Finite Complexes with Vanishing Lines of Small Slope
Jeffrey H Smith
Purdue University
x1 Introduction
The purpose of this paper is to construct finite CW-complexes whose mod-p coh*
*omology
have vanishing lines of small slope. A left module M over a connected Fp-algebr*
*a A has
a vanishing line over A of slope m if there is an intercept b such that Exts;tA*
*(M; Fp) = 0
for s > m(t - s) + b (we are using Adams spectral sequence indexing).
To state the main result, give the dual Steenrod algebrasthe basis of monomia*
*ls in
Milnor's generators (see 3.2). Let Ptsbe dual to pt and (if p 6= 2) let Qt be *
*dual to
ot. For a CW-complex X let H*X be the mod-p cohomology of X, let Hev denote the
cohomology in even degrees, and let X^N denote the N-fold smash power X ^X ^. *
*.^.X.
For an element a of the mod-p Steenrod algebra let |a| denote the degree of a a*
*nd let
aH*X denote the image of multiplication by a.
Theorem 1.1. Let A be a sub-Hopf algebra of the mod-p Steenrod algebra and let *
*X be
a p-local CW-complex with H*X a finite dimensional Fp-vector space. Then there *
*is an
integer NX (see 2.5) depending on H*X as a graded vector space and:
(1)For p = 2, if PtsH*X 6= 0 for all Pts2 A with s < t, then X^NX has a non*
*-trivial
stable summand Y such that H*Y is A-free.
(2)For p 6= 2, if PtsHevX 6= 0 for all Pts2 A with s < t and QtH*X 6= 0 for *
*all Qt 2 A,
then X^NX has a non-trivial stable summand Y such that H*Y is A-free.
(3)For p = 2, if PtsH*X 6= 0 for all Pts2 A such that s < t and |Pts| d, th*
*en X^NX
has a non-trivial stable summand Y such that H*Y has a vanishing line ove*
*r A of
slope 1=d.
(4)For p 6= 2, if PtsHevX 6= 0 for all Pts2 A such that s < t and p|Pts| 2d*
* and
QtH*X 6= 0 for all Qt 2 A such that |Qt| d, then X^NX has a non-trivial*
* stable
summand Y such that H*Y has a vanishing line over A of slope 1=d.
Next, we state some corollaries of the main result. For p = 2 let An be then*
*sub-
algebra of the mod-2 Steenrod algebra which is generated by Sq1; Sq2; : :;:Sq2 *
* and for
p 6= 2 let An be the subalgebranof-the1mod-p Steenrod algebra which is generate*
*d by
fi; P 1; P p; : :;:P p .
Theorem 1.2.
(1)For p = 2 and n > 0, if Pn0H*X 6= 0 then X^NX has a stable summand Y su*
*ch
that H*Y is An-1-free and therefore (see 4.1) H*Y has a vanishing line ov*
*er the
Steenrod algebra of slope ____1____2(2n-1)-1for n > 1 and of slope 1=2 fo*
*r n = 1.
(2)For p = 2 and n > 1, if Pn0H*X 6= 0 and Pn1H*X 6= 0 then X^NX has a stab*
*le
summand Y such that H*Y has a vanishing line over the Steenrod algebra of*
* slope
__1____
2(2n-1).
Typeset by AM S-*
*TEX
(3)For p 6= 2 and n > 0, if Qn-1H*X 6= 0, then X^NX has a stable summand Y *
*such
that H*Y is An-1-free and therefore (see 4.1) H*Y has a vanishing line ov*
*er the
mod-p Steenrod algebra of slope ___1___2(pn-1).
Let k(n) denote the n'th connective Morava K-theory at the prime p. For a spa*
*ce X,
the k(n)-homology k(n)*X is a module over the coefficient ring k(n)* = Fp[vn] w*
*hich is
the polynomial algebra on a generator of degree 2(pn - 1).
Theorem 1.3. [Mit]For each prime p and integer n 1 there is a finite CW-comple*
*x Y
such that H*Y has a vanishing line over the Steenrod algebra of slope ___1___2(*
*pn-1)and k(n)*Y
is vn torsion free.
Remark 1.4. The k(n)-homology of X is vn-torsion free if and only if QnH*X = 0 *
*and
the classical Adams Spectral Sequence converging to ss*k(n) ^ X collapses at E2*
*. On the
other hand if H*X has a vanishing line over the Steenrod algebra of slope less *
*than ___1___2(pn-1)
then vn k(n)*X = 0.
Let Ap denote the mod-p Steenrod algebra. For p 6= 2 there is an algebra spli*
*tting
Ap = E[Q0; Q1; : :]: Pp
where Pp is the polynomial part of Ap. Every Ap-module is a Pp-module by restri*
*ction.
For p = 2 let
P2 = A2=
where is the two sided ideal generated by Sq1. Every A2-module M such th*
*at
Sq1M = 0 is a P2-module in a naturalnway. For p 6= 2, let Pn denote the subalge*
*bra of
Pp generated by P 1; P p; : :;:P p and for p = 2, let Pn be the quotient of the*
* sub-algebra
An+1 A2.
Theorem 1.5. For each prime p and integer n 0 there is a finite CW-complex Y s*
*uch
that the integral cohomology of Y is torsion free and H*Y is a free Pn-module.
Remark. These finite complexes are of interest since they have an Adams-Novikov*
* E2-
term with a vanishing line of small slope.
We also give a proof of the following algebraic result.
Theorem 1.6. [Mit] Let B denote one of the algebras An or Pn. Then B admits a l*
*eft
module structure over the mod-p Steenrod algebra extending its left B-module st*
*ructure.
Stable splittings of X^n for X a p-local CW-complex can be constructed by a s*
*tandard
technique using idempotents in the group ring Z(p)Sn. In x2 we construct idempo*
*tents in
QSn. In x3 we recall the idempotent splitting technique and prove the theorems*
* stated
above. The crux of the proof is Theorem 3.4 analyzing the summand MNM eM of *
*MNM ,
for M a module over a sub-Hopf algebras of the Steenrod algebra and eM 2 FpSNM*
* an
idempotent in the Fp group ring of the symmetric group. The proof of Theorem 3*
*.4 is
given in x4.
2
x2 The Idempotents
In this section we recall the construction of idempotents in the rational gro*
*up ring of
the symmetric group. The method is due to Young. For more details see [J-K].
For n > 0, let Sn denote symmetric group of permutations of the set {1; 2; : *
*:;:n}. For
A {1; 2; : :;:n} let SA be the subgroup of permutations which leave the comple*
*ment of
A pointwise fixed.
A partition of the positive integer n is a sequence ff = (ff1; ff2; : :;:ffk)*
* of integers such
that
ff1 ff2 . . .ffk > 0 and ff1 + ff2 + . .+.ffk = n:
The Young diagram [ff] of the partition is the array of n nodes * arranged in k*
* rows with
ffi nodes in the i'th row and with all rows beginning in the first column. For *
*example the
Young diagram of [4; 3; 1] is:
* * * *
* * *
*
An ff tableau tffis an array constructed by replacing the nodes * of the Youn*
*g diagram
[ff] by the integers 1; 2; : :;:n. For example one of the 8! tableau on the Yo*
*ung diagram
[4; 3; 1] is:
1 5 8 7
(2.1) 4 6 3
2
The row group Rffof the tableau tffis the subgroup of Sn consisting of those *
*permutations
for which the rows of tffare invariant sets. Similarly, the column group Cffof*
* tffis the
subgroup of Sn consisting of those permutations for which the columns of tffare*
* invariant
sets. For example the row and column groups of tableau 2.1 are
R(4;3;1)= S{1;5;7;8}x S{3;4;6}x S{2} and
C(4;3;1)= S{1;2;4}x S{5;6}x S{3;8}x S{7}
Let sgn: Sn ! 1 denote the sign homomorphism
Theorem 2.2. (Young),[J-K, 3.1.10] Let Rffand Cffbe the row and column groups of
the Young tableau tff. There is an integer kffwhich depends only on the Young d*
*iagram
[ff] such that
X
eff= _1_k sgn(o)oeo
offe2Rff
o2Cff
is an idempotent in the rational group ring QSn.
Remark. The different tableau tffon [ff] give conjugate idempotents.
Next we recall the explicit formula for the integer kff. The (i; j)-node of *
*the Young
diagram [ff] is the node in the i'th row and j'th column. The (i; j)-hook is th*
*e set consisting
3
of the (i; j)-node, all nodes to the right of it in the i'th row, and all nodes*
* below it in the
j'th column. The hook length h(i; j) is the cardinality of the (i; j)-hook. For*
* example, if
each node of the Young diagram (4; 3; 1) is replaced by its hook length the res*
*ult is
6 4 3 1
4 2 1
1
Theorem 2.3. [J-K,2.3.2 and 3.1.10] The integer kffis given by
Y
kff= h(i; j)
(i;j)2[ff]
For example
k[4;3;1]= 6 . 4 . 4 . 3 . 3 . 2 . 1 . 1
Corollary 2.4. If all the hook lengths of [ff] are relatively prime to p then t*
*he idempotent
effis in the p-local group ring Z(p)Sn
In the next section, we use idempotents in Z(p)Sn to construct stable splitti*
*ngs of X^n
for p-local CW- complexes X. The idempotent used for a particular X depends onl*
*y on
H*X as a graded Fp-vector space.
Definition 2.5. Let V be a finite dimensional graded Fp-vector space. Let Vev*
* be the
subspace of even degree elements and let Vod be the subspace of odd degree elem*
*ents. Now
define h i
(1)nV = dim Vev+ dimVod_p-1where [x] denotes the greatest integer function,
n +1
(2)NV = (p - 1) V2 ,
(3)NX = NH*X for X a CW-complex with H*X finite dimensional.
(4)ffV is the partition (p - 1)n; (p - 1)(n - 1); (p - 1)(n - 2); : :;:p - 1*
* of NV ,
(5)tV is the unique ffV -tableau such that the integers in each row are in *
*increasing
order from left to right and every integer in row i is less than every in*
*teger in row
i + 1.
For example, let p = 3 and let V be concentrated in even degrees with dim V =*
* 3. Then
(1)nV = 3
(2)NV = 12
(3)ffV = (6; 4; 2)
(4)
1 2 3 4 5 6
tV = 7 8 9 10
11 12
Proposition 2.6. The idempotent effVcorresponding to the tableau tV is in the g*
*roup
ring Z(p)SNV
Proof: Use Corollary 2.4.
Definition 2.7. Let eV 2 FpSNV be the idempotent which is the mod-p reduction*
* of
effV2 Z(p)SNV .
4
x3 The Constructions
Idempotent splittings. We begin by recalling the construction of stable splitti*
*ngs using
idempotents. Let g : X ! X be a self map of a CW-complex X. The mapping telesco*
*pe
of g is the homotopy direct limit of the diagram
g g g
X!- X!- X!- : :::
Let __g: X ! Xg denote the natural inclusion.
Proposition 3.1. Assume that X is a double suspension, so that
[X; X]
is naturally a ring, and let g : X ! X be a homotopy idempotent. Then
(1)Id - g is homotopy idempotent, where Id is the identity map
(2)the natural map
_g_____Id-g
X -----! Xg _ X(Id - g)
is a weak equivalence,
(3)The induced map g* : H*X ! H*X is a projection,
(4)The splitting of X induces a splitting in cohomology as
H*(Xg) = H*X g* H*X (Id - g*)
If a finite group G acts on a double suspension X, there is a homomorphism ZG*
* !
[X; X]. If X is p-local, this extends to a homomorphism Z(p)G ! [X; X]. So any *
*idem-
potent e 2 Z(p)G gives a splitting
X ' Xe _ X(1 - e):
Splitting the n-fold Smash. The symmetric group Sn acts on the n-fold smash pow*
*er
X^n of any CW- complex X by permutting the factors. We use the right action
(x1; x2; : :;:xn)oe = (xoe(1); xoe(2); : :;:xoe(n))
For any graded vector space V , there is a (signed) permutation action on V n .*
* Again we
use the right action
v1 v2; : :;:vn . oe = voe(1) voe(2) . . .voe(n)
The permutation action of Sn on X^n induces the (signed) permutation on (H*X)n *
* .
If X is p-local, any idempotent e 2 Z(p)Sn gives a splitting
2X^n ' 2X^n e _ 2X^n (1 - e)
5
and in cohomology
H*(Xn e) = (H*X)n e
Dual Steenrod algebra 3.2. [Mil]Recall that the dual Steenrod algebra for p = 2*
* is
A* = F2[1; 2; : :]: |i| = 2i- 1
and for p 6= 2 is
[o0; o1; : :]: Fp[1; 2; : :]: |oi| = 2pi- 1 |i| = 2(pi- 1):
Let A be a sub-Hopf algebra of the Steenrod algebra. The dual, A*, is a quoti*
*ent of the
dual Steenrod algebra and by [A-M] it is of the form
e(1) 2e(2)
A* = A2=<21 ; 2 ; : :>: forp = 2
or e(1) e(2)
A* = Ap=: forp 6= 2
where 0 e(i) 1 and (for p 6= 2) 0 k(i) 1. The sequences e and k must satisfy
certain conditions imposed by the diagonal of Ap. The sub-Hopf algebra An corre*
*sponds
to the sequence
e = (n + 1; n; : :;:0; 0; : :):
for p = 2, and for p 6= 2 to the sequences
e = (n; n - 1; : :;:1; 0; 0; : :):
k = (1; 1; : :;:1; 0; 0; : :): with 1 repeated n + 1 times
Give the dual Steenrod algebra the basis of monomials in the elements iand (f*
*or p 6= 2)
oi. Then the Adams-Margolis elements are
s
Ptsdual topt and for(p 6= 2) Qt dual toot
Proposition 3.3.
(1)For all primes p and s < t, (Pts)p = 0.
(2)For p 6= 2, Q2t= 0.
t
Proof: It can be checked that Fp[t]=ptis a quotient Hopf algebra of A*. So the *
*elements
Pstfor s < t lie in a divided polynomial sub-Hopf algebra of the Steenrod algeb*
*ra, and
therefore (Pst)p = 0. For similar reasons Q2t= 0.
Main Algebraic Result. The main algebraic result of this paper is
6
Theorem 3.4. Let A be a sub-Hopf algebra of the Steenrod algebra and let M be an
A-module which is finite dimensional over Fp. Let NM be the integer and let eM*
* be the
idempotent given in Defintion 2.5.
(1)Then MNM eM 6= 0.
(2)For p = 2, if PtsM 6= 0 for all Pts2 A with s < t, then MNM eM is A-fr*
*ee.
(3)For p 6= 2, if PtsMev 6= 0 for all Pts2 A with s < t, and QtM 6= 0 for al*
*l Qt 2 A,
then MNM eM is A-free.
(4)For p = 2, if PtsM 6= 0 for all Pts2 A such that s < t and |Pts| d, then*
* MNM eM
has a vanishing line of slope 1=d
(5)For p 6= 2, if PtsMev 6= 0 for all Pts2 A such that s < t and p|Pts| 2d *
*and
QtM 6= 0 for all Qt 2 A such that |Qt| d, then MNM eM has a vanishing*
* line of
slope 1=d
The proof will be given in x4. As a corollary we have the
Proof of the Main Theorem: The theorem follows from Theorem 3.4 and the prop-
erties of idempotent splittings.
Remark. Notice that Theorem 3.4 applies to any sub-Hopf algebra of the Steenrod*
* al-
gebra so is stronger than is needed for the proof of 1.1.
Proof of 1.2. One has the following commutator relations in the Steenrod algebra
Pts= [Pts+1-1; P1s] = [P1s+n-1; Pts-1]
For p 6= 2 Qt = [Q0; Pt0-1] = [Qt-1; P1t-1]
So for a module M over the Steenrod algebra, if Qn-1M 6= 0 (Pn0M 6= 0 for p = 2*
*) then
PtsM 6= 0 for all Pts2 An-1 with s < t and (for p 6= 2) QtM 6= 0 for all Qt 2 A*
*n-1. Now
use Theorem 1.1. This takes care of (1) and (3). Part (2) is similar.
Proof of Theorem 1.3. For p 6= 2, let
X = sk2pn-1BZ=pZ
Qn-1H*X is non-trivial and therefore by 1.2 X^NX has a stable summand Y such t*
*hat
H*Y has a vanishing line of slope ___1___2(pn-1). The k(n)-homology k(n)*X is v*
*n-torsion free
for dimensional reasons. Therefore k(n)*XNX and k(n)*Y are vn-torsion free.
For p = 2 let n n
X = RP 2 ^ CP 2
where RP nis real projective space of dimension n and CP nis complex projective*
* space
of complex dimension n. We have Pn0H*X 6= 0 and Pn1H*X 6= 0. Therefore by 1.2
X^NX has a stable summand Y such that H*Y has a vanishing line of slope ___1__*
*_2(2n-1). The
k(n)-homology k(n)*RP 2nis vn-torsion freenfor dimensional reasons and k(n)*CP *
*2nis
vn-torsion free since all the cell of CP 2 are in even degrees. Therefore k(n)**
*XNX and
k(n)*Y are vn-torsion free.
7
Proof of 1.5. For p 6= 2 let n
X = CP p
Clearly PtsH*X 6= 0 for all Pts2 Pn with s < t. By 1.2 the p-localization of XN*
*X has a
stable summand Y such that H*Y is Pn-free. Now Y is a p-local CW-complex and H*Y
is finite but Y need not be a finite CW-complex. By a variant of the CW-approxi*
*mation
theorem there is a finite CW-complex Y 0and a map Y 0! Y which induces an isomo*
*rphism
in mod-p cohomology.
The case p = 2 is similar.
Proof of 1.6. Let Cn be the cyclic sub-module of
H*sk2pnBZ=pZ
generated by a non-trivial class in degree 1. Then QnCn 6= 0 (Pn0+1for p = 2) a*
*nd as in
the proof of 1.2 it follows using 3.4 that
M = CNCnn eCn
is a non-trivial An-free module. The module Cn has the smallest possible vecto*
*r space
dimension for a module over the Steenrod algebra with a non-trivial Qn multipli*
*cation.
The proof is completed by showing that there is only one copy of An. Let min an*
*d max
be the smallest the largest degrees for which the graded vector space V is non*
*-zero. It
follows from Proposition 4.3 (5) that the integer max - min is equal to the top*
* non-trivial
degree of the graded algebra An. Thus there is a set of generators for the An-m*
*odule M
in degree min. But it also follows from Proposition 4.3 (5) that dim Mmin = 1. *
*The proof
is finished.
x4 Proof of the Theorem 3.4
Theorem 3.4 follows by combining a result of Miller and Wilkerson with an alg*
*ebraic
Lemma.
Vanishing Lines. Miller and Wilkerson give criteria for freeness and for the ex*
*istence of
vanishing lines over sub-Hopf algebras of the mod-p Steenrod algebra.
An elementary Hopf algebra is a Hopf algebra having one of the following simp*
*le algebra
structures:
T ypeE : E = E[x] = Fp[x]=x2 |x| odd orp = 2
T ypeD : D = D[x] = Fp[x]=xp |x| even and p 6= 2
By Proposition 3.3 the Steenrod algebra has many elementary subalgebras. For *
*p = 2
and s < t, let E[Pts] be the type E elementary subalgebra generated by Pts. For*
* p 6= 2 and
s < t, let D[Pts] be the type D elementary subalgebra generated by Ptsand let E*
*[Qt] be
the elementary subalgebra generated by Qt.
8
Theorem 4.1. [A-D][M-P][M-W]Let A be a finite sub-Hopf algebra of the mod-p Ste*
*en-
rod algebra and let M be a connective A-module.
(1)For p = 2, if M is a free E[Pts]-module for all Pts2 A with s < t, then M*
* is A-free.
(2)For p 6= 2, if M is a free D[Pts]-module for all Pts2 A with s < t and if*
* M is a free
E[Qt]-module for all Qt 2 A, then M is A-free.
(3)For p = 2, if M is a free E[Pts]-module for all Pts2 A such that s < t an*
*d |Pts| d,
then M has a vanishing line of slope 1=d.
(4)For p 6= 2, if M is a free D[Pts]-module for all Pts2 A such that s < t a*
*nd p|Pts| 2d
and M is a free E[Qt]-module for all Qt 2 A such that |Qt| d, then M has*
* a
vanishing line of slope 1=d.
Let Ds[x] (Es[x] for p = 2) be the Hopf algebra dual of Fp[x]=xps+1with |x| e*
*ven (|x|
arbitrary for p = 2). As an algebra,
For p 6= 2 Ds[x]~= si=0D[xi] with |xi| = pi|x| and
For p = 2 Es[x]~= si=1E[xi] with |xi| = 2i|x|
This brings us to the key technical point.
Lemma 4.2. For a finite dimensional graded vector space V .
(1)The vector space V NV eV is non-zero.
(2)If V is a Ds[x]-module (ES [x]-module for p = 2) and xsVev 6= 0 (xsV 6=*
* 0 for
p = 2) then V NV eV is D[xs]-free (E[xs]-free for p = 2).
(3)If V is an E[x]-module and xV 6= 0 then V NV eV is E[x]-free.
Before proving this lemma we give the
Proof of 3.4: The vector space V NV eV is non- zero by proposition 4.2(1). We *
*finish
the proof by using Theorem 4.1
For assub-Hopf+algebra1A of the Steenrod algebra, if Pts2 A with s < t, then *
*the dual of
Fp[]=pt is a sub-Hopf algebra of A which is isomorphic to Ds[x] (Es[x]forp=2)*
* with
xs = Pts. If PtsVev 6= 0 (PtsV 6= 0 for p = 2) then by lemma 4.2(2) V NV eV is*
* D[Pts]-free
(E[Pts]-free for p = 2). And (for p 6= 2) if QtV 6= 0 then V NV eV is E[Qt]-fr*
*ee by 4.2(3).
Now use Theorem 4.1 to finish the proof of 3.4
Proof of Lemma 4.2: Let
BV = {v1; v2 : :;:vd}
be an ordered homogeneous basis of V . For a function
f : {1; 2; : :;:n} ! BV
let
f = f(1) f(2) . . .f(n) 2 V n
The collection of all tensors of this form gives a basis of V n . Notice that f*
*or oe 2 Sn
f . oe = f O oe
9
Now let n = NW . A function
f : {1; 2; : :;:NW } ! BV
is standard if
(1)for each integer i, the restriction of f to the set of integers in the i'*
*th row of tW is
order preserving,
(2)for each integer j, the restriction of f to the set of integers in the j'*
*th column of
tW is monotonic and order preserving,
(3)for each integer i and basis vector v 2 BV , if |v| is even there are at *
*most p - 1
integers k in the i'th row of tW with f(k) = v, and if |v| is odd there *
*is at most one
integer k in the i'th row of tW with f(k) = v .
For example, let p = 3, V = W , dim Vev = 2 and dim Vod = 2. Let BV = {v1; v2*
*; v3; v4}
be a homogeneous basis of V with |vi| even for i = 1; 2 and odd for i = 3; 4. R*
*eplacing
each integer of the tableau tW by its image in BV we display an example of a s*
*tandard
function.
v1 v1 v2 v2 v3 v4
v2 v2 v3 v4
v3 v4
Proposition 4.3. Let V and W be finite dimensional graded Fp-vector spaces and *
*let f
be a function
f : {1; 2; : :;:NW } ! BV
where BV is an ordered basis of V .
(1)If there is an integer i and a basis vector v 2 BV such that |v| is even *
*and there
are p different integers k in the i'th row of tW with f(k) = v or such t*
*hat |v| is odd
and there are two integers k in the i'th row of tv with f(k) = v, then
feW = 0:
(2)If nV < nW ,
V NW eW = 0:
(3)If f is standard then
feW 6= 0:
(4)If nV nW then
V NW eW 6= 0:
(5)Suppose that nV nW . Choose a homogeneous basis
BV = {v1; v2 : :;:vd}
such that the map i 7! |vi| is order reversing and let the integers min a*
*nd max be
respectively the smallest and the largest degrees in which the graded vec*
*tor space
V NW eW is non-zero. Then the set
{feW |fstandard and | f| = min}
10
is a basis of the degree min homogenous subspace of V NW eW . Similarly
{feW |fstandard and | f| = max}
is a basis of the degree max homogenous subspace of V NW eW .
Proof: For (1), by hypothesis either, there is a set J of p different integers *
*in the i'th
row of tW such that f(J) = {v} with v an even degree element of the basis or t*
*here is a
set J of two integers in the i'th row of tW such that f(J) = {v} with v an odd*
* degree
element of the basis. Then for oe 2 SJ
ae f for|v| even
f . oe =
sgn(oe) f for|v| odd
and it follows that X
f oe = 0
oe2SJ
therefore
X
f oe = 0 and
oe2RffW
feW = 0
For (2), notice that
(p - 1)nV (p - 1) dimVev+ dimVod < (p - 1)(nV + 1) (p - 1)nW
and so by the pigeon hole principle, any function
f : {1; 2; 3; : :;:NW } ! BV
satisfies the hypothesis of (1). Therefore
f eW = 0
Part (2) now follows immediately.
For a standard function f, a permutation oe in the row group of tW , and a pe*
*rmutation
o in the column group of tW . If
f oeo = f
then o = Id and oe is in the isotropy group of f. So the coefficient of the bas*
*is element
f in the sum feW is |G| where G is the group of permutations oe in the row gro*
*up such
that foe = f. The order |G| is relatively prime to p since G is a product of g*
*roups
isomorphic to Sp-1. Therefore
feW 6= 0
11
If nV nW then standard functions exist, proving (4).
Part (5) is more of the same and is left to the reader.
Lemma 4.2(1) follows from proposition 4.3(4)
To prove lemma 4.2(2), let V be an E[x]-module with xV 6= 0 It follows that t*
*here is a
splitting as E[x]-modules
V = E[x] K
There is an equivariant splitting
V NV = F KNV
as E[x]-modules and therefore a splitting
V NV eV = F eV KNV eV :
of E[x]-modules. The module F is E[x]-free since it is a direct sum of modules *
*of the form
E[x] W . The summand F eV is E[x]-free since E[x] is a local ring. But by Prop*
*osition
4.3(2)
KNV eV = 0
since eK < eV .
We use a filtration argument to prove 4.3(3). We will construct a filtration *
*of V NV
0 = W0 W1 W2 : : :Wm = V NV
by Ds[x]-modules (Es[x]-modules for p = 2) such the associated graded module
E0V NV = mi=1Wi=Wi-1
is D[xs]-free (E[xs]-free for p = 2) which implies that V NV is D[xs]-free (E[*
*xs]-free for
p = 2)
Let I be the kernel of the homomorphism Ds[x] ! D[xs] (Es[x] ! E[xs] for p = *
*2).
Let v 2 Vev (v 2 V for p = 2) be an element such that xsv 6= 0 and (xs)2v = 0. *
*Filter the
Ds[x]-module (Es[x]-module for p = 2) V by
W1 = Iv
W2 = Ds[x]v (Es[x]vforp = 2)
W3 = V
We have
W2=W1 = W = D[xs]=(xs)2 (E[xs] forp = 2)
and the associated graded module is
E0V = W U whereU = W1 W3=W2
Then the tensor filtration of V NV has associated graded module
E0V NV = (E0V )NV = (W U)NV
and we finish the argument by showing that
(4.4) (W U)NV eV
is D[xs]-free (E[xs]-free for p = 2).
The case p = 2 is covered by 4.2(2). We now assume p 6= 2. We need two Propos*
*itions.
12
Proposition 4.5. Let V and W be Ds[x]-modules. If V is D[xs]-free then V W *
*is
D[xs]-free.
Proof: The statement is obvious when W = Fp. Now proceed by induction on dim W .
Proposition 4.6. For a Hopf algebra A which is a local ring, let M be an A-modu*
*le
such that M = F K and F is a free A-module. If P : M ! M is an A-linear projec*
*tion
such that
K KerP
then the image of P is a free A-module.
Proof: We have
F = M=K = ImP KerP=K
since K KerP . So ImP is a summand of a free module and therefore it is free s*
*ince A
is a local ring.
Now we continue with the proof that 4.4 free.
Let BV = {v1; v2 : :;:vd} be a homogeneous basis of V such that {v1; v2} is a*
* basis of
W and {v3; : :;:vd} is a basis of U. Let
R1 = {1; 2; : :;:(p - 1)nV }
be the set of integers in the first row of the tableau tV . For J R1 let
UJ = <{f|f : {1; 2; : :;:NV } ! BV and f-1 {v1; v2} \ R1 = J}>
where denotes the vector space generated by the set A.
Proposition 4.7.
(1)For subsets J and K of R1. If J 6= K then UJ \ UK = 0
(2)V NV = JR1 UJ
If |J| p - 1, let sJ be the set consisting of the p - 1 smallest integers in*
* J and let
X
gJ = - oe 2 FpSNV :
oe2SsJ
The element gJ is an idempotent. Now define
ae U if|J| < p - 1
KJ = J
UJ(1 - gJ) if|J| p - 1
ae 0 if|J| < p - 1
FJ =
UJgJ if|J| p - 1
Proposition 4.8. For J R1
(1)UJ = KJ FJ
(2)KJeV = 0
(3)FJ is a free D[xs]-module
13
Proof: Part (1) is clear since gJ is an idempotent. For oe 2 SR1 we have
oeeV = eV and so gJeV = eV
Then (2) follows from the identity (1 - gJ)eV = 0.
The zero module is free and so for (3) we may assume that |J| p - 1. Then th*
*ere is a
Ds[x]-module R such that
UJ = W p-1 R as Ds[x]text- - modules :
The idempotent gJ acts on the factor W p-1giving a splitting
W p-1 = W p-1 gJ W p-1 (1 - gJ)
Now W p-1 gJ is a free D[xs]-module on one generator. Then by Proposition 4.5
UJgJ = W p-1 gJ R
is D[xs]-free.
Now let
X
F = FJ and
JR1
X
K = KJ
JR1
The proof is finished by combining 4.6 with
Proposition 4.9.
(1)F is a free D[xs]-module and
(2)KeV = 0
Proof: F is free D[xs]-module since it is a direct sum of free D[xs]-modules. K*
*eV =0
since KJeV = 0 for all J R1.
This completes the proof that 4.4 is D[xs]-free and the proof of Lemma 4.2.
14
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15