THE BROWN-PETERSON COHOMOLOGY of the CLASSIFYING SPACES OFTHEPROJECTIVEUNITARYGROUPSPU(P) and EXCEPTIONAL LIE GROUPS

Home , Projective unitary group

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 5, May 2008, Pages 2265–2284 S 0002-9947(07)04425-X Article electronically published on December 20, 2007

THE BROWN-PETERSON COHOMOLOGY OF THE CLASSIFYING SPACES OFTHEPROJECTIVEUNITARYGROUPSPU(p) AND EXCEPTIONAL LIE GROUPS

MASAKI KAMEKO AND NOBUAKI YAGITA

Abstract. For a ﬁxed prime p, we compute the Brown-Peterson cohomologies of classifying spaces of PU(p) and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that BP ∗(BPU(p)) and K(n)∗(BPU(p)) are even dimensionally generated.

1. Introduction Let p be a ﬁxed odd prime and denote by BP ∗(X)(resp. P (m)∗(X)) the Brown- ∗ Peterson cohomology of a space X with the coeﬃcient ring BP = Z(p)[v1,v2, ···] ∗ k (resp. P (m) = Z/p[vm,vm+1, ···]) where deg vk = −2p +2.Wedenoteby PU(n) the projective unitary group which is the quotient of the unitary group U(n) by its center S1. Recall that the cohomologies of PU(p) and the exceptional Lie groups F4,E6,E7,E8 have odd torsion elements. In this paper, we compute the Brown-Peterson cohomologies of classifying spaces BG of these Lie groups G as BP ∗-modules using the Adams spectral sequence. Let us write H∗(X; Z/p)by simply H∗(X)andletA be the mod p Steenrod algebra. Ourmainresultisasfollows: Theorem 1.1. Let (G, p) be one of cases (G = PU(p),p) for an arbitrary odd prime p and G = F4,E7 for p =3,andG = E8 for p =5. Then the E2-terms of the Adams spectral sequences abutting to BP ∗(BG) and P (m)∗(BG) for m ≥ 1, s,t ∗ ∗ s,t ∗ ∗ ExtA (H (BP),H (BG)), ExtA (H (P (m)),H (BG)), have no odd degree elements. An immediate consequence is as follows: Corollary 1.2. For (G, p) in Theorem 1.1, the Adams spectral sequences abutting ∗ ∗ to BP (BG) and P (m) (BG) in the previous theorem collapse at the E2-level. In particular BP odd(BG)=P (m)odd(BG)=0. ∗ ∼ ∗ ∗ Recall K(m) (X) = K(m) ⊗P (m)∗ P (m) (X)istheMoravaK-theory. From the above theorem and corollary, we see K(m)odd(BPU(p)) = 0. Then we have the following corollary ([Ko-Ya], [Ra-Wi-Ya]).

Received by the editors November 2, 2005. 2000 Mathematics Subject Classiﬁcation. Primary 55P35, 57T25; Secondary 55R35, 57T05. Key words and phrases. BP-theory, PU(p), exceptional Lie groups.

c 2007 American Mathematical Society 2265

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Corollary 1.3. For (G, p) in Theorem 1.1, the following holds: ∗ ∗ ∗ ∗ ∼ (1) BP (BG) is BP -ﬂat for BP (BP)-modules, i.e., BP (BG × X) = ∗ ∗ BP (BG) ⊗BP ∗ BP (X) for all ﬁnite complexes X. ∗ ∼ ∗ ∗ (2) K(n) (BG) = K(n) ⊗BP ∗ BP (BG). ∗ ∼ ∗ ∗ (3) P (n) (BG) = P (n) ⊗BP ∗ BP (BG). We give the BP ∗-module structure of BP ∗(BPU(p)) more explicitly in this introduction. For the exceptional Lie group cases, the explicit formulas are given in §5. Theorem 1.4. There exists a BP ∗-algebra exact sequence ∗ ∗ ∗ 0 → BP ⊗M → grBP (BPU(p)) → BP ⊗IN/(f0,f1) → 0 where ∼ (1) M = Z [x ,x ,...,x ] as Z -modules (but not Z -algebras). ∼ (p) 4 6 2p (p) (p) (2) IN =Z(p)[x2p+2,x2p(p−1)]{x2p+2}, the principal ideal of Z(p)[x2p+2,x2p(p−1)] generated by x2p+2. 2 (3) Relations f0,f1 are given with modulo (p, v1,v2, ···) : ≡ − p−1 ··· ≡ − ··· f0 v0 v2x2p+2 + ,f1 v1 v2x2p(p−1) + .

Remark. In the above theorem, subscript i of xi means its degree. The alge- ∗ ∗ bra BP (BPU(p)) does not contain the subalgebra BP ⊗Z(p)[x4,...,x2p], but contains a subalgebra which is isomorphic as BP ∗-modules to the above BP ∗- subalgebra. For an algebraic group G over C, Totaro [To] deﬁnes its Chow ring and con- ∗ ∼ ∗ jectures that BP (BG) ⊗BP ∗ Z(p) = CH (BG)(p). Recall that PGL(p, C)isthe algebraic group over C corresponding to the Lie group PU(p). Theorem 1.5. There exists an isomorphism ∗ ∼ ∗ BP (BPU(p)) ⊗BP ∗ Z(p) = CH (BGL(p, C))(p). Hence there exists an additive isomorphism ∗ ∼ CH (BGL(p, C))(p) = Z(p)[x4,x6, ··· ,x2p] ⊕ Fp[x2p+2,x2p(p−1)]{x2p+2}. Remark. Vistoli [Vi] also determined the additive structure of the Chow ring and integral cohomology of BPGL(p, C) by using stratiﬁed methods of Vezzosi [Ve] (see also [Mo-Vi]). Moreover he shows that for G = PGL(p, C), ∗ ∗ H (G; Z) → H (BT; Z)WG(T ) is epic.

∗,∗ Let MGL (X) be the motivic cobordism ring deﬁned by V. Voevodsky [Vo1] 2∗,∗ 2i,i and MGL (X)= i MGL (X). 2∗,∗ ∼ ∗ Corollary 1.6. MGL (BPGL(p, C))(p) = MU (BPU(p))(p). We prove Theorem 1.1 by using the Adams spectral sequence converging to the Brown-Peterson cohomology. The E1-term of the spectral sequence could be given by ∞ ∗ Fp[v0,v1, ···]⊗H (X)withd1x = vkQkx k=0

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where the Qk’s are Milnor’s operations. (Here we identify Fp[v0 = p, v1, ···]= P (0)∗.) By the change-of-rings isomorphism, the E2-term is ∗ ∗ ∼ ∗ ExtA(H (BP),H (X)) = ExtE (Fp,H (X)) ∗ where E = Λ(Q0,Q1, ···). The E∞-term is given by grBP (X). ∗ To state the cohomology H (BPU(p)), we recall the Dickson algebra. Let An be an elementary abelian p-group of rank n,and ∗ ∼ H (BAn) = Fp[t1, ..., tn] ⊗ Λ(dt1, ..., dtn)withβ(dti)=ti. The Dickson algebra is

GL(n,Fp) ∼ Dn = Fp[t1, ..., tn] = Fp[cn,0, ..., cn,n−1] n i with |cn,i| =2(p − p ). The invariant ring under SL(n, Fp) is also given: F − − F SL(n, p) ∼ { p 2} p 1 SDn = p[t1, ..., tn] = Dn 1,en, ..., en with en = cn,0.

We also recall Mui’s ([Mu]) result by using Qi according to Kameko and Mimura [Ka-Mi]

∗ SLn(Fp) ∼ grH (BA) = SDn/(en) ⊕ SDn ⊗ Λ(Q0, ..., Qn−1){un}

where un = dt1...dtn and en = Q0...Qn−1un. Theorem 1.7. There exists a short exact sequence 0 → M/p → H∗(BPU(p)) → N → 0 where M/p is the trivial E-module given in Theorem 1.4 and ∼ N = SD2 ⊗ Λ(Q0,Q1){u2} = Fp[x2p+2,x2(p2−p)] ⊗ Λ(Q0,Q1){u2}

identifying x2p+2 = e2 and x2(p2−p) = c2,1. This theorem is proved by using the following facts. The group G = PU(p) has just two conjugacy classes of maximal elementary abelian p-subgroups, one of ∗ which is toral and the other is nontoral A of rankp = 2. The cohomology H (BG) is detected by these two subgroups. The restriction image to the nontoral subgroup ∗ ∗ ∼ ∗ SL(2,Fp) is iA(H PU(p)) = H (BA) . Similar (but not the same) facts also hold for the exceptional Lie groups given in Theorem 1.1. The algebraic main result in this paper is as follows: ∗ Theorem 1.8. For m ≥ 0, deﬁne f , ..., f − in P (m) ⊗SD by 0 n 1 n d1un = vkQk(un)=f0Q0un + ···+ fn−1Qn−1un. k≥m ∗ Then the sequence f0, ..., fn−1 is a regular sequence in P (m) ⊗SDn. With the notation in this theorem, we prove that the complex ∗ C =(P (m) ⊗SDn ⊗ Λ(Q0,Q1,...,Qn−1){un},d1) − with the diﬀerential d u = n 1 f Q u is a Koszul complex. This means that 1 n i=0 i i n ∗ P (m) ⊗SDn{en}/(f0,...,fn−1) for i =0 Hi(C, d1)= 0 for i ≥ 1. Thus Theorem 1.1 follows from the above theorem.

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Remark about the convergence of the Adams spectral sequence. By The- orem 15.6 in Boardman’s paper [Bo2], since H∗(BP) is of ﬁnite type, the above Adams spectral sequence is conditionally convergent. Moreover, since we prove the above Adams spectral sequence collapses at the E2-level, by the remark after Theorem 7.1 in [Bo1], the above Adams spectral sequence is strongly convergent, so that we know the Brown-Peterson cohomology up to group extension. We now give an outline of the paper. In §2, we note the relation of the Koszul complex and the E1-term of the Adams spectral sequence for Λ(Q0, ..., Qn−1)-free ∗ H (X). Section 3 is stated about the Dickson invariant rings and the Milnor Qi- operation on the cohomology of classifying spaces of elementary abelian p-groups. In §4, we compute H∗(BG), MU∗(BG), CH∗(BG)forG = PU(p). The similar computations are done in §5 for exceptional groups. The last section is devoted to the proof of Theorem 1.8.

2. The Koszul complex and the Adams spectral sequence For spaces X, Y , the Adams spectral sequence converging to the stable homotopy {X, Y } has the E2-term ∗,∗ ∼ ∗ ∗ ⇒{ } E2 = ExtA(H (Y ),H (X)) = X, Y . We want to consider the case Y = P (m). ∼ ∗ ∗ ∼ Recall that A = H (BP) ⊗E with E =Λ(Q0,Q1, ...). Similarly H (P (m)) = ∗ H (BP) ⊗ Λ(Q0, ..., Qm−1). Hence the E2-term of the Adams spectral sequence abutting to P (m)∗(X)isgivenby ∗ ∗ ∼ ∗ ∗ ExtA(H (P (m)),H (X)) = ExtH∗(BP)⊗E (H (BP) ⊗ Λ(Q0, ..., Qm−1),H (X)) ∼ F ∗ E ··· = ExtEm ( p,H (X)) where m =Λ(Qm,Qm+1, ). We will study the case 0 → M → H∗(X) → N → 0

where M is E-trivial and N is a free Λ(Q0,...,Qn−1)-module having the following properties. (For example, X = BPU(p); recall Theorem 1.7.) For a Z/p-algebra S, suppose that E acts on

N = S ⊗ Λ(Q0,...,Qn−1)

such that Qi acts as a derivation and Qn(s ⊗ t)=s ⊗ Qi(t)fors ∈ S, t ∈ N.Then for each k ≥ 0, there exist α0,k, ..., αn−1,k ∈ S such that

Qkx = α0,kQ0x + ···+ αn−1,kQn−1x for all x ∈ N (compare the proof of Proposition 3.3 below). Let R = P (m)∗⊗S. For i =0,...,n − 1, let

fi = vmαi,m + vm+1αi,m+1 + ···∈R. F The derived functor ExtEm ( p,N) is the cohomology of a cochain complex ∗ E1 = P (m) ⊗N = R ⊗ Λ(Q0,...,Qn−1) with a diﬀerential d1(x)= vkQk(x)=f0Q0(x)+···+ fn−1Qn−1(x). k≥m

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Proposition 2.1. With the notation above, if f0, ..., fn−1 is a regular sequence in R = P (m)∗⊗S,then ∗ F ∼ ExtA(H (P (m),N) = ExtEm ( p,N) = R/(f0,...,fn−1). Hence Theorem 1.1 (for G = PU(p)) follows from Theorem 1.7, 1.8 and the above proposition. (We will show Theorem 1.7 in §4 and Theorem 1.8 in §3, 6.) To see the above proposition, we recall the Koszul complex. Let R be a commutative algebra and let g0,...,gn−1 be a sequence of R.LetJ be the ideal of r r R generated by g0, ..., gn−1.LetΛ =Λ (x0,...,xn−1)ber-homogeneous parts with deg(xi) = 1. The Koszul complex is a chain complex

0 → R ⊗ Λn −→···→d R ⊗ Λ1 −→d R ⊗ Λ0 → 0 with the diﬀerential r ··· − j+1 ··· ··· d(xi1 xir )= ( 1) gij xi1 xij xir . j=1 ∗ Let us denote by Hk(R ⊗ Λ ,d) the induced homology. The following proposition is well known.

Proposition 2.2. If g0,...,gn−1 is a regular sequence in R,then ∗ ∼ R/J for k =0, Hk(R ⊗ Λ ,d) = 0 for k ≥ 1.

i Proof of Proposition 2.1. Let gi =(−1) fi. It is clear that g0,...,gn−1 is a regular sequence if and only if f0,....,fn−1 is a regular sequence. Let ∗ µ : R ⊗ Λ(Q0,...,Qn−1) → R ⊗ Λ be an R-module isomorphism deﬁned by ··· ··· ··· ··· µ(Q0 Qi1 Qir Qn−1)=xi1 xir

where 0 ≤ i1 < ···

− i1 ··· ··· ··· ··· =(1) fi1 Q0 Qi1 Qi2 Qir Qn−1 − − i2 1 ··· ··· ··· ··· +( 1) fi2 Q0 Qi1 Qi2 Qir Qn−1 . . − − − ir (r 1) ··· ··· ··· ··· +( 1) fir Q0 Qi1 Qi2 Qir Qn−1,

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we get r − ◦ ··· ··· ··· − ij j+1 ··· ··· µ d1(Q0 Qi1 Qir Qn−1)= ( 1) fij xi1 xij xir j=1 r − j+1 ··· ··· = ( 1) gij xi1 xij xir . j=1 Therefore, µ is an isomorphism of chain complexes, so that by Proposition 2.2, Proposition 2.1 holds.

3. The Dickson invariant and Milnor Qi operation Let A be an elementary abelian p-group of rank n.Themodp cohomology H∗(BA) is isomorphic to the polynomial tensor exterior algebra

Fp[t1,...,tn] ⊗ Λ(dt1,...,dtn)

as a graded algebra. It is also clear that the ﬁnite general linear group GL(n, Fp) ∗ and the ﬁnite special linear group SL(n, Fp)actonA, BA and H (BA). Dick- son computed the ring of invariants of Fp[t1,...,tn] with respect to the action of GL(n, Fp). In the case of G = GL(n, Fp), the ring of invariants is a polynomial algebra

Dn = Fp[cn,0,...,cn,n−1] whose generators are given by the equation n−1 pn pj On(X)= (X + t)=X + cn,j X .

t∈Fp{t1,··· ,tn} j=0

n−i Remark. In [Ka], [Ka-Mi], notation cn,i is used as (−1) cn,i in this paper. For the invariant ring under SL(n, Fp), we have { p−2} p−1 SDn = Dn 1,en, ..., en with en = cn,0.

Mui computed the ring of invariants of Fp[t1,...,tn] ⊗ Λ(dt1,...,dtn) with respect to the action of SL(n, Fp). Of course un = dt1...dtn is invariant under SL(n, Fp). In terms of Milnor’s operation, we may state Mui’s result in the following form. Theorem 3.1 (Mui [Mu], Kameko-Mimura [Ka-Mi]).

∗ SL(n,Fp) ∼ H (BA) = Fp[cn,1, ..., cn,n−1] ⊕ SDn ⊗ Λ(Q0, ..., Qn−1){un} ∼ = SDn/(en) ⊕ SDn ⊗ Λ(Q0, ..., Qn−1){un}

where Q0...Qn−1un = en.

In this section, we compute f0, ..., fn−1 in §2 for the case S = SDn. We study some important relations between Qi and cn,j . Lemma 3.2. For x ∈ H∗(BA), it follows that n n−i (Qn + cn,iQi)(x)=0,Q0...Qˆi...Qn(un)=(−1) cn,ien. i=1

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pi Proof. Note Qi(dtj)=tj .Whenx = dtj ,wesee pn pi O (Qn + cn,iQi)(dtj)=tj + cn,itj = n(tj)=0.

Since Qi is a derivation, we get the first equation. Applying Q0...Qî...Qn−1 to the first equation, we get the second equation.

pk By using the relation Qk+1 =[Qk,℘ ] and the above lemma, we may deﬁne an E-module structure on SDn ⊗ Λ(Q0,...,Qn−1){un}. Let us write (recall the deﬁnition in §2)

Qkun = α0,kQ0un + ···+ αn−1,kQn−1un. With the following proposition, we verify that there is an algebra homomorphism E→SDn ⊗ Λ(Q0,...,Qn−1) such that the evaluation at un induces an E-module ∗ homomorphism SDn ⊗ Λ(Q0,...,Qn−1) → H (BA). ∗ Proposition 3.3. For each x in SDn ⊗ Λ(Q0,...,Qn−1){un}⊂H BA,

Qkx =(α0,kQ0 + ···+ αn−1,kQn−1)x. ··· − r ··· Proof. It is immediate that QkQi1 Qir =( 1) Qi1 Qir Qk and so ··· − r ··· QkQi1 Qir un =(1) Qi1 Qir Qkun − r ··· ··· =(1) Qi1 Qir (α0,kQ0un + + αn−1,kQn−1un) ··· ··· =(αk,0Q0 + + αn−1,kQn−1)Qi1 Qir un.

The following propositions are only used in the last section to prove Theorem 1.8.

Proposition 3.4. For i ≥ 1, αi,0 =0and α0,0 =1. Letting α−1,k =0, we have p − p αi,k+1 = αi−1,k αn−1,kcn,i.

pk Proof. Since Qk+1 =[Qk,℘ ], we have n−1 n−1 pk pk p Qk+1un = ℘ Qkun = ℘ αi,kQiun = αi,kQi+1un. i=0 i=0 k i The last equation follows from deg αi,k =2p − 2p and pk pk−pi · pi p ℘ αi,kQiun = ℘ αi,k ℘ Qiun = αi,kQi+1un. We also get n−1 n−2 p p p αi,kQi+1un = αi,kQi+1un + αn−1,kQnun i=0 i=0 n−1 n−1 p − p = αi−1,kQiun αn−1,kcn,iQiun. i=1 i=0 Here the last equation follows from Lemma 3.2. Hence the last element is equal to Qk+1u = αi,k+1Qiun.Thuswegettheequationinthisproposition.

Proposition 3.5. For k ≤ n − 1, αi,k = δi,k.

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Proposition 3.6. We have αi,n = −cn,i and αn−1,k =0 for k ≥ n;inparticular,

pk+···+p+1 αn−1,n+k = cn,n−1 + terms lower with respect to cn,n−1.

In particular, α0,n = −cn,0,...,αn−1,n = −cn,n−1, αi,j = δi,j for i, j = 0,..., n − 1. That is, fi is expressed as ⎧ ⎪ f0 = v0 − vncn,0 + ··· ⎪ ⎪ ⎨ f1 = v1 − vncn,1 + ··· ⎪ . . ⎪ . .. ⎪ . ⎩⎪ fn−1 = vn−1 − vncn,n−1 + ···

4. Projective unitary groups PU(p) The additive structure of the cohomology H∗(BPU(p)) was first determined by the first author computing the PoincaréseriesoftheE2-term of the Rothenberg- Steenrod spectral sequence. (A. Vistoli [Vi] also gets the same results by completely different methods.) Indeed, we have ∗ i i PS(H (BPU(p),t)) = rankpH (BPU(p))t 1 t2 + t3 + t2p+1 + t2p+2 = + . (1 − t4)(1 − t6)...(1 − t2p) (1 − t2p+2)(1 − t2p(p−1)) HoweverwegiveitherebyashortargumentusingtheresultsofVavpetiˇcand Viruel [Va-Vi] and Kono and Yagita [Ko-Ya]. Let G be a compact Lie group. Quillen showed that the following restriction map r is an F-isomorphism: ∗ → ∗ r : H (BG) lim←− H (BA) A where A runs over a set of conjugacy classes of elementary abelian p-subgroups of G. Here an F-isomorphism means that its kernel is generated by nilpotent elements s and for each x ∈ H∗(BG)thereiss ≥ 0 such that xp ∈ Im(r). Recently A.Vavpetiˇc and A.Viruel gave the following short proof of the fact that the map r is injective for G = PU(p). There is only one conjugacy class of nontoral abelian groups A of rankp =2. Let TG be a maximal torus of G and NG = NG(T ) the normalizer of the torus TG. Let Np(G)bethep-normalizer of TG, namely, the preimage of a p-Sylow subgroup in the Weyl group WG = NG/TG. For example, it is known that ∼ 1 ∼ i ∗ Np(U(p)) = S Z/p = {diag(z1, ..., zp)P ∈ U(p)|zj ∈ C , 0 ≤ i ≤ p − 1} where P ∈ U(p) is a permutation matrix of order p. ∼ Lemma 4.1. Np(SU(p)) = Np(PU(p)). ∗ ∼ The cofiber sequence (C =)S1 → BSU(p) → BU(p) induces the Gysin exact sequence

∗−2 ∗ ∗ → H (BNp(U(p))) → H (BNp(U(p))) → H (BNp(SU(p))) → .

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Considering restrictions to A and tori of U(p)andSU(p), VavpetiˇcandViruel prove : ∗ Lemma 4.2 ([Va-Vi]). H (BNp(SU(p))) is detected by A and TSU(p). ∗ Theorem 4.3 ([Va-Vi]). H (BPU(p)) is detected by A and TPU(p). Proof. Consider the restrictions i i ∗ NPU ∗ Np ∗ ∼ ∗ H (BPU(p)) → H (BNPU(p)) → H (BNp(PU(p))) = H (BNp(SU(p))).

Recall the transfer tr and Gottlieb transfer Gtr so that Gtr.iNPU = id and tr.iNp = id. Hence the above restrictions are injective.

Note that WG(A)=SL(2, Fp)forG = PU(p). Next by using arguments ([Ko-Ya]), we will show for G = PU(p) that the following map piA is epic:

∗ iA ∗ WG(A) piA : H (BG) → H (BA) = SD2/(e2) ⊕ SD2 ⊗ Λ(Q0,Q1){u2} proj. → SD2 ⊗ Λ(Q0,Q1){u2}. Let E be the subgroup of SU(p) generated by c = diag(ξ, ..., ξ),a= diag(1, ξ, ..., ξp−1),b= P (permutation matrix) where ξ is the p-th primitive root of 1. Then we see that ap = bp = cp =1, [a, b]=c ∈ center of E. 1+2 3 Hence E is isomorphic to the extraspecial p-group p+ of order p .Weconsider the commutative diagram for ﬁberings Z/p −−−−→ SU(p) −−−−→ PSU(p)=PU(p)

⏐ ⏐ ⏐ =⏐ ⏐ ⏐

Z/p = c−−−−→ E −−−−→ A = a, b. ∗,∗ → ∗,∗ The above diagram induces the map fr : E(U)r E(E)r from the spectral sequence converging to H∗(BSU(p)) to that converging to H∗(BE). ∗,∗ The diﬀerential of E(E)r is well known [Te-Ya]. Let us write ∗,∗ ∼ ∗ ∗ Z ∼ F ⊗ E(E)2 = H (BA; H (B /p)) = p[t1,t2,w] Λ(dt1,dt2,dw) with βdw = w. Then it is known that

d2(dw)=dt1dt2 = u2. ∈ ∈ ∗ Since dw Im(f2), we see that u2 Im(f3)andsoinIm(piA). ∈ ∗ Here we also see that c2,1,c2,0 Im(iA). Let ρ be the canonical representation of SU(p)and λ = ρ ⊗ ρ−1 : SU(p) → SU(p2). Since ρ ⊗ ρ−1|Z/p is trivial, we can identify λ as the representation of PU(p). For i j ∼ 2 i j 2 a b ∈ A = (Z/p) , we easily compute χλ(a b )=p if i = j =0and=0otherwise. This means that λ|A is the regular representation. So the total Chern class is

| 2 c(λ A)=Π(λ1,λ2)∈(Z/p) (1 + λ1t1 + λ2t2)=1+c2,1 + c2,0.

Hence we have cp2−p(λ)|A = c2,1 and cp2−1(λ)|A = c2,0.

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Lemma 4.4. The map piA is epic. p−1 (From Lemma 3.2, we also have e2 = Q0Q1u2, e2 = c2,0 and Q2Q0(u2)= c2,1e2.) Thus we get the short exact sequence ∗ 0 → Ker(piA) → H (BPU(p)) → SD2 ⊗ Λ(Q0,Q1){u2}→0. By the theorem by Vavpetiˇc and Viruel, we have the injection ∗ ∗ iT × iA : Ker(piA) ⊂ H (BPU(p)) → H (BTPU(p)) × SD2/(e2).

In particular Ker(piA) is generated by even-dimensional elements. Now we consider the Bockstein spectral sequence ∗ ∗ E1 = H (X)=⇒ Fp ⊗ H (X; Z)/(torsions).

The ﬁrst diﬀerential is the Bockstein operation Q0.SinceIm(piA)isQ0-free, so H(Im(piA); Q0)=0.Hencewehave ∗ ∼ E2 = H(H (BPU(p); Q0)) = Ker(piA).

Since Ker(piA) is generated by even-dimensional elements, we get ∼ ∼ E∞ = E2 = Ker(piA). It is well known that ∗ ∼ ∗ ∼ H (BPU(p); Q) = H (BSU(p); Q) = Q[c2, ..., cp].

i i Since rankp(Fp ⊗ (H (X; Z)/(torsions))) = rankQH (X; Q), we get Lemma 4.5. There exist additive isomorphisms ∼ ∗ ∼ Ker(piA) = Fp ⊗ H (BPU(p); Z)/(torsions) = Fp[x2, ..., xp] with |xi| =2i. Remark. This is not a ring isomorphism; e.g., for p =3wehavetheringisomor- phism ∼ F 3 − 2 ∼ F Ker(piA) = 3[c2,c3,c6]/(c2 c3) = 3[c2,c3]. Thus we have Theorem 1.7 in the introduction. Theorem 4.6. There exists a short exact sequence of rings ∗ 0 → M/p → H (BPU(p)) → SD2 ⊗ Λ(Q0,Q1){u2}→0 ∼ with additively (not rings) M/p = Fp[x2, ..., xp]. Now we consider the Adams spectral sequence converging to P (m)∗(BPU(p)). By the change of rings, Exts,t(H∗(P (m),H∗(X)) ∼ Exts,t (F ,H∗(X)). A = Em p From the above theorem, Theorem 1.8 implies Lemma 4.7. There exists a short exact sequence → ∗⊗ˆ → F ∗ → ∗⊗ˆ { }→ 0 P (m) M ExtEm ( p,H (BPU(p)) P (m) SD2/(f0,f1) e2 0.

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Thus we have Theorem 1.1 in the introduction for G = PU(p). ∼ 1 Next we consider the Chow ring for BPU(p). Recall that Np(U(p)) = S Z/p. By Totaro, it is known that CH∗(S1 Z/p)/p is detected by CH∗((BS1)×p)/p and CH∗(B(S1 × Z/p))/p (see page 271 in [To]) and so by elementary p-subgroups. We also know that the restriction map ∗ 1 ∗ 1 ×p Z/p ∼ Z/p CH (B(S Z/p)) → CH ((BS ) ) /p = Z/p[y1, ..., yp] is epic. Moreover he showed (his conjecture) ∗ ∼ ∗ (∗) CH (BG) = BP (BG) ⊗BP ∗ Z(p) for this group G = S1 Z/p.ByusingCH∗(−)/p (and BP ∗(−)) version of the Gysin sequence given just before Lemma 4.2, we also see (∗)forG = Np(SU(p)). ∗ ∗ Since Chow rings CH (−)(p) (and BP (−) ⊗BP ∗ Z(p)) have transfer (and Gottlieb transfer), the isomorphism (∗)forG = PU(p) follows from the arguments of the proof of Theorem 4.3. Recall PGL(p, C) is the algebraic group corresponding to PU(p). ∗ ∼ ∗ Theorem 4.8. CH (BPGL(p, C))(p) = BP (BPU(p)) ⊗BP ∗ Z(p). Thus we get Theorem 1.5 in the introduction. ∗ ∗ Remark. Vezzosi first studied CH (BPGL(3, C))(3) = CH (PU(3))(3).Heshowed 6 (Theorem 1.1 in [Ve]) if some element χ ∈ CH (PU(3))(3) is zero, then we have the isomorphism in the above theorem. Indeed, we have seen χ =0. ∗,∗ Let H (−; Z) be the motivic cohomology defined by Suslin and Voevodsky 2∗,∗ Z ∼ ∗ 2∗,∗ Z 2i,i Z so that H (X; ) = CH (X). Here H (X; )= i H (X; ). Recall that MGL∗,∗(−) is the motivic cohomology defined by Voevodsky ([Vo1]) by using spectrum MGL in the A1 stable homotopy category. Of course there is the natural map MGL2∗,∗(X) → MU2∗(X).

2∗,∗ ∼ 2∗ Corollary 4.9. MGL (BGL(p, C))(p) = MU (BPU(p))(p). Proof. Let X = BPGL(p, C). Consider the motivic Atiyah-Hirzebruch spectral sequence ∗,∗ ∗ ∗,∗ E2 = H (X; MU )=⇒ MGL (X). When X is smooth and m>2n,itisknownthatHm,n(X; Z) = 0. This implies ((1.2) in [Ya]) 2∗,∗ ∼ 2∗,∗ ∼ ∗ MGL (X) ⊗MU∗ Z = H (X; Z) = CH (X). From the above theorem, we only need to prove that there is a relation such that 2∗,∗ f1x2p+2 = v1e2 + .... =0 alsoinMGL (X)

(x2p+2 = e2 in the notation of Theorem 1.4 and Theorem 4.6). By the solution of the Bloch-Kato conjecture of degree 2 by Merkurjev-Suslin, we get 2,2 3,2 u2 ∈ H (X; Z/p)andQ0(u2) ∈ H (X; Z(p)), 2p+2,p+1 p+1 and hence e = Q1Q0(u2) ∈ H (X; Z(p))=CH (X)(p). The ﬁrst diﬀeren- tial of the Atiyah-Hirzebruch spectral sequence is given by ((4.6) in [Ya])

dp−1(Q0u2)=v1 ⊗ Q1Q0(u2)=v1 ⊗ e2. 2∗,∗ Hence v1e2 =0ingrMGL (X).

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5. The exceptional Lie groups cases With the result on Poincar´e series of cotorsion products, the ﬁrst author proves that the Rothenberg-Steenrod spectral sequence for the mod p cohomology of BG collapses at the E2-level and the Quillen homomorphism r is a monomorphism in the cases G = E8 and p =5,G = F4,E6,E7 and p = 3. In these cases, each exceptional Lie group has two conjugacy classes of maximal elementary abelian p-groups. One is the subgroup of a maximal torus and the other is a nontoral A. Let us write iA : BA → BG and iT : BTG → BG for the induced maps from the inclusions. (However note that for G = E8 and p = 3, there are two conjugacy classes of nontoral maximal elementary abelian 3-subgroups.)

Theorem 5.1 ([Ka]). In the case G = E8 and p =5, G = F4,E6,E7 and p = ∗ × ∗ ∗ → ∗ × ∗ 3, the induced homomorphism iA iT : H (BG) H (BA) H (BTG) is a monomorphism. ∗ Moreover Im(iA) is also known for these groups.

Case I. Consider the cases G = E8 and p =5, G = F4 and p =3. In these cases WG(A)=SL(3, Fp). Let

∗ ∗ WG(A) piA : H (BG) → H (BA) → SD3 ⊗ Λ(Q0,Q1,Q2){u3}.

Let K = Ker(piA)andIM = Im(piA). The map piA is not epic, indeed u3 ∈ IM, and we can show Lemma 5.2. There exists a short exact sequence

0 → IM → SD3 ⊗ Λ(Q0,Q1,Q2){u3}→Fp{u3}→0. Thus we get F ∼ ∗⊗ˆ { }⊕ ∗{ } | | ExtEm ( p,IM) = P (m) SD3/(f0,f1,f2) e3 P (m) δu3 , δu3 =4.

Since K = Ker(piA) is a trivial Em-module generated by even-dimensional elements, we get Theorem 1.1 for these groups. F ∗ Proposition 5.3. For G in case I, ExtEm ( p,H (BG)) is isomorphic to ∗ ∗ ∗ P (m) ⊗ˆ SD3/(f0,f1,f2){e3}⊕P (m) {δu3}⊕P (m) ⊗ˆ K. Next consider the Bockstein spectral sequence ∗ ∗ E1 = H (BG)=⇒ Fp ⊗ H (BG; Z)/(torsions).

Note IM is not Λ(Q0)-free but ∗ ∼ E2 = H(H (BG); Q0) = K ⊕ Fp{Q0u3}.

Hence the Poincar´eseriesPSR(−,t)(forgradedR-vector spaces) is given by 4 PSZ/p(K ⊕ Z/p{δ3},t)=PSZ/p(K, t)+t ∗ = PSZ/p(K ⊕ Z/p{Q0u3},t)=PSQ(H (BG; Q),t). Thus we get Theorem 5.4. For the case I, there exists a P (m)∗-algebra exact sequence ∗ ∗ ∗ 0 → P (m) ⊗ˆ M → grP(m) (BG) → P (m) ⊗ˆ SD3/(f0,f1,f3){e3}→0.

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The mod 3 cohomology of BF4 is completely determined by Toda. ∗ Theorem 5.5 ([Toda]). The cohomology H (BF4) is isomorphic to Z ⊗ Z ⊗{ 2 } Z ⊗ ⊗{ } /3[x36,x48] ( /3[x4,x8] 1,x20,x20 + /3[x26] Λ(x9) 1,x20,x21,x25 )

where the above two terms have the intersection {1,x20}.

Indeed, we see that x26|A = e3,x36|A = c3,1,x48|A = c3,2,x4|A = Q0(u3),x8|A = Q1(u3),x20|A = Q2(u3), x9|A = Q0Q1(u3),x21|A = Q0Q2(u3),x25|A = ∗ Q1Q2(u3). The Brown-Peterson theory BP (BG) is also computed in [Ko-Ya] by using the Atiyah-Hirzebruch spectral sequence. The ring structure of M in Theorem 5.4 is quite complicated ∼ Proposition 5.6 ([Ko-Ya]). M = D3/(c3,0) ⊗ (Z(3){1, 3x4}⊕E) with Z { | ∈{ }} ⊂ Z { 2 } E = (3)[x4,x8] ab a, b x4,x8,x20 (3)[x4,x8] 1,x20,x20 .

4 8 The Poincar´eseriesof(Z/3{1, 3x4}⊕E/3) is a/(1 − t )(1 − t )with a =1+t20 + t40 − (t8 + t20)(1 − t4)(1 − t8) =1− t8 + t12 + t16 − t20 + t24 + t28 − t32 + t40 =(1− t8 + t16)(1 + t12 + t24). Then we have 1 a PZ (D /(c ) ⊗ (Z/3{1, 3x }⊕E),t)= × /3 3 3,0 4 (1 − t36)(1 − t48) (1 − t4)(1 − t8) a(1 + t8) = . (1 − t12)(1 + t12 + t24)(1 − t24)(1 + t24)(1 − t4)(1 − t16) Here we see that (1 + t8)a =(1+t8)(1 − t8 + t16)(1 + t12 + t24)=(1+t24)(1 + t12 + t24). Hence the above Poincar´e series is indeed equal to

12 24 4 16 PZ/3(M/3,t)=1/(1 − t )(1 − t )(1 − t )(1 − t ). Proposition 5.7. Let cl : CH∗(X)/p → H∗(X) be the mod p cycle map. For G ∗ of Case I, we have piA · cl(CH (BG)) = SD3{e3}. Proof. From Lemma 9.6 in [Ya] and the aﬃrmative answer to the Bloch-Kato con- 4,3 jecture, there is an element x ∈ H (BG; Z/p)withcl(x)=Q0e3.Herenote

2pi+2pj +2,pi+pj +1 pi+pj +1 QiQj(x) ∈ H (BG; Z/p)=CH (BG)/p.

Thus e3,c3,1e3,c3,2e3 are in the image of the cycle map.

Case II. The groups G are E6 and E7 for p =3. For each of these cases, G contains a maximal nontoral elementary abelian 3- ∼ 4 subgroup A = (Z/3) . It is known (e.g., [A-G-M-V]) that WG(A) is the subgroup

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of SL(4, F ) generated by matrices of the form 3 ⎛ ⎞ 0 ⎜ ⎟ ⎜ SL(3, F3)0⎟ ⎝ 0 ⎠ ∗∗∗

where ∗∈F3, =1(resp. = ±1) for G = E6 (resp. G = E7). Recall that

O 3 3(X)=Π(λ1,λ2,λ3)∈(Z/3) (X + λ1t1 + λ2t2 + λ3t3).

Let us denote by O simply O3(t4)sothate4 = Q0Q1Q2Q3(u4)=e3O. Then the invariant ring is computed by Kameko and Mimura [Ka-Mi]: SD /(e )[O] ⊕ SD [O] ⊗ Λ(Q ,Q ,Q ){u ,u } for G = E ∗ WG(A) ∼ 3 3 3 0 1 2 3 4 6 H (BA) = 2 2 SD3/(e3)[O ] ⊕ SD3[O ] ⊗ Λ(Q0,Q1,Q2){u3, Ou4} for G = E7.

At ﬁrst consider the case G = E6.LetuswriteN = SD3[O]⊗Λ(Q0,Q1,Q2){u3,u4} and let the projection be

∗ ∗ WG(A) piA : H (BG) → H (BA) → N.

Lemma 5.8. Ou3 =(Q3 + c3,2Q2 + c3,1Q1 + c3,0Q0)(u4). Proof. From Lemma 3.2, we have (Q3 + c3,iQi)(u4)=(Q3 + c3,iQi)(dt4) × dt1dt2dt3 = O(t4)u3. Hence we have the decomposition

N = SD3 ⊗ Λ(Q0,Q1,Q2){u3}⊕SD3[O] ⊗ Λ(Q0,Q1,Q2,Q3){u4}.

We can see that f0, ..., f3 is not regular in SD3[O] (while it is regular in SD4)for general m ≥ 1.Henceweassumeherem =0.TakeExtE (Fp, −).

Lemma 5.9. The module ExtE (Fp,N) is isomorphic to ∗ ∗ BP ⊗ˆ SD3{e3}/(v1,v2) ⊕ BP ⊗ˆ SD3[O]{e3O}/(v1,v2,v3).

It is known that the image IM = Im(piA) is an algebra over A generated by Q0Q1u4,Q0u3 and SD3, O;namely, ∼ N/IM = SD3[O]{u3,u4,Q0u4,Q1u4,Q2u4,Q3u4}. ∗ The above fact is also known in Kono-Mimura [Ko-Mi]; indeed, H (BE6) is still computed in [Ko-Mi].

Lemma 5.10. ExtE (Fp,N/IM) is isomorphic to ∗ P (0) ⊗ˆ SD3[O]{u3,Q0u4,Q1u4,Q2u4,Q3u4}/(3Q0u4), ∗ which is generated by odd-dimensional elements where P (0) = Z/3[3,v1, ...].

From the above lemmas, we know that ExtE (Fp,Im(piA)) is generated by even- dimensional elements. Of course, so is ExtE (Fp,Ker(piA)), and hence we have

Theorem 5.11. For (E6,p=3), there exists a short exact sequence ∗ ∗ 0 → BP ⊗ˆ M → grBP (BE6) → IN → 0. ∼ ∗ ∗ ∗ (1) IN = BP ⊗ˆ SD3[O]{δQ0u4}⊕BP ⊗ˆ SD3{e3}/(v1,v2) ⊕ BP ⊗ˆ SD [O]{e O}/(v ,v ,v ). 3 3∼ 1 2 3 (2) M = Z(3)[x4,x10,x12,x16,x18,x24] as additively.

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Next consider the case (G = E7,p=3).Itisknownthat ∗ 2 2 iA(H (BG)) ⊂ SD3/(e3)[O ] ⊕ SD3[O ] ⊗ Λ(Q0,Q1,Q2){u3}

(but not in the submodule SD3[O] ⊗ Λ(Q0,Q1,Q2){Ou4}). Let us write N = 2 SD3[O ] ⊗ Λ(Q0,Q1,Q2){u3} and let the projection

∗ ∗ WG(A) piA : H (BG) → H (BA) → N. We also know that ∼ 2 N/Im(piA) = SD3[O ]{u3}. Hence we get Theorem 1.1 by an argument similar to (but more easily than) the case E6.

Theorem 5.12. For (G = E7,p=3), there exists a short exact sequence ∗ ∗ ∗ 2 0 → P (m) ⊗ˆ M → grP(m) (BE7) → P (m) ⊗ˆ SD3[O ]{e3}/(f0,f1,f2) → 0 ∼ where M = Z(3)[x4,x12,x16,x20,x24,x28,x36] as additively.

6. Proof of Theorem 1.8 ∗ For the sake of notational ease, we write Rm,n for P (m) ⊗SDn and J for the ∗ ideal (f,...,fn−1) ⊂ P (m) ⊗Dn.

Proof of Theorem 1.8 in the case m =0. It is clear that fi = vi − vncn,i + ··· for i = 0, ..., n − 1. The above f0, ..., fn−1 is a regular sequence in R0,n.Let ρ : R0,n → R0,n be a ring homomorphism deﬁned by

ρ(vk)=fk for 0 ≤ k ≤ n − 1, ρ(vk)=vk for k ≥ n, ρ(cn,i)=cn,i for 0 ≤ i ≤ n − 1. − p Then, it is clear that for k = 0, ..., n 1, ρ (vk)=vk + pfk = vk where fk = vnβk,n + vn+1βk,n+1 + ···. So the ring homomorphism ρ is a ring automorphism. It is also clear that v0, ..., vn−1 is a regular sequence. Hence, ρ(v0),..., ρ(vn−1) is also a regular sequence. Thus, f0, ..., fn−1 is a regular sequence. Proof of Theorem 1.8 in the case n =1. In the case n = 1, the statement of Theo- − − pm 1+···+1 pm+pm 1+···+1 ··· rem 1.8 is that f0 = vmcn,0 + vm+1cn,0 + is not a zerodivisor. It is clear that R(m, 1) is an integral domain, so that Theorem 1.8 holds in the case n =1. Lemma 6.1. Let ∆ be the determinant of the matrix m ⎛ ⎞ α0,m ··· α0,m+n−1 ⎜ . . . ⎟ Am = ⎝ . .. . ⎠ . αn−1,m ··· αn−1,m+n−1

Then ∆0 =1and p ∆m+1 =∆mcn,0 for m ≥ 0. In particular, the above matrix has an inverse as a matrix of coeﬃcients −1 in Rm,n[cn,0]. This lemma will be proved in the last parts of this section. −1 − Lemma 6.2. The ring (Rm,n/J)[cn,0] is an integral domain for =0,...,n 1.

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Proof. Let ··· fi = vm+nαi,m+n + vm+n+1αi,m+n+1 + for i =0,...,n − 1. Let −1 → −1 ρm : Rm,n[cn,0] Rm,n[cn,0] be a ring homomorphism deﬁned by ≤ ≤ − ρm(vm+k)=vm+k + fk for 0 k n 1, ρm(vm+k)=vm+k for k ≥ m + n, ρm(cn,i)=cn,i for 0 ≤ i ≤ n − 1. p As in the above proof of Theorem 1.8 in the case m =0,ρm =1,sothatρm is a ring automorphism. We deﬁne a ring homomorphism −1 → −1 ψm : Rm,n[cn,0] Rm,n[cn,0] by

ψm(vm+k)=vmαk,m + ···+ vm+n−1αk,m+n−1 for 0 ≤ k ≤ n − 1, ψm(vm+k)=vm+k for n ≤ k, ψm(cn,i)=cn,i for 0 ≤ i ≤ n − 1. −1 By Lemma 6.1, in Rm,n[cn,0], the inverse of the matrix Am exists. Hence, ψm also has an inverse, that is, ψm is also a ring automorphism. Therefore, ψm ◦ ρm is also a ring automorphism and ψm ◦ ρm maps vm, ..., vm+n−1 to f0, ..., fn−1.Itis clear that ψm ◦ ρm induces an isomorphism −1 → −1 Rm,n[cn,0]/(vm+,...,vm+n−1) (Rm,n/J)[cn,0]. −1 It is clear that Rm,n[cn,0]/(vm+,...,vm+n−1) is an integral domain and so is −1 (Rm,n/J)[cn,0].

Lemma 6.3. Let φ : Rm,n → Rm−1,n−1 be a ring homomorphism deﬁned by

φ(vk)=vk−1, φ(cn,i)=cn−1,i−1, φ(cn,0)=0.

Then, φ maps fk to fk−1 for k =1, ..., n − 1 and φ(f0)=0. Proof. It suﬃces to show that (1) φ(α0,n)=0, (2) φ(αi,k)=αi−1,k−1 for i ≥ 1. It is clear from the fact that α0,n = −cn,0 for k ≥ 1 that (1) holds. We prove this lemma by induction on k.Fori ≥ 1, p − p φ(αi,k)=φ(αi−1,k−1 αn−1,k−1cn,k) p − p = ai−2,k−2 αn−2,k−2cn−1,k−1 = αi−1,k−1

in Rm−1,n−1.

Proposition 6.4. If Theorem 1.8 holds for Rm−1,n−1,thencn,0 is not a zerodivisor in Rm,n/J for 1 ≤ ≤ n − 1.

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Proof. Recall J =(f, ..., fn−1). We prove this proposition by induction on n−1− . In the case n − 1 − = 0, the proposition holds because Rm,n is an integral domain. Suppose that cn,0a = af + ···+ an−1fn−1 (1)

in Rm,n. Then, φ(a)φ(f)+···+ φ(an−1)φ(fn−1)=0

in Rm−1,n−1. By the induction hypothesis, φ(f),...,φ(fn−1) is a regular sequence in Rm−1,n−1. Therefore,

φ(a)=b+1φ(f+1)+···+ bn−1φ(fn−1)

for some b+1, .... bn−1. Since the kernel of φ is a principal ideal (cn,0), ··· a = b+1f+1 + + bn−1fn−1 + cn,0b (2) ∈ − for some bk, b Rm,n such that φ(bk)=bk for k = +1,..., n 1. From (1), (2), ··· cn,0a = cn,0bf +(a+1 + b+1f)f+1 + +(an−1 + bn−1f)fn−1

in Rm,n. Hence, cn,0(a − bf) ≡ 0

in Rm,n/J+1.Sincecn,0 is not a zerodivisor, a − bf =0inRm,n/J+1. Hence, a =0inRm,n/J.

Remark. Since f0 is divisible by cn,0, cn,0 is a zerodivisor in Rm,n/J0.

Proposition 6.5. If Theorem 1.8 holds for Rm−1,n−1,thenRm,n/J is an integral domain for 1 ≤ ≤ n − 1.

Proof. For 1 ≤ ≤ n − 1, since cn,0 is not a zerodivisor in Rm,n/J the induced homomorphism → −1 Rm,n/J (Rm,n/J)[cn,0] −1 is a monomorphism. Since (Rm,n/J)[cn,0] is an integral domain by Lemma 6.2, Rm,n/J is also an integral domain. This proposition completes the proof of Theorem 1.8. Now we prove Lemma 6.1. Firstly, we prove that − n−1 p det Am+1 =( 1) cn,0 det Am for m ≥ n. For the sake of notational ease, let

α = αn−1,m ···αn−1,m+n−1.

Note that αn−1,m =0for m ≥ n from Proposition 3.6. We write βi,j for the quotient αi,m+j/αn−1,m+j

and γi,j for βi,j − βi,0 where j =1,..., n − 1. By deﬁnition, α0,m α0,m+1 ··· α0,m+n−1 . . . . . . det Am = . αn−2,m αn−2,m+1 ··· αn−2,m+n−1 αn−1,m αn−1,m+1 ··· αn−1,m+n−1

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Dividing the j-thcolumnbyαn−1,m+j−1,wehave β0,0 β0,1 ··· β0,n−1 . . . . . . det Am/α = . βn−2,0 βn−2,1 ··· βn−2,n−1 11··· 1 Subtracting the ﬁrst column from the j-thcolumnforj =2,...,n,wehave β0,0 γ0,1 ··· γ0,n−1 . . . . . . det Am/α = . βn−2,0 γn−2,1 ··· γn−2,n−1 10··· 0 Hence, γ0,1 γ0,2 ··· γ0,n−1 − n−1 . . . det Am =(1) α . . . . γn−2,1 γn−2,2 ··· γn−2,n−1

On the other hand, by Proposition 3.4, det Am+1 is equal to − p ··· − p αn−1,mcn,0 αn−1,m+n−1cn,0 p − p ··· p − p α0,m αn−1,mcn,1 α0,m+n−1 αn−1,m+n−1cn,1 . . . . . .. . p − p ··· p − p αn−2,m αn−1,mcn,n−1 αn−2,m+n−1 αn−1,m+n−1cn,n−1 p Dividing the j-thcolumnbyαn−1,m+j−1 for each j =1,..., n,wehave − − ··· − cn,0 cn,0 cn,0 p − p − ··· p − β0,0 cn,1 β0,1 cn.1 β0,n−1 cn,1 det A /αp = . m+1 . . .. . . . . . p − p − ··· p − βn−2,0 cn,n−1 βn−2,1 cn,n−1 βn−2,n−1 cn,n−1 Subtracting the ﬁrst column from the j-thcolumnforj =2,...,n,wehave − ··· cn,0 0 0 p − p ··· p β0,0 cn,1 γ0,1 γ0,n−1 det A /αp = . m+1 . . .. . . . . . p − p ··· p βn−2,0 cn,n−1 γn−2,1 γn−2,n−1 Hence, p γ0,1 γ0,2 ··· γ0,n−1 − p . . .. . det Am+1 = cn,0α . . . . . γn−2,1 γn−2,2 ··· γn−2,n−1 Therefore, we have n p det Am+1 =(−1) cn,0(det Am ) for m ≥ n.

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Secondly, we deal with the case 0 ≤ m ≤ n − 1. It is clear from Proposition 3.5 that det A0 = 1. Suppose that 1 ≤ m ≤ n − 1. From Proposition 3.5, we have αi,k = δi,k for k ≤ n − 1. ··· α0,n α0,n+m−1 . . 0m,n−m . . αm−1,n ··· αm−1,n+m−1 det Am = αm,n ··· αm,n+m−1 . . I − . . n m αn−1,n ··· αn−1,n+m−1

where 0m,n−m is the m × (n − m) zero matrix and In−m is the (n − m) × (n − m) identity matrix. So we get α0,n ··· α0,n+m−1 − (n−1)(n−m) . . det Am =(1) . . . αm−1,n ··· αm−1,n+m−1 Using Proposition 3.5 and Proposition 3.6, subtracting the ﬁrst column multiplied p by α − − from the j-th column, we have n 1,n+j 1 − ··· cn,0 0 0 − p ··· p cn,1 α0,n α0,n+m−2 det A =(−1)(n−1)(n−m−1) m+1 . . . . . . − p ··· p cn,m−1 αm−2,n αm−2,n+m−2 p α0,n ··· α0,n+m−2 − (n−1)(n−m−1)+1 . . =(1) cn,0 . . αm−2,n ··· αm−2,n+m−2

n p =(−1) cn,0(det Am ). This completes the proof of Lemma 6.1.

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Faculty of Regional Science, Toyama University of International Studies, Toyama, Japan E-mail address: [email protected] Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan E-mail address: [email protected]

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