RATIONAL COHOMOLOGY OF WITT GROUPS Chariya Peterson and Nobuaki Yagita July 9, 1993 Let k be an algebraically closed field of characteristic p and for each n > 0 l* *et W (n) denote the group of Witt vectors of length n. W (n) is a commutative algebraic * *group. For reference, see Jacobson [2], Serre [6]. One of the important properties of* * the Witt groups is the following: Every commutative algebraic k-group whose underlying v* *ariety is an affine space is a homomorphic image of products of W (n). We compute the * *rational cohomology of W (n) for n 2. H*(W (n); k) = S((V n-1*)-1fiL# ) E(Rn-1*L# ); where fi is the Bockstein, V , the shift and R the restriction homomorphism and* * where L# is the graded dual of the restricted Lie algebra End(Ga) identified with the* * first co- i] homology group H1(Ga; k) ~=kx[p. We also show the existence of the higher Bocks* *tein for 1-dimensional cohomology classes of algebraic groups. As an application, we* * compute the rational cohomology of a family of commutative unipotent groups V (n) and d* *iscuss the connection of these cohomology rings with that of the Steenrod algebra. 1 The ring of Witt vectors Let W = Q(xi; yj; zk); 0 i; j; k < m be a polynomial Q algebra and let Wn = W * *x. .x.W be an n-fold product of W with componentwise addition " + " and multiplication * *" . ". Define a new addition " " and multiplication " " on Wn as follows: a b = OE-1(OEa + OEb) (1) a b = OE-1(OEa . OEb); P r i pr-i where, for a = (a0; : :;:am-1), OEa = (OEa0; : :;:OEam-1) with OEar = 0p ai * *: It's inverse P r-1 i -* *1 pr-i OE-1 is defined inductively as: OE-1a0 = a0 and OE-1ar = _1_pr(ar - i=0p (OE * * ai)i ). The triple (Wn; ; ) is a commutative ring over Q with 1 = (1; 0; : :;:0) as identit* *y and 1 (0; : :;:0) as zero element. The map OE : Wn ! Wn is a ring isomorphism from (W* *n; ; ) onto (Wn; +; .). Consider generic vectors x = (x0; : :;:xm-1) and y = (y0; : :;:ym-1), with xi a* *nd yi inde- terminate as above, then each component of x y and x y are in fact a polynomi* *al with integral coefficient. (x y)r, (x y)r 2 Z[x0; y0; : :;:yr] for 0 r < m. For e* *xample: (x y)0 = x0 + y0 ! 1p-1X p i p-i (x y)1 = x1 + y1 - __ x0y0 (2) p 1 i (x y)0 = x0y0 (x y)1 = xp0y1 + x1yp0+ px1y1: For an arbitrary commutative ring A of characteristic p, let Wm (A) be the set * *of m- tuples (a0; : :;:am-1) with ai 2 A and with addition and multiplication defined* * via the polynomials (x y)r and (x y)r as follows: For any three elements a; b; c 2 Wm* * (A), let s : Z[xi; yi; zi] ! Wm (A) be the map sending xi; yi; zi to ai; bi; ci * *respectively. Then Wm (A) becomes an associative commutative ring of characteristic p with (a* * b)r = s(x y)r and (a b)r = s(x y)r), called the ring of Witt vectors of length m. * *In fact, Wm is a functor from commutative rings of characteristic p to commutative rings* *. The prime ring of Wm (A) is isomorphic to Fpm. It consists of Witt vectors with coe* *fficients in Fp, the prime ring of A. 2 Witt groups The underlying abelian group of Wn, denoted by W (n) is a commutative algebraic* * group. It is commonly known as the Witt group of dimension, or length, n. There are na* *tural homomorphisms among W (n) for various n 1: (1) The Frobenius homomorphism: F : Wm ! Wm : F (a) = (ap0; : :;:apn-1), (2) The restriction homomorphism: R : Wm ! Wm-1 : R(a) = (a0; : :;:an-2), (3) The shift homomorphism V : Wm ! Wm+1 : V (a) = (0; a0; : :;:an-1* *). R, F and V commute with each other and their product RF V is multiplication by * *p. Similar to the ring Wn, the Hopf algebra associated to W (n) is constructed fir* *st in charac- teristic 0, then followed by reduction mod p. Over the field of rational number* *s Q, consider the associated algebra Q[y0; . .;.yn-1] of the additive Q-vector group Gna. For* * 0 j < n let j-1 xj = (yi) = pjyj+ pj-1ypj-1+ . .+.yp0 : The Z lattice Z[x0; : :;:xn-1] of Q[V * *] generated 2 by the xi's is closed under comultiplication, counit and antipode. That is is* * an automor- phism on Q[y0; : :;:yn-1] whose restriction to the Z-lattice Z[y0; : :;:yn-1] i* *nduces a Hopf algebra structure on its image Z[x0; : :;:xn-1]. For any field k of characteris* *tic p > 0, W (n) is defined to be the algebraic group associated to Z[x0; : :;:xn-1] k = k[x0; * *: :;:xn-1]. The generator xi is the function xi(a) = ai for a 2 W (n)(A). The first few exa* *mples are 4x0 = x0 1 + 1 x0 and from (2) ! 1X p i p-i 4x1 = x1 1 + 1 x1 - __ x0 x0 : (3) p i 3 Cohomology of W (n) Let G be an algebraic group defined over a field k and k[G] be its coordinate a* *lgebra. For a G module M, the rational cohomology H*(G; M) is the homology of the cobar com* *plex. Cn(G; M) = M In; I is the augmentation idealkof[G]; with the coboundary @i : Ci(G; M) ! Ci+1(G; M) Xi @i(f0: :f:i) = (-1)jf0: :(:4(fj)-fj1-1fj): :f:i+ f0: :f:i1; (4) j=0 Let k[Ga] = k[x] be the associted algebra of the additive algebraic group Ga. T* *he rational cohomology of Ga is given (see Cline, Parshall, Scott and van der Kallen, 4.1 i* *n [1]), ( # # H*(Ga; k) ~= S(fiL# ) E(L ) for p 3 S(L ) for p = 2, where L is the restricted Lie algebra End(Ga), which can be identified with the* * infinite sum i pi 1i=0kxp . Let x(i) denotes the dual basis to x and identify it with the first * *cohomology i 1 class of 1 xp 2 C (Ga; k). S(-) and E(-) are the symmetric and exterior algebr* *a and fi denotes the (algebraic) Bockstein induced from the map fi: C1(Ga; k) ! C2(Ga* *; k). For any monomial xi p-1X ! fixi= -1_ p xij xi(p-j) (5) p j=1 j Remark 3.1 For p = 2 we have fix(i) = x(i)2. However for p 3 fi is not the * *usual Bockstein "fiin the ordinary cohomology, which is induced from the long exact s* *equence from the extension 0 ! k ! W (2)(k) ! k ! 0; 3 but it is "fiP 0(for detail see the appendix A1.5.2 in Ravenel [5]). Indeed, fo* *r H*(Ga; k), P 0is the Frobenius homomorphism in L, P 0x(i) = x(i + 1) and fix(i) = "fix(i +* * 1). In terms of x(i) and fix(i) := y(i + 1) we write aeN 1 k[y(i + 1)] E(x(i))for p 3 H*(Ga; k) ~= N i=01 i=0k[x(i)] for p = 2. Now we consider the cohomology of W (n). For each pair of positive integers n; * *m, the homomorphisms R and V induce an extension 0 ! W (m) ! W (n + m) ! W (n) ! 0: In particular, for n - 1 and 1 we have the extension 0 ! Ga ! W (n) ! W (n - 1) ! 0 (6) which corresponds to the coextension of Hopf algebras: k[xn-1] k[x0; : :;:xn-1] k[x0; : :;:xn-2]; To compute H*(W (n); k) for n 2 we apply the Hochschild-Serre's spectral seque* *nce E*;*2(n) = H*(W (n - 1); H*(Ga; k)) =) H*(W (n); k): For n = 2 and p 3 O1 E*;*2(2) ~= k[y0(i + 1); y1(i + 1)] E(x0(i); x1(i)) i=0 The differential in C*(W (n); k) is given by (3), (4) and (5) i pi pi pi pi @1xp1= 4x1 - (x1 1 - 1 x1 ) = fix0 : So the induced differential in the spectral sequence is d2x1(i) = y0(i + 1). He* *nce O1 E*;*3(2) ~= k[y1(i + 1)] E(x0(i)): i=0 By Cartan-Serre's transgression theorem (see the appendix A.1.5.2 in [5]) d3y1(i + 1) = d3(f"iP 0x1(i)) = "fiP 0d2x1(i) = "fiP 0y0(i + 1) = "fiy0(i* * + 2) = 0: Therefore E*;*3(2) ~=E*;*1(2) and we have just proved the following theorem for* * n = 2. 4 Theorem 3.2 (Compare VII, 9, Lemma 4 in [6]). For any integer n 1, O1 H*(W (n); k) ~= k[yn-1(i + 1)] E(x0(i)) forp 3 (7) i=0 O1 ~= k[x2n-1(i)] E(x0(i)) forp = 2: (8) i=0 Proof: The map of extensions 0 ! Ga -V! W (2) -! Ga ! 0 flfl ? ? fl ?yV n-2 ?yV n-1 n-1 0 ! Ga V-! W (n) -! W (n - 1) ! 0 induces a map of spectral sequences E*;*2(n)~= H*(W (n - 1); H*(Ga; k))=) H*(W (n); k) ?? ? yV n-2* ?yV n-2* E*;*2(2)~= H*( Ga ; H*(Ga; k)) =) H*(W (2); k): By induction, we assume O1 H*(W (n - 1); k) ~= k[yn-2(i + 1)] E(x0(i)): i=0 Since V n-2*yj(i + 1) = yj-n+2(i + 1), and V n-2*xj(i) = xj-n+2(i), where yj(i * *+ 1) = xj(i) = 0 for j < 0, we get d2xn-1(i) = yn-2(i + 1) modulo the ideal(x0(i)); from the naturality and from the result for n = 2. Hence E*;*3(n) is isomorphi* *c to the formular in the theorem, and we see that E*;*3(n) ~=E*;*1(n) by the same reason* * as in the case n = 2. The proof for the case p = 2 is by similar arguments exchanging yj(i + 1) with * *xj(i)2. Corollary 3.3 The map F *on H*(W (n); k) induced from the Frobenius map is inje* *ctive. Proof: This follows from the Theorem since F *xj(i) = xj(i+1) and F *yj(i+1) = * *yj(i+2). 5 4 Higher Bockstein operations Recall that H*(W (n); k) is generated by yn-1(i + 1) and x0(i). We may and will* * hereafter assume that yn-1(i + 1) 2 H2(W (n); k) has a representative in C2(W (n + 1); k)* * of the form i i i i Y = @1xpn= 4xpn- (xpn 1 + 1 xpn) i 3 since V n-1*(4xn) = 4x1 and @2@1(xpn) = 0 in C (W (n + 1); k), so Y is a cocycl* *e. For n = 1 we have the Bockstein fix0(i) = y0(i + 1). For n 1 we define the higher * *Bockstein fin for W (n) by: finx0(i) = yn-1(i + 1), setting fi = fi1. In general Definition 4.1 Let G be an algebraic group defined over k. For an element x 2 H* *1(G; k) and an integer n 1 we define the higher Bockstein of x to be an element finx =* * y in H2(G; k) if there is a map q : G ! W (n) of algebraic k-groups such that the in* *duced map q* : H*(W (n); k) ! H*(G; k) satisfies q*x0(0) = x and q*yn-1(1) = y. Theorem 4.2 Let G be an algebraic k-group. For each element x 2 H1(G; k) such* * that fi1(x) = . .=.fin(x) = 0, then fin+1(x) is defined. Proof: For an element x 2 H1(G; k), let "x2 C1(G; k) be a representative of x. * * Then @1"x= 0 implies that "xis primitive and we get a Hopf algebra homomorphism: k[Ga] ~=k["x] ,! k[G] which induces a homomorphism of algebraic groups q : G ! Ga such that q*x(0) = * *x. Hence the theorem is true for n = 1. Now suppose fi1x = . .=.finx = 0. The last equality implies there is an algebra* *ic group homomorphism q : G ! W (n) with q*x0(0) = x and q*yn-1(1) = 0 in H*(G; k). Let "x2 C1(G; k) be such that @1"xrepresents q*yn-1(1) in C2(G; k). Define a map OE : k[W (n + 1)] ! k[G] as follows: OE|k[x0;:::;xn-1]= q and OE(xn) = "x. The map OE is a map of Hopf a* *lgebra such that OE*x0(0) = x and OE*yn(1) := fin+1x. This finishes the proof of the theore* *m. As a consequence of this Theorem, we can explicitly write down fin in the cobar* * complex. For any sequence I = (i0; : :):, with is 0, for all s 0, let aI denote ai00ai* *11... .Take IJr2 k such that X (a b)r = ar+ br+ IJraIbJ: 6 IfPx 2 H1(G; k) and fi1x = . .=.finx = 0, then there are x1; : :;:xn such that * *dxr = IJrxI xJ for 1 r n and we can define X fin+1x = IJnxI xJ: QUESTION It is still an open question whether the higher Bockstein fin can be e* *xtended to all of H*(G; k). We have the following nonvanishing lemma for the higher Bockstein. Lemma 4.3 Let G be an algebraic k-group. Consider the spectral sequence induc* *ed from a central extension 0 ! Ga ! G !ssG0! 1. For any integer n 1, if in the Hochsc* *hild- Serre's spectral sequence, d2x(0) = fin(x0) 6= 0 for x(0) 2 H1(Ga; k) and x02 H* *1(G0; k). Then fin+1(ss*x0) 6= 0 in H*(G; k). Proof: Since fin(ss*x0) = 0 in H*(G; k), there exists a map qn : G ! W (n) indu* *cing a map of extensions 0 ! Ga -! G -! G0 ! 0 ?? ? ? yqa ?yqn ?yqn-1 0 ! Ga -! W (n) -! W (n - 1) ! 0 with q*n-1x0(0) = x0. Since q*nyn-2(1) = fin-1(x0) 6= 0 in the E2 term of the * *spectral sequence associated to the first extension, we know that q*axn-1(0) 6= 0 in H*(* *Ga; k) since d2xn-1(0) = yn-2(1) 2 H2(W (n-1); k). Hence q*ayn-1(1) = q*afixn-1(0) 6= 0 in E* **;*2. Since yn-1(1) is permanent, so is q*ayn-1(1) which is fin(ss*x0). 5 The group V (n) Every commutative affine algebraic group over k whose underlying varity is an a* *ffine n- space is isogeneous to a product of Witt groups. I.e. it is an extension of a p* *roduct of Witt groups by a finite abelian group. Those groups that are of interestQto us* * in this work are thePones that are isomorphic as algebraic group to a product miW (ni* *), when ni ni+1and ni= n. For n = m we get the additive vector group Gnaand for m = 1 we get W (n). See [6]. For each integer n 2, let V (n) be the commutative linear algebraic group isom* *orphic to a subgroup of the unipotent group U(n) consisting of nxn upper triangular matri* *ces such 7 that each entry along an off diagonal is constant. More precisely, a matrix [ai* *;j] 2 V (n) if ai;j= ffii;jfor i j, and ai;j= ai+r;j+rfor i < j and 0 r n - i. The coordina* *tePalgebra k[V (n)] is a polynomial algebra k[a1; : :;:an-1] with comultiplication 4ai= * *ij=0ajai-j, where, by convension, a0 = 1. V (n) is the so called big Witt group of length n* *, or Witt group at all prime simultaneously. It isomorphic as an algebraic group to a pr* *oduct of Witt groups. Y V (n) ~= W (ri); (9) p-i where for each i, ri is the smallest positive integer such that pri n=i. See [6* *] chapter 5. This decomposition, together with the rational cohomology of W (n) computed * *in the previous section immediately yield H*(V (n); k). However we can compute H*(V (* *n); k) directly. Using the higher Bockstein operation we will prove (9) by showing tha* *t there is a tensor decomposition of H*(V (n); k) in terms of H*(W (m); k). Like in W (n), there exist the Frobenius, the restriction and the shift homomor* *phisms for V (n) and we will also denote them by F , R and V respectively. These maps * *induce various extensions, in particular 0 ! Ga ! V (n + 1) ! V (n) ! 0: (10) with the associated Hochschild-Serre's spectral sequence Ep;q2(n + 1) = Hp(V (n); Hq(Ga; k)) =) Hp+q(V (n + 1); k): (11) Let us denote by S(n) (resp. E(n)) the symmetric algebra S(kyn(i + 1)) (resp. e* *xterior algebra E(kxn(i))). For p = 2, let yn(i + 1) = xn(i)2. Theorem 5.1 For all n 2, Q (a)V (n) ~= p-iW (ri) N n-1 r -1 (b)H*(V (n); k) ~= p-i=1S(p i i) E(i); where ri is the smallest integer such that prii n and firixi(j) = ypri-1i(j + * *1). The proof of the Theorem follows from the following Lemmas which may be useful * *for other results. Let G be a unipotent algebraic group obtained from an extension* * of a product of Witt groups by Ga. mY 0 ! Ga ! G ! W (si) ! 0: (12) i=1 If we write k[W (si)] = k[xi;0; : :;:xi;si-1] and k[Ga] = k[x], then their coho* *mologies are H*(W (si); k) = 1j=0k[yi;si-1(j + 1)] E(xi;0(j)), and H*(Ga; k) = 1j=0k[y(j + * *1)] E(x(j)) respectively, by Theorem 3.2. 8 Lemma 5.2 In the spectral sequence induced from the extension (12); Q (1)If d2x(0) = 0, then G ~= W (si) x Ga, Q (2)If d2x(0) = yj;sj-1(1) for some 1 j m, then G ~= j6=iW (si) x W (sj+ * *1): Proof: Consider the coextension associated to the extension (12) k[xn] k[G] k[W (si)]: If d2x(0) = 0, then 0 6= x(0) 2 H1(G; k) induces a map Y ss : G ! W (si) x Ga; which induces an epimorphism in the coordinate algebras. Since k[G] is polynom* *ial, it also induces an isomorphism of groups by dimension counting argument. Next consider the case d2x(0) = yj;sj-1(1). Since yj;sj-1(1) = fisjxj;0(0), by* * Lemma 4.3 fisj+1xj;0(0) 6= 0 in H2(G; k). Let : G ! W (sj + 1) be the map defining fisj* *+1xj;0(0). We get Y G -ss! W (si) x W (sj+ 1): i6=j Since d2x(0) = yj;sj-1(1) = fisjxj;0(0). In the cobar complex C2(G) we have @1x = ss2(fisjxj;0) = ss2(@1xj;sj) = @1ss1xj;sj: Therefore @1(x-ss1xj;sj) = 0 but d2x(0) 6= 0. Hence x = i1ss1xj;sjin k[Ga], for* * i : Ga ! G. This means ss* is surjective and hence ss is an isomorphism of groups. Lemma 5.3 Let G be a commutative unipotent group defined in (12). Then in the* * asso- ciated spectral sequence X d2x(0) = i(s)yi;si-1(s); i(s) 2 k: Proof: Suppose p 3. Write X X d2x(0) = i;j(k; l)xi;0(k)xj;0(l) + i(s)yi;si-1(s); for i;j(k; l) and i(s) 2 k. This means that there is an element a 2 C1(G) such * *that a belongs to the ideal (xi;j), i.e. the image of a in C1(Ga) = 0 and X pk pl X s @1(x - a) = i;j(k; l)xi;0xj;0+ i(s)(fisi-1xi;0)p ; 9 in C2(G). Since G is a commutative group, the coboundary @1 must be cocommutati* *ve. This implies that @1(x - a) is invariant under the twist, o(c d) = d c, in C2* *(G).Q Therefore i;j(k; l) = j;i(l; k). But xi;0(k)xj;0(l) = -xj;0(l)xi;0(k)Pin H2( * *W (si); k), which forces i;j(k; l) = 0 for all i; j; k; l. Hence d2x(0) = i(s)yi;si-1(s). The case p = 2 is proved by replacing yi;0(k + 1) by xi;0(k) xi;0(k) and use s* *imilar argument as in the case p > 2. Proof of Theorem 5.1. Assume p 3. It is clear that the lemma is true for n =* * 2. Assume true for n 2 and induct on n. The group V (n + 1) can be obtained from V (n) by extension by Ga, i.e. it is the extension (12) with the following rep* *lacements: G V (n + 1), si ri, p - i, ri the smallest positive integer such that prii * * n - 1, xi;j xipjand x xn. Recall that the weight w(xi(j)) = w(yi(j)) = ipj, which,* * of course, is preserved by the differential. From Lemma 5.3, we have ae yn_(1) if p|n, d2xn(0) = p (13) 0 otherwise, because the other elements of the same degree are also of the same weight, henc* *e they are all of the form y_n_ps(s) for some s 2. But these elements do not appear in th* *e assumption (b) for n - 1. We will now show that 6= 0. First take n = p, we will show that V (p + 1) AE * *Gpa= Ga x . .x.Ga. For simplicity in the notation, we denote a matrix [aij] 2 V (n)* * by its first row entries: [ai;j] = (1; a1; : :;:an-1). For n + 1 = p + 1 conside* *r the matrix A = [aij] = (1; 1; 0; : :;:0) 2 V (p + 1). Then Ap = (1; 0; : :;:0; 1) 6= I, wi* *th the non trivial entries in position 1 and p + 1. Hence V (p + 1) is not a product of Ga. Now, i* *f = 0, by induction and Lemma 4.3 implies that V (p + 1) is a product of Ga, which lea* *ds to a contradiction. Let n + 1 = mp + 1 and let : V (p + 1) ,! V (mp + 1) be an inclusion of V (p +* * 1) into V (mp + 1) defined as (a1; : :;:ap) = (1; 0;_:_:;:0;_a1-z____"; 0;_:_:;:0;_a2-z____"; :* * :;:0; : :;:0;)ap * * _____-z____" m m * * m By the naturality with respect to of the spectral sequences, d2xp(0) = y1(1) i* *nduces d2xmp(0) = ym (1). The Frobenius F *then implies ae n_(i + 1)if p|n, d2xn(i) = yp (14) 0 otherwise. This proves Theorem 5.1 (b) for the case n + 1. The Bockstein is given by Lemma* * 4.3. Part (a) follows from Lemma 4.3. The case p = 2 is proved similarly by replacin* *g yj(i+1) with xj(i)2. 10 s Remark 5.4 The subalgebra k[x0; xp1] k[x0; x1] = k[W (2)] is a Hopf subalgeb* *ra. Hence there is a group Ws(2) isogenic to W (2). For the extension. 0 ! Ga ! Ws(2) ! Ga ! 0 the differential of the induced spectral sequence is d2x1(0) = y0(s + 1). And h* *ence H*(Ws(2); k) ~= si=1S(y0(i)) 1j=s+1S(y1(j)) 1k=0E(x0(k)); with y0(i) = fix0(i - 1) and y1(i) = fi2x0(j - 1). 6 Frobenius Kernel and the Steenrood Algebra Let r be a positive integer and let Gr be the rthFrobenius kernel of an algebra* *ic k-group G, i.e. it is the kernel of the rthpower of the Frobenius homomorphism r 0 -! Gr -! G-F! G -! 0: It is easy to obtain the similar results as in Sections 3 to 5 for the rational* * cohomology H*(Gr; k). For example r-1O H*(W (n)r; k) ~= k[yn-1(i + 1)] E(x0(i)); i=0 and fin(x0(i) = yn-1(i + 1). Let G(n) be the subgroup of the unipotent group U(n) such that a matrix [ai;j] * *2 G(n) r if ai;j= ffii;jfor i j and api;j= ai+r;j+rfor i < j and 0 r n - i. The coord* *inate ring k[G(n)] is a polynomial algebra k[a1; : :;:an-1] with the comultiplication Xi j 4ai= aj api-j: j=0 On the otherhand, let P (n) be the finite dimensional subalgebra of the Steenro* *d algebra 0 pn generated by the reduced powers P p; : :;:P . Its dual Hopf algebra is n+1 pn p P (n)* ~=k[1; : :;:n+1]=(p1 ; 2 : :;:n+1); 11 P i pj with 4i= j=0j i-j: There is a Hopf algebra epimorphism by (3.3) in [4]. k[G(n)n-1] ! P (n - 2)*: Therefore H*(G(n)n-1; k) is important in homotopy theories. However the computa* *tions seem difficult except for p = 2 and n 3 which we now show. Consider the spectral sequence arises from the extension 1 ! Ga2! G(3)2 ! Ga2! 1 E*;*2~=k[x1(0); x2(0); x1(1); x2(1)]; with d2x2(0) = x1(0)x1(1) and d2x2(1) = x1(1)x1(2) = 0. Therefore we have E*;*3~=k[x1(0); x1(1)]=(x1(0)x1(1)) k[x2(0)2; x2(1)]: The next differential is (see A1, 5.2 in [5]) 1 1 d3x2(0)2 = d3fSqx2(0) = fSq(x1(0)x1(1)) 1 0 0 1 = fSqx1(0)Sfqx1(1) + fSqx1(0)Sfqx1(1) = x1(1)3: Therefore we get i * * j E*;*4~=k[x2(0)4; x2(1)] k[x1(0); x1(1)]=(x1(0)x1(1); x1(1)3) k[x1(0)]x1(0)* *x2(0)2 ; and this is isomorphic to E*;*1. This result is essentially obtained by Liuevic* *ius. See for example, 3.1.24 in [5], where their notation is the following h10 = x1(0), h11 * *= x1(1), w = x2(0)4 and v = x1(0)x2(0)2, and H*(G(3)2; k) ~=ExtP(1)*(k; k) k[x2(1)]: References [1]E. Cline, B. Parshall, L. Scott, van de Kallen, Rational and generic cohomol* *ogy, Invent. Math. 39 (1977), 143-163. [2]N. Jacobson, Basic Algebra II. 12 [3]M. Kaneda, N. Shimada, M. Tezuka and N. Yagita, Cohomology of infinitesimal * *alge- braic groups, Math. Z. 205 (1990), 61-95. [4]M. Kaneda, N. Shimada, M. Tezuka and N. Yagita, Representations of the Steen* *rod algebra, to appear in J. of Algebra. [5]D. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, (1986), Academic Press inc. [6]J. P. Serre, Algebraic Groups and Class Fields. Nobuagi Yagita Chariya Peterson Musashi Institute of Technology Department of Mathematics Tamazutsumi, Setagaya-Ku Northwestern University Tokyo 158, JAPAN Evanston, IL 60208-2033 yagita@math.titech.ac.jp chariya@math.nwu.edu 13