a Galois field IF
and let A * enjoy an unstable action of the Steenrod algebra. Let S # A * \ 0
be a multiplicatively closed subset 5 . In this section we want to determine
the units and the degree zero part in Un(S -1 A * ) and establish some functorial
properties. Some of the results of this section are known. They are explicitely
stated and proved here, because, as often in the case of folkloristic results, they
are hard to find in the literature -- if at all.
First, recall from Lemma 3.1.2. in [11] that, the unstable part of a localization
can be easily calculated:
Un(S -1 A * ) = n a
s
ÎS -1 A * such that P(n)(s) # P(n)(a) o .
If a
s
ÎUn(S -1 A * ) (0) has degree zero then by definition
deg(a) = deg(s) =: d.
Moreover, since this fraction is unstable we have that
s + P 1 (s)n + · · · + s q n d = P(n)(s) # P(n)(a) = a + P 1 (a)n + · · · + a q n d ,
hence there exists an element f ÎA * such that
f P(n)(s) # P(n)(a).
4 All algebras in this paper are assume to be graded, commutative IFalgebras equipped with an
action of the Steenrod algebra. We reserve the letter H * for non negatively graded connected
unstable algebras. If we drop (one of) these assumptions we use letters A * and B * . However, we
often list the properties of the algebra in question, just to make sure that there is no confusion.
Moreover, further properties, e.g., Noetherianess are always explicitely stated when needed.
5 Since S -1 A * = 0 Û 0 ÎS we always assume that S does not contain zero. Moreover, we assume
that 1 ÎS.
3
MARA D. NEUSEL
Since a and s have the same degree, f has degree zero. Moreover,
f s = a and f s q = a q ,
i.e.,
a
s = f = a
s q
.
We are now prepared to determine the units in our localization.
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1 (folklore) : Let A * be a non negatively graded unstable
algebra over the Steenrod algebra P P P P P * . Let S # A * \ 0 be a multiplicatively
closed subset. Then the units in Un(S -1 A * ) are precisely the nonzero elements
in the ground field IF.
PROOF PROOF PROOF PROOF PROOF : Take a unit
a
s ÎUn(S -1 A * )
where a ÎA * and s ÎS. Then s
a is an unstable element in the localization, so
P(n)(s) # P(n)(a) # P(n)(s),
hence both elements must have the same degree d, i.e.,
a
s
ÎUn(S -1 A * ) (0) .
The preceding observations give us an element f ÎA * of degree zero such that
H
a
s = f = a q
s q .
This implies
a q-1 - s q-1 = 0.
Therefore ks, for any k ÎIF × , are the (q - 1)st roots of the polynomial
X q-1 - s q-1 ÎA * [X ].
Since a is also a root of this polynomial it must be a multiple of s
a = ks for some k ÎIF × .
This means
a
s = k ÎIF ×
as claimed .
REMARK REMARK REMARK REMARK REMARK : Note that this means also that the units in A * itself are precisely
the nonzero elements in the ground field.
We can now determine the complete degree zero part of Un(S -1 A * ).
4
THE LOST CHAPTER
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2 (folklore) : Let S # A * \ 0 be a multiplicatively closed
subset of a non negatively graded unstable algebra over P P P P P * . Then the degree
zero part
Un(S -1 A * ) (0)
is a qBoolean algebra consisting of zerodivisors and the ground field IF. More
over zerodivisors exist if and only if the set S contains a zerodivisor or there
are zerodivisors in the degree zero part, A *
(0) , of A * .
PROOF PROOF PROOF PROOF PROOF : We have seen that in degree zero our algebra is
Un(S -1 A * ) (0)
= a
s ÎS -1 A * such that P(n)(s) # P(n)(a) and deg(s) = deg(a)
= a
s
ÎS -1 A *
# a
s = P 0 ( a
s ) = a q
s q .
Therefore any element in Un(S -1 A * ) (0) is equal to its qth power, i.e., Un(S -1 A * ) (0)
is a qBoolean ring. Next choose a nonzero element a
s of degree zero that is not
a unit. Then
a
s a
s q-1
- 1 ! = 0.
Since a
s
is not a unit no power of it can be 1 and hence we have that
a
s q-1
- 1 #= 0.
Therefore a
s
is a zerodivisor. However, this means that there exists an element
a #
s # such that
a
s
a #
s # =
0
1 .
Therefore aa # = 0, hence a is a zerodivisor. Finally we have
s f = a for some f ÎA *
(0) .
Hence s ÎS or f ÎA *
(0) is a zerodivisor.
On the other hand, if s ÎS -1 is a zerodivisor in A * , and let a ÎA * \ 0 such that
sa = 0. Then
s
s Î Un(S -1 A * ) (0)
is a zerodivisor, because
s
s
a
1 = 0.
5
MARA D. NEUSEL
Finally if a ÎA *
(0) is a zerodivisor then so is
a
1 ÎUn(S -1 A * ) (0) .
That's all we claimed .
REMARK REMARK REMARK REMARK REMARK : In particular the preceding result implies that any prime ideal
p of the degree zero part Un(S -1 A * ) (0)
is maximal and the quotient ring
Un(S -1 A * ) (0)
/ p is a field with q elements.
REMARK REMARK REMARK REMARK REMARK : Note also that this result also implies that A * itself in degree
zero is a qBoolean algebra.
We close this section with a technical lemma which will be useful later.
LEMMA 1.3
LEMMA 1.3
LEMMA 1.3
LEMMA 1.3
LEMMA 1.3 :
(1) Let A * be an algebra over the Steenrod algebra. Let S # A * \ 0 be a
multiplicatively closed subset not containing zerodivisors. Then
S -1 (Un(S -1 A * ))) = S -1 A * .
(2) Let IK * be a field 6 over the Steenrod algebra. Then
Un(FF(Un(IK * ))) = Un(IK * ),
where FF(-) denotes the field of fractions functor.
PROOF PROOF PROOF PROOF PROOF :
AD AD AD AD AD (1) : Since S contains no zerodivisors, nor zero, we have inclusions
A *
# Un(S -1 A * ) # S -1 A * .
Hence by localizing with respect to S we get
S -1 A *
# S -1 Un(S -1 A * ) # S -1 A * ,
so the rings are all equal.
AD AD AD AD AD (2) : We have by construction
Un(IK * ) # FF(Un(IK * )) # IK * ,
and taking unstable parts leads to
Un(IK * ) = Un(Un(IK * )) # Un(FF(Un(IK * ))) # Un(IK * ),
which gives the desired result .
6 See Chapter 2 in [11] for an introduction to field theory over the Steenrod algebra.
6
THE LOST CHAPTER
§2. The case where S contains no zerodivisors
In this section we restrict our attention to the case where the multiplicatively
closed subset S contains no zerodivisors.
For a ring extension
u : A * # B *
denote by A *
B * the integral closure of A * in B * , i.e., the maximal subring of B *
which is integral over A * .
The main goal of this section is to show that if H * is Noetherian (and, since we
use the letter ``H'', also non negatively graded, connected and unstable) then
Un(S -1 H * ) = H * S -1 H * . This is the key result that correlates the two concepts;
integral dependence and unstability.
First we show that 7 the integral closure A *
S -1 A * of A * in S -1 A * is again an
unstable algebra over the Steenrod algebra
PROPOSITION 2.1
PROPOSITION 2.1
PROPOSITION 2.1
PROPOSITION 2.1
PROPOSITION 2.1 : Let S # A * \ 0 be a multiplicatively closed subset not
containing any zerodivisor. Then the integral closure
A *
S -1 A *
of A * in S -1 A * is an algebra over the Steenrod algebra. If the action of P
P P P P * on
A * is unstable then it is also unstable on the integral closure.
PROOF PROOF PROOF PROOF PROOF : If A * is an integral domain, then this is the contents 8 of Proposi
tion 3.2 in [15]. If we allow A * to have zerodivisors, but assume that in S are
none, then the same proof works by replacing the field of fractions by the total
ring of fractions, RF(A * ), of A * . One needs to note
A *
# S -1 A *
# RF(A * )
and
A *
S -1 A * = A * Ç S -1 A * ,
which is all we need .
REMARK REMARK REMARK REMARK REMARK : Just to make sure that nobody miss the point: the preceding
proposition means that
A *
S -1 A * # Un(S -1 A * ),
if Steenrod's action on A * is unstable.
We append an obvious corollary.
7 In this and the following two statements it is enough to consider arbitrarily graded commutative
IFalgebras over P
P P P P *
. Therefore we use letters ``A'' and ``B''.
8 Yes, C. W. Wilkerson considers only connected algebras over the prime field IF p . However, this
doesn't make any difference at this point.
7
MARA D. NEUSEL
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2 : Let S # A * \ 0 be a multiplicatively closed subset not
containing any zerodivisors. Then the unstable part Un(S -1 A * ) is integrally
closed in S -1 A * .
PROOF PROOF PROOF PROOF PROOF : By the preceding Proposition 2.1 we have
Un(S -1 A * ) # Un(S -1 A * ) S -1 A * # Un(S -1 A * ),
what give the desired result .
We need a technical result.
PROPOSITION 2.3
PROPOSITION 2.3
PROPOSITION 2.3
PROPOSITION 2.3
PROPOSITION 2.3 : Let
A *
# B *
be an integral extension of algebras over the Steenrod algebra. Then the
restriction of the ring extension
Un(A * ) # Un(B * )
is again integral.
PROOF PROOF PROOF PROOF PROOF : For the case of fields A * and B * over the Steenrod algebra the
statement of this proposition is proven in Proposition 2.2 [15]. Again, C. W.
Wilkerson considers only the prime field as ground field, so see Lemma 3.1.1 in
[11] for a proper generalization to arbitrary Galois fields. Moreover, in Chapter
2 of [11] the notion of separability and inseparability for arbitrary ground fields
is introduced. Equipped with this the same case checking argument as used in
[15] or [11] yields the desired result .
We can next prove the main result 9 of this section.
THEOREM 2.4
THEOREM 2.4
THEOREM 2.4
THEOREM 2.4
THEOREM 2.4 : Let H * be a non negatively graded connected unstable
Noetherian algebra. Then the unstable part of its ring of fractions is precisely
the integral closure, H * , of H * (i.e., in its total ring of fractions RF(H * ))
H * = Un(RF(H * )).
PROOF PROOF PROOF PROOF PROOF : The Big Imbedding Theorem in [11], Theorem 8.1.8 hands us a ho
mogeneous system of parameters h 1 , . . . , h n ÎH * such that the polynomial sub
algebra IF[h 1 , . . . , h n ] is again an unstable algebra over the Steenrod algebra.
Hence we have
IF[h 1 , . . . , h n ] # u H *
# #
Un IF(h 1 , . . . , h n ) # Un RF(H * )
# #
IF(h 1 , . . . , h n ) # F
RF(H * ),
9 For the special case of integral domains over the prime field the following result is the contents
of Proposition 1.2 in [16].
8
THE LOST CHAPTER
where the ring extension u and the induced map F are integral and finite. By
Proposition 2.3 the restricted map
F# : Un IF(h 1 , . . . , h n ) # Un RF(H * )
is also integral. By Lemma 3.1.3 in [11]
Un IF(h 1 , . . . , h n ) = IF[h 1 , . . . , h n ],
hence we have an integral ring extension
IF[h 1 , . . . , h n ] # Un(RF(H * )).
This map factorizes through H * via the natural inclusions. Therefore
H *
# Un(RF(H * ))
is also integral, so Proposition 2.1 yields
Un(RF(H * )) = H * ,
as claimed .
COROLLARY 2.5
COROLLARY 2.5
COROLLARY 2.5
COROLLARY 2.5
COROLLARY 2.5 : Let H * be Noetherian and let S # H * \ 0 be a multiplica
tively closed subset not containing any zerodivisor. Then the unstable part of
S -1 H * is precisely the integral closure, H * S -1 H * , of H * in S -1 H *
H * S -1 H * = Un(S -1 H * ).
PROOF PROOF PROOF PROOF PROOF : Let H * be Noetherian. By Proposition 2.1 and Theorem 2.4 we
have an integral extension of rings
H *
# H * S -1 H * # Un(S -1 H * ) # Un(RF(H * )) = H * .
Therefore the ring extension
H *
S -1 H * # Un(S -1 H * )
is integral and these two rings have the same total ring of fractions by Lemma
1.3. The smaller ring is integrally closed. Therefore both rings are equal .
§3. The Integral Closure
Let S # H * \ 0 be a multiplicatively closed subset, not containing any zerodi
visor, of a non negatively graded connected unstable algebra over the Steenrod
algebra P
P P P P * . The unstable part, Un(S -1 H * ) is by definition again a non neg
atively graded unstable algebra H * over P P P P P * . It is also connected by Proposi
tion 1.2. We have seen that it coincides with the appropriate integral closure
H * S -1 H * , if we had started with a Noetherian algebra H * . So, of course we want
to know: Is it again Noetherian? We start this section with a result that gives
(together with some kind of converse) an affirmative answer to this question,
if H * is reduced. 10
10 The following Proposition bears the same wall flower fate as many other results in graded con
9
MARA D. NEUSEL
THEOREM 3.1
THEOREM 3.1
THEOREM 3.1
THEOREM 3.1
THEOREM 3.1 (Wall Flower) : Let H * be a non negatively graded connected
unstable algebra over the Steenrod algebra. Let H * (or equivalently H * ) be
reduced. Then H * is Noetherian if and only if its integral closure H * is so.
PROOF PROOF PROOF PROOF PROOF : H * is a non negatively graded connected commutative unstable
algebra over P P P P P * if and only if H * is. For connectivity use Proposition 2.1 to
conclude that
H *
# H * # Un(RF(H * )).
Therefore
H *
(0) # H * (0) # Un(RF(H * )) (0) ,
and use Proposition 1.2.
So, assume that H * is Noetherian. If H * is an integral domain then its integral
closure is Noetherian by a classical result due to Emmy Noether, see, e.g., p.
261 in [7]. If H * has zerodivisors we have to work a bit:
Take a minimal prime q # H * lying over p := q ÇH * # H * . We have
H * / p #
integral
H * / q # H * / p.
Since H * is Noetherian, so is H * / p. So, by Emmy's theorem the ring extension
H * / p # H * / p is finite, hence so is H * / p # H * / q. Therefore
H * / q
is Noetherian for any minimal prime q. If we can prove that there are only
finitely many, say l, height zero prime ideal in H * , then we have
H * / Nil (H * )
is Noetherian, because
Nil (H * ) = q 1 Ç · · · Ç q l
is a finite intersection. Since we assume that the nil radical is trivial, it follows
that H * is Noetherian.
To prove that there are only finitely many height zero prime ideals, note that
any prime ideal p # H * of height zero consists of zerodivisors. Therefore its
extension to the total ring of fractions
p e
# RF(H * )
is again a height zero prime ideal, and conversely any height zero prime ideal
in the total ring of fraction is an extended one. Hence the number of height
zero prime ideals in H * is equal to their number in RF(H * ). Denote by
u : H * # H * # RF(H * )
nected commutative algebra: it's ignored and abadoned from the literature, because algebraists
don't seem to take the graded connected case seriously.
10
THE LOST CHAPTER
the canonical inclusion, and by
u * : Proj (RF(H * )) ## Proj (H * ) ## Proj(H * )
the induced map on the homogeneous prime spectra. Then u * restricted to the
subset, Proj (RF(H * )) (0) , of height zero prime ideals is surjective onto Proj(H * ) (0) ,
i.e.,
ïProj (RF(H * )) (0) ï ³ ïProj (H * ) (0) ï ³ ïProj(H * ) (0) ï,
which leads to the desired result.
For the converse, assume H * is Noetherian. Therefore, H * is finitely gener
ated 11 as an algebra over IF, and a fortiori as an algebra over H * . This means
that the ring extension
H *
# H *
is integral and finite, i.e., H * is finitely generated as a module over H * . So, take
a homogeneous system of parameters
h 1 , . . . , h n ÎH * .
Since H * is a finite H * module there exists a natural number l ÎIN such that
IF[h l
1 , . . . , h l
n ] # H *
# H * ,
where, the overall ring extension is finite. Therefore the ring extension
IF[h l
1 , . . . , h l
n ] # H *
is also finite, which in turn implies that H * is Noetherian.
REMARK REMARK REMARK REMARK REMARK : Note that the assumption on the nil radical is used only for the
''only if'' case.
REMARK REMARK REMARK REMARK REMARK : The argumentation for the ''if'' case gives a quick proof of the
Little ImbeddingTheorem, Theorem 7.3.3 in [11]. However, in [11] more proven
than is stated: namely the algorithm given there shows that a (fractal of a)
Dickson algebra in H * lifts to one in H * , no matter whether the one or the other
algebra is Noetherian, provided only that we can lift the top Dickson class.
Let us first note that we have answered the question at the start of this section.
COROLLARY 3.2
COROLLARY 3.2
COROLLARY 3.2
COROLLARY 3.2
COROLLARY 3.2 : Let H * be a non negatively graded connected unstable
algebra over P
P P P P * . If H * is Noetherian and reduced, then so is H * = Un(RF(H * )).
PROOF PROOF PROOF PROOF PROOF : Combine Theorems 2.4 and 3.1 .
This pair of results generalizes as follows.
11 Yes, it's unbelievable how nicely the graded connected case behaves.
11
MARA D. NEUSEL
COROLLARY 3.3
COROLLARY 3.3
COROLLARY 3.3
COROLLARY 3.3
COROLLARY 3.3 : Let H * be a non negatively graded connected unstable
algebra over P
P P P P * . Suppose H * is reduced, and let S #H * \0 be a multiplicatively
closed subset containing no zerodivisors. Then H * is Noetherian if and only if
H * S -1 H * is.
PROOF PROOF PROOF PROOF PROOF : Let H * be Noetherian. By Theorem 2.4 and Corollary 2.5 we have
integral extensions of rings
H *
# H * S -1 H * = Un(S -1 H * ) # Un(RF(H * )) = H * ,
where the smallest and the largest ring are Noetherian by Theorem 3.1. Hence
by standard tictactoe 12 H * S -1 H * is also Noetherian of the same Krull dimen
sion, and is finite over H * .
For the converse, assume that
H * S -1 H *
is reducedand Noetherian. Then by Theorem 3.1 so is H * and again by Theorem
3.1 so is H * .
§4. Excursion to the DFinite World
We have seen so far that
H * Noetherian H * reduced
Û H * Noetherian Þ H * = Un(RF(H * ))
(and the analog statements are true if we invert only but not every non zero
divisor in H * ). In this section we establish some technical (sorry for that!)
results that will be needed in the following section to show when all this three
statements are equivalent, i.e., for the investigation when the converse of
Theorem 2.4, resp. Corollary 2.5, are valid.
In particular we prove that every unstable integral domain H * of finite tran
scendence degree (i.e., trzdeg(FF(H * ) / IF) < # ) is Dfinite and conversely, every
Dfinite integral domain has finite transcendence degree. Moreover, this is
equivalent to saying that Un(FF(H * )) is Noetherian. Besides this, we have a
look at the case where H * is not an integral domain.
Recall some terminology and results from Chapter 1 of [11]. We have that the
inductively defined elements of the Steenrod algebra
P D 1 := P 1 ,
P D i := [P D i-1 , P q i-1
], i ³ 2,
12 Remember, we are in the graded connected world; to be specific: H * is a Noetherian nonneg
atively graded connected commutative IFalgebra. Therefore it is finitely generated as an algebra
over IF and a fortiori as an algebra over H * . Since the ring extension H * # H * is integral the
integral closure H * is finitely generated as a module over H * . Since H * is a Noetherian ring any
H * submodule of H * is finitely generated. In particular Un (S -1 H * ) is finitely generated as an
H * module, hence finitely generated as an IFalgebra and therefore Noetherian.
12
THE LOST CHAPTER
are primitive derivations, see e.g. [8] Corollary 5 in Section 6. In addition we
have a derivation
P D 0 (h) := deg(h)h, " h ÎH *
(that is not in P P P P P * ).
Recall from Section 1.2 in [11] the following definition.
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION : Let H * be an unstable algebra over the Steenrod algebra
with Nil (H * ) = (0) and let D(H * ) be the H * module of derivations generated by
P D i , " i ÎIN 0 . If there exists an m ÎIN 0 such that some m + 1 derivations (and
hence by Proposition 1.1.7 in [11] all m + 1) are linearly dependent, then H * is
called Dfinite. Moreover, call the relation of linear dependence
h 0 P D 0 + · · · + hmP Dm = 0
(where m is chosen to be minimal w.r.t. this property) the Drelation of H *
and m(H * ) ÎIN the Dlength.
So, take a non negatively graded connected unstable integral domain H * over
the Steenrod algebra of finite transcendence degree t. First we want to show
that H * is Dfinite.
If we can show that any set of m + 1 elements h 0 , . . . , hm ÎH * is algebraically
independent, for some m ÎIN, we are done as the following proposition shows
(which is just a closely revised version of Corollary 1.2.2 in [11]).
PROPOSITION 4.1
PROPOSITION 4.1
PROPOSITION 4.1
PROPOSITION 4.1
PROPOSITION 4.1: Let A * be an algebra over the Steenrod algebra. Let any
set of m + 1 element of A * be algebraically dependent. Then there are elements
a 0 , . . . , am ÎA * , not all zero, such that
a 0 P D 0 + · · · + amP Dm = 0.
PROOF PROOF PROOF PROOF PROOF : By assumption any set of m + 1 elements a 0 , . . . , am ÎA * is al
gebraically dependent. Hence the determinant of the generalized Jacobian
matrix vanishes or is a zero divisor, see Theorem A.4.1 in [11],
det P D i (a j ) i, j=0 ,..., m
= a 0 P D 0 (a 0 ) + · · · + amP Dm (a 0 )
for any a 0 ÎA * , where we get this equation from the Lagrange expansion using
the first column. Hence for some a ÎA * \ {0} we get a relation of the form
0 = adet P D i (a j ) i, j=0 ,..., m
= a a 0 P D 0 (a 0 ) + · · · + amP Dm (a 0 )
However, if this is the trivial relation, i.e.,
aa 0 = · · · = aam = 0,
then, since
a k = det P D i (a j ) i, j=0 ,..., “
k ,..., m
,
13
MARA D. NEUSEL
where we adopt the topologist's convention that
b- means that this element is
omitted, the generalized Jacobian matrix
am = det P D i (a j ) i, j=0 ,..., m-1
vanishes or is a zero divisor. So, we would have by Lagrange expansion a
relation of length at most m - 1
”
a ”
a 0 P D 0 (a 0 ) + · · · + ”
am-1 P Dm-1 (a 0 ) = 0,
for all a 0 ÎA * . Proceeding this way we end up with a nontrivial relation
0 = Ÿ
adet P D i (a j ) i, j=0 ,..., k
= Ÿ
a Ÿ
a 0 P D 0 (a 0 ) + · · · + Ÿ
a k P D k (a 0 )
for some k , 0 £ k £ m .
The next result ensures that the condition in the preceding proposition is given
if the transcendence degree of FF(H * ) over the ground field IF is finite.
PROPOSITION 4.2
PROPOSITION 4.2
PROPOSITION 4.2
PROPOSITION 4.2
PROPOSITION 4.2 : Let H * be a non negatively graded connected unstable
integral domain over a field IF of q = p s elements. Let trzdeg(FF(H * ) / IF) = t <#.
Then the Dlength of H * is at most t.
PROOF PROOF PROOF PROOF PROOF : Since the transcendence degree of FF(H * ) over IF is t, any t + 1
elements in FF(H * ) are algebraically dependent. By Proposition 4.1 we can
therefore find elements a 0 , . . . , a t ÎFF(H * ), not all zero, such that
H a 0 P D 0 + · · · + a t P D t = 0
is zero when evaluated on an element of FF(H * ). Since
a i = l i
k i
" i = 0 , . . . , t,
where l i , k i ÎH * for all i, we mupltiply equation H with k 1 · · · k t and get a non
trivial relation
h 0 P D 0 + · · · + h t P D t = 0
with coefficients in H * . Since H * has no nil potent elements this relation can
be reduced to a minimal Drelation by Proposition 1.1.6 in [11] and therefore
the Dlength of H * is at most t .
It is possible to generalize the preceding result in the following way.
PROPOSITION 4.3
PROPOSITION 4.3
PROPOSITION 4.3
PROPOSITION 4.3
PROPOSITION 4.3: Let H * be a non negatively graded connected reduced un
stable IFalgebra. Let H * have finitely many height zero prime ideals p 1 , . . . , p l .
If trzdeg(FF(H * / p i ) / IF) £ t ÎIN 0 for any i = 1 , . . . , l then the Dlength of H * is
at most t.
14
THE LOST CHAPTER
PROOF PROOF PROOF PROOF PROOF : First note that the preceding result hands us a non trivial relation
h 0 (i)P D 0 + · · · + h t (i)P D t = 0
on H * / p i of length t for any i = 1 , . . . , l, i.e.,
h 0 (i)P D 0 + · · · + h t (i)P D t Îp i
when evaluated on an element in H * . Note carefully that, however, the coeffi
cients are not in the respective prime ideals p i , because otherwise this would
give trivial relations on H * / p i . So, take an element h 1 ÎH * such that
h 0 (1)P D 0 (h 1 ) + · · · + h t (1)P D
t (h 1 ) Î p 1 Ç p i 1
Ç · · · Ç p i h 1 \ S i#=1,i 1 ,..., i h 1
p i .
Without loss of generality assume that our indices are consecutively numbered
and set j 1 = 1 and j 2 = i h 1
+ 1, so that the above statement becomes
h 0 (1)P D 0 (h 1 ) + · · · + h t (1)P D t (h 1 ) Î p 1 Ç · · · Ç p i h 1 \ 0 @
l
[
i=j 2
p i
1 A .
Take another element h 2 ÎH * such that
h 0 (j 2 )P D 0 (h 2 ) + · · · + h t (j 2 )P D t (h 2 ) Î p j 2 Ç · · · Ç p i h 2 \ 0 @
l
[
i=j 3
p i
1 A ,
where we again ordered the set of indices consecutively and set j 3 = i h 2
. Pro
ceeding this way, say a times, we end up with apartition of the set of indices
{1 , . . . , l} =
a
G j=1
{j j , . . . , j j+1 - 1},
where we set j j+1 - 1 = l. Note carefully that in the final step we get that
h 0 (j a )P D 0 + · · · + h t (j a )P D
t Î p j a Ç · · · Ç p l
evaluated on any h ÎH * . We multiply all the stuff together and set
0 @
a-1
Y i=1
h 0 (j i )P D 0 (h i ) + · · · + h t (j i )P D t (h i ) 1 A = h
so that
h h 0 (j a )P D 0 + · · · + h t (j a )P D t
= 0
is a non trivial relation on H * , which we can reduce to a minimal Drelation
by Proposition 1.1.6 in [11], because we assumed that H * has no nil potent
elements .
15
MARA D. NEUSEL
We will see later that indeed none of the assumptions on H * can be removed,
see Example .
We need some more technics.
First recall from Chapter 4 in [11] the notion of P P P P P * inseparability. In particular
an unstable algebra H * is called P
P P P P * inseparably closed if whenever h ÎH * is an
element such that all Steenrod derivations vanish on h
P D i (h) = 0 " i ³ 0
then its a pth power, i.e., there exists an h # ÎH * such that
h #p = h.
The next result shows that taking the integral closure (-) commutes with taking
the P P P P P * inseparable closure # -.
LEMMA 4.4
LEMMA 4.4
LEMMA 4.4
LEMMA 4.4
LEMMA 4.4: Let H * be a non negatively graded connected unstable algebra
over the Steenrod algebra P P P P P * . Then
p H * = p H * .
PROOF PROOF PROOF PROOF PROOF : First observe that
RF p H * = RF H * P P P P P * -insep
= RF(H * ) P P P P P * -insep
= RF( p H * ),
where we twice made use of Proposition 4.2.6 in [11] 13 . Next note that
p H *
is integrally closed, because H * is, by Proposition 4.2.1 and the following remark
in [11]. Hence we have a chain of integral ring extensions
p H * # p H * # p H * ,
where all rings have the same total ring of fractions and the one in the middle
is integrally closed. Hence
p H * = p H * ,
as claimed .
The next lemma establishes some kind of natural functorial behavior of Un(-),
# - and quotients.
13 The original statement considers only integral domains. The generalization to algebras with
zero divisors is evident.
16
THE LOST CHAPTER
LEMMA 4.5
LEMMA 4.5
LEMMA 4.5
LEMMA 4.5
LEMMA 4.5 : Let A * be a non negatively graded connected algebra over the
Steenrod algebra and let I # A * be a P P P P P * invariant 14 ideal. Then
(1) # Un(A * ) = Un( # A * ).
(2) Un(A * / I) = Un(A * ) / I Ç Un(A * ) .
PROOF PROOF PROOF PROOF PROOF :
AD AD AD AD AD (1) :
a ÎUn( # A * )
if and only if a Î # A * is unstable if and only if a q
l
ÎA * is unstable (for some
l ÎIN 0 ) if and only if a q
l
ÎUn(A * ) what is equivalent to saying that
a Î p Un(A * ).
AD AD AD AD AD (2) : Consider the canonical projection
u : Un(A * ) # Un(A * / I), a # a MOD MOD MOD MOD MOD I .
Its kernel is obviously I Ç Un(A * ) .
PROPOSITION 4.6
PROPOSITION 4.6
PROPOSITION 4.6
PROPOSITION 4.6
PROPOSITION 4.6 : If H * is a non negatively graded connected Dfinite un
stable integral domain over the Steenrod algebra. Then Un(FF(H * )) contains
integrally a fractal, D * (n) q l
, of the Dickson algebra.
PROOF PROOF PROOF PROOF PROOF : First we replace H * by its P P P P P * inseparable closure p H * . Since,
H * # p H *
is an integral extension, also
FF(H * ) # FF( p H * )
is integral and hence by Proposition 2.3 so is
Un(FF(H * )) # Un(FF( p H * )).
If we detect a fractal D * (n) q l
of the Dickson algebra in Un(FF( p H * )) such that
D * (n) q
l
# Un(FF( p H * ))
is integral, then, since by Lemma 4.5
Un(FF( p H * )) = p Un(FF(H * )
we can lift it to Un(FF(H * )) by taking high enough qth powers, compare
Lemma 7.3.1 in [11].
14 An ideal I # A * is called P P P P P * invariant, if it is closed under the action of the Steenrod algebra,
see Chapter 11 in [14] for an introduction and for further references.
17
MARA D. NEUSEL
So without loss of generality we can assume that H * = p H * is P P P P P * inseparably
closed.
The constructions of Section 5.3. in [11], in particular Lemma 5.3.2, Theorem
5.3.4 and Lemma 5.3.1 (in this order), imply that the Dlength of H * is equal to
its Krull dimension. Hence Theorem 5.1.8 in [11] shows that
FF(D * (n)) # FF(H * )
is a finite and separable field extension. Therefore Proposition 2.3 yields
D * (n) = Un(FF(D * (n))) #
integral
Un(FF(H * )),
as we claimed .
We summarize.
THEOREM 4.7
THEOREM 4.7
THEOREM 4.7
THEOREM 4.7
THEOREM 4.7 : Let H * be a non negatively graded connected unstable
integral domain over P P P P P * . Then the following statements are equivalent:
(1) trzdeg(FF(H * )) = t < #.
(2) H * is Dfinite of Dlength m.
(3) There exists an integral ring extension
D * (n) q l
# Un(FF(H * )).
Moreover 15 n = t ³ m.
PROOF PROOF PROOF PROOF PROOF : The implication (1)Þ(2) is the contents of Proposition 4.2. (2)Þ(3)
is shown in Proposition 4.6. It remains to show that (3) Þ (1). So assume that
we have an integral ring extension
(H) D * (n) q l
# Un(FF(H * )),
then the Krull dimensions of both rings are equal and hence
n (1) = dim D * (n) q l
(2)
= trzdeg FF(D * (n) q
l
) / IF
(3) = trzdeg FF Un(FF(H * )) / IF
(4) = trzdeg FF(H * ) / IF ,
= t
(5)
³ m,
where (1) follows by definition, (2) is true, because the Dickson fractal is affine,
(3) is valid, because the ring extension (H) is integral, (4) follows from Lemma
1.3 and (5) from Proposition 4.2 .
15 Note that trzdeg FF(H * ) / IF ³ dim(H * ) anyway, see e.g. Theorem 20.9 in [6].
18
THE LOST CHAPTER
REMARK REMARK REMARK REMARK REMARK : Note that the preceding theorem shows also that a Dfinite
integral domain has finite Krull dimension. It remains open whether the
converse is true or not.
REMARK REMARK REMARK REMARK REMARK : We will see in the next section in which way we can generalize
the preceding result to algebras H * with zero divisors.
§5. The Converse and Consequences
In this section we want to investigate when the converse of Theorem 2.4, resp.
Corollary 2.5 is valid, as it is described at the beginning of the preceding section.
Our first proposition adds another property to the list in Theorem 4.7 of equiv
alent properties of an integral domain H * .
PROPOSITION 5.1
PROPOSITION 5.1
PROPOSITION 5.1
PROPOSITION 5.1
PROPOSITION 5.1 : Let H * be a non negatively graded connected unstable
integral domain over the Steenrod algebra. Then H * has finite transcendence
degree t if and only if the unstable part Un(FF(H * )) is Noetherian of Krull
dimension t .
PROOF PROOF PROOF PROOF PROOF : By Theorem 4.7 we find a diagram as follows
FF(D * (t) q l
)
algebraic
# FF(H * )
# #
D * (t) q l
integral
# Un(FF(H * )).
There field extension is finite by Theorem 5.1.8 in [11]. Hence by Emmy
Noethers classical result that we already used it follows that
D * (t) q l
FF( q H * )
is Noetherian. By Theorem 2.4
D * (t) q l
FF( q H * )
= Un(FF(H * )).
Conversely, if Un(FF(H * )) is Noetherian of Krull dimension t, then
trzdeg FF(Un(FF(H * ))) / IF = dim Un(FF(H * )) = t,
because Un(FF(H * )) is an affine IFalgebra. It follows that
trzdeg FF(H * ) / IF = trzdeg FF(Un(FF(H * ))) / IF = t,
by Lemma 1.3 .
We generalize this result to
19
MARA D. NEUSEL
PROPOSITION 5.2
PROPOSITION 5.2
PROPOSITION 5.2
PROPOSITION 5.2
PROPOSITION 5.2 : Let H * be a non negatively graded connected unstable
reduced algebra over the Steenrod algebra. Let H * have only finitely many
height zero prime ideals p 1 , . . . , p l . Then trzdeg FF(H * / p i ) / IF
£ t for all
i = 1 , . . . , l if and only if the unstable part Un(RF(H * )) is Noetherian of Krull
dimension at most t .
PROOF PROOF PROOF PROOF PROOF : Consider the natural inclusion
H *
# Un(RF(H * )) # RF(H * )
and denote by p e
i # RF(H * ) the height zero prime ideals lying over p i , i =
1 , . . . , l, and note that these are all height zero prime ideals in the total ring
of fractions. Hence we have also in the unstable part precisely l height zero
prime ideals, q 1 , . . . , q l , and without loss of generality
q i ÇH * = p i p e
i Ç Un(FF(H * )) = q i
for i = 1 , . . . , l. Note also that
FF(RF(H * ) / p e
i ) = FF(H * / p i ),
hence RF(H * ) / p e
i # FF(H * / p i ) for all i = 1 , . . . , l. However, the little ring is
actually a field containing H * / p i . Therefore
RF(H * ) / p e
i = FF(H * / p i ) " i = 1 , . . . , l.
Hence we have
Un(FF(H * / p i )) = Un(RF(H * ) / p e
i ) = Un(RF(H * )) / q i ,
" i = 1 , . . . , l, where the last eqaulity follows from Lemma 4.5. Hence by the
preceding Proposition 5.1 all these rings are Noetherian of Krull dimension at
most t. Therefore 16
Un(RF(H * )) / q 1 Ç · · · Ç q l = Un(RF(H * ))
is Noetherian of Krull dimension at most t.
Conversely, if the unstable part Un(RF(H * )) is Noetherian of Krull dimension
at most t, then so is Un(RF(H * )) / q i for any i = 1 , . . . , l. Since this is an affine
ring we have that
t ³ trzdeg FF Un(RF(H * )) / q i = trzdeg FF H * / p e
i
for any i = 1 , . . . , l .
Again none of the assumptions on H * can be removed -- a bit more patient, see
Example .
We are now able to generalize Theorem 4.7. Summarizing we have:
16 Note that
(0) = Nil (H * ) # Nil (Un (RF(H * ))) = q 1 Ç · · · Ç q l # Nil (RF(H * )) = (0).
20
THE LOST CHAPTER
THEOREM 5.3
THEOREM 5.3
THEOREM 5.3
THEOREM 5.3
THEOREM 5.3 : Let H * be a non negatively graded connected reduced un
stable algebra over P P P P P * . Let H * have only finitely many height zero prime ideals
p 1 , . . . , p l . Then the following statements are equivalent:
(1) trzdeg(FF(H * / p i )) £ t ÎIN for all i = 1 , . . . , l.
(2) H * / p i is Dfinite of Dlength at most t for all i = 1 , . . . , l.
(3) There exists an integral ring extension
D * (n i ) q l
# Un FF H * / p i ,
where t ³ n i for all i = 1 , . . . , l.
(4) Un(RF(H * )) is Noetherian of Krull dimension n £ t.
Moreover if one of these statements (and hence all) is true then it follows that
H * itself is Dfinite of Dlength at most t and that there exists an integral ring
extension
D * (n) q
l
# Un RF(H * ) ,
where t ³ n.
PROOF PROOF PROOF PROOF PROOF : The equivalence of (1), (2) and (3) is the contents of Theorem 4.7
and that (1) Û (4) is the contents of Proposition 5.2.
Moreoever, if, e.g., (4) is valid then the Little Imbedding Theorem, Theorem
7.3.3. in [11] hand us an integral extension
D * (n) q
l
# Un RF(H * )
,
where t = n.
Finally Proposition 4.3 implies that H * is Dfinite of Dlength at most t if (1) is
valid .
The following examples shows that Theorem 5.3 (resp. Propositions 5.2 and
4.3) is the best possible in the sense that none of the assumptions on H * cna be
removed.
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1 : Take a polynomial algebra in infinitely many linear genera
tors x 1 , x 2 , · · · over IF and mod out the ideal generated by x 1 x i for all i ³ 2.
Then
H * := IF[x 1 , x 2 , x 3 , · · ·] / (x 1 x 2 , x 1 x 3 , · · ·)
is reduced and has finitely many prime ideals
(0) = (x 1 ) Ç (x 2 , x 3 , · · ·).
However,
H * / (x 1 ) = IF[x 2 , x 3 , · · ·]
fails to be Noetherian, its field of fractions has infinite transcendence degree
over IF and
H * = Un(RF(H * ))
21
MARA D. NEUSEL
is also not Noetherian. Moreover, H * has Dlength 2, namely
x q
1 P D 0 - x 1 P D 1 = 0
is a minimal Drelation on H * , although H * / (x 1 ) has infinite transcendence
degree. In particular this example shows that Dfiniteness does not imply that
the Krull dimension is finite, since H * has infinite Krull dimension:
(x 1 ) # (x 1 , x 2 ) # (x 1 , x 2 , x 3 ) # · · ·
is an infinite proper chain of prime ideals.
Here is another one.
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2 We : take again a polynomial ring over IF in infinitely many
linear generators x 1 , x 2 , · · ·. This time we mod out the ideal generated by all
products x i x j for all pairwise distinct i, j. Then
H * := IF[x 1 , x 2 , · · ·] / x i x j #i #= j
is reduced with minimal prime ideals of the form
p i = (x 1 , . . . ,
b
x i · · ·).
Hence
H * / p i = IF[x i ]
is Noetherian and so its field of fractions has finite transcendence degree over
IF. However,
H * = Un(RF(H * ))
is not Noetherian, because there are infinitely many minimal prime ideals.
The following result establishes a partial converse of Theorem 2.4. It is for the
case of integral domains over prime fields the contents of Proposition 1.2 in
[16].
COROLLARY 5.4
COROLLARY 5.4
COROLLARY 5.4
COROLLARY 5.4
COROLLARY 5.4 : Let H * be a non negatively graded connected reduced
unstable algebra over P P P P P * . Let H * have only finitely many height zero prime
ideals p 1 , . . . , p l and let trzdeg(FF(H * / p i )) £ t ÎIN for any i = 1 , . . . , l. If H * =
Un(FF(H * )) then H * is Noetherian.
PROOF PROOF PROOF PROOF PROOF : Combine Proposition 5.1, resp. Proposition 5.2 with Theorem 3.1
.
In contrast, the following example shows that Corollary 2.5 has no converse
even for integral domains.
22
THE LOST CHAPTER
EXAMPLE 3
EXAMPLE 3
EXAMPLE 3
EXAMPLE 3
EXAMPLE 3 : Consider the polynomial algebra over IF in three linear gen
erator x, y, z and look at the sub algebra H * generated by x, y, yz i for all
i ÎIN. Let S := {x i #i ÎIN} be a multiplicatively closed subset. We have
H * := IF < x, y, yz, yz 2 , yz 3 , · · · > # IF[x, y, z]
# #
S -1 H * = IF < x, x -1 , y, yz, yz 2 , · > # IF < x, x -1 , y, z]
# #
FF(H * ) = IF(x, y, z)
Note that H * is an integrally closed non Noetherian integral domain of Krull
dimension 3. By Lemma 3.1.3 in [11] we have
Un(IF(x, y, z)) = IF[x, y, z].
Hence
H * = H *
= S -1 H * Ç IF[x, y, z]
= S -1 H * Ç Un(F(x, y, z))
= Un(S -1 H * ),
but it is not Noetherian.
§6. The Odd Case
We turn our attention to the case where our multiplicatively closed set S #
H * \ 0 contains zerodivisors. This causes problems because the canonical map
u : H *
# S -1 H * , h # h
1
is no longer injective. We will see in a minute how we get around this.
First we need a generalization of Proposition 2.1.
PROPOSITION 6.1
PROPOSITION 6.1
PROPOSITION 6.1
PROPOSITION 6.1
PROPOSITION 6.1: Let H * be an unstable algebra over the Steenrod algebra,
and let A * be an arbitrary algebra over P
P P P P * containing H * . Then the integral
closure, H * A * , of H * in A * is again an unstable algebra over the Steenrod algebra.
PROOF PROOF PROOF PROOF PROOF : The proof bases on a carefull analyse of the proof of Proposition
3.2 in [15]. We add it hoping to make the read easier.
First, H * A * inherits an action of the Steenrod algebra from A * , because if
a l + h l-1 a l-1 + · · · + h 1 a + h 0 = 0
is a relation of integral dependence of a ÎA * over H * for some l ÎIN and
h 0 , . . . , h l-1 ÎH * , then applying the giant Steenrod operation yields to a re
lation of integral dependence
(H) P(n)(a) l + P(n)(h l-1 ) P(n)(a) l-1 + · · · + P(n)(h 1 )P(n)(a) + P(n)(h 0 ) = 0
23
MARA D. NEUSEL
of P(n)(a) over H * [n], i.e., all coefficients of (H) as polynomials in n vanish. This
means that for a ÎA * integral over H * , P 1 (a) is integral over H * < a > and hence
over H * and successively P i (a) is integral over H * < a, P 1 (a) , . . . , P i-1 (a) > and
hence over H * for all i ÎIN 0 . Therefore H * A * inherits the action of the Steenrod
algebra from A * .
Next we show that this action is also unstable. To this end note that the
equation H tells us also that
P(n)(a) ÎH * [n].
However the integral closure H * [n] is nothing else but H * [n] 17 . This means
that
P(n)(a) ÎH * [n]
is a polynomial in n, i.e., a is an unstable element .
We are now ready to prove.
THEOREM 6.2
THEOREM 6.2
THEOREM 6.2
THEOREM 6.2
THEOREM 6.2 : Let S # H * \ 0 be a multiplicatively closed subset. Let
H * be Noetherian. Then the unstable part of S -1 H * is again Noetherian and
moreover integrally closed.
PROOF PROOF PROOF PROOF PROOF : Since the case where S contains no zerodivisors is the contents of
Theorem 2.4 we can assume without loss of generality that
S = {s, s 2 , s 3 , · · ·}
consists of one zero divisor s and its powers. The canonical mapu : H *
## S -1 H *
induces maps
H * u
# u(H * ) w
# S -1 H * ,
where wis the inclusion and therefore injective. The image u(H * ) is Noetherian,
because H * is. Since u commutes with the action of the Steenrod algebra the
image it again an unstable algebra over the Steenrod algebra. Therefore by
Proposition 6.1
u(H * ) S -1 H * # Un(S -1 H * ).
Observe that
S -1 H * = u(S) -1 u(H * )
so that we have
u(H * ) S -1 H * # Un u(S) -1 u(H * ) .
Finally note that the kernel of the map u is the union of the annihilators of the
powers of the element s
ker(u) = [ i
(0 : s i ).
17 Do Exercise 4.17 on page 137 of [5] if you have doubts.
24
THE LOST CHAPTER
Therefore u(S) contains no zerodivisors of u(H * )
u(S) Ç Zero(u(H * )) = Æ.
So, we can apply our Theorem 2.4 and conclude that
u(H * ) S -1 H * = Un(S -1 H * )
and therefore Un(S -1 H * ) is Noetherian of the same Krull dimension as u(H * )
over which it is finite and integral by Theorem 3.1 .
The following example shows that the converse of the preceding result is not
true.
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1 : Consider the algebra
A * := IF[x 1 , x 2 , · · ·] / (x 1 x 2 , x 1 x 3 , x 1 x 4 , · · ·)
where the generators have degree one. Let S := {x i
1 #i ³ 1}. Then
S -1 A * = IF(x 1 )
and hence
Un(S -1 A * ) = IF[x 1 ].
Direct computation yields that also
A * / ker(u) = IF[x 1 ],
where u : A * ## S -1 A * denotes the canonical map. So everything is Noetherian
and fine, exept A * itself.
25
MARA D. NEUSEL
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26