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 [A-D] D.W. Anderson and D.W. Davis, A vanishing line in homological algebra, Co* *mm. Math. Helv 48 (1973), 318-327. [A-M1] J. Adams and H.R. Margolis, Sub-Hopf algebras of the Steenrod algebra, P* *roc. Camb. Phil. Soc. 76 (1974), 45-52. [A-M2] J. Adams and H.R. Margolis, Modules over the Steenrod algebra, Topology * *10 (1971), 271-282. [J-K] G. James and A. Kerber, "The representation theory of the Symmetric group* *," Encyclopedia of Mathematics and its Applications vol. 16, Addison Wesley, Rea* *ding, Mass, 1981. [M-W] H. Miller and C. Wilkerson, Vanishing lines for modules over the Steenrod* * alge- bra, Journal of Pure and Applied Algebra 22 (1981), 293-307. [Mil] J. Milnor, The Steenrod algebra and its dual, Ann. Math. 67 (1958), 150-1* *71. 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