arXiv:1510.08135v2 [math.AT] 4 Apr 2016 › pdf › 1510.08135.pdf · 2018-10-30 ·...

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arXiv:1510.08135v3 [math.AT] 1 May 2020 ALGEBRAIC COBORDISM AND FLAG VARIETIES N.YAGITA Abstract. Let X be an algebraic variety over k such that ¯ X = X k ¯ k is cellular. We study torsion elements in the Chow ring CH * (X) which corre- sponds to v i y in the algebraic cobordism Ω * ( ¯ X) where 0 = y CH * ( ¯ X)/p and v i is the generator of BP * with |v i | = 2(p i 1). In particular, we try to compute CH * (X) from Ω * ( ¯ X) when X are twisted complete flag varieties. 1. Introduction Let X be a smooth algebraic variety over a field k of ch(k) = 0 such that ¯ X = X k ¯ k is cellular. For a fixed prime p, let CH (X )= CH (X ) (p) be the Chow ring generated by algebraic cycles modulo rational equivalence, localized at p. Let us write Ω (X )= MGL 2,(X ) MU BP the BP -version of the algebraic cobordism defined by Voevodsky ([Vo1], [Le- Mo1,2], [Ya2,6]) with the coefficient ring Ω = BP = Z (p) [v 1 ,v 2 , ...], where |v n | = 2(p n 1). The relation between these theories are given ([Vo1],[Le-Mo1,2]) CH (X ) (X ) Ω Z (p) . In this paper, we study CH (X ) by the restriction map res Ω (X ) Ω ( ¯ X ). Let (y i ) be a Z (p) -base of CH ( ¯ X) with the degree |y i |≤|y i+1 | so that CH ( ¯ X) = i Z (p) {y i }, Ω ( ¯ X) = i Ω {y i }. (Here A{y i } is the A-free module generated by y i .) Define the filtration F i = Ω {y j |j i}⊂ Ω ( ¯ X) and the associated graded algebra gr Ω( ¯ X )= i F i /F i+1 CH ( ¯ X ). Let us write Res Ω (X )= gr (Im(res Ω ) gr ( ¯ X )). An ideal J BP is called invariant if r(J ) J for all (Landweber-Novikov) cohomology operations r in Ω (X ) theory. By using the Cartan formula for the Landweber-Novikov operation, it is almost immediate ; Lemma 1.1. Let (y i ) be a Z (p) -base of CH ( ¯ X ). Then there are invariant ideals J (y i ) in Ω such that Res Ω (X ) = i J (y i ){y i }⊂ Ω CH ( ¯ X ). In the above lemma, let us write J (y i )=(a i1 , ..., a is ) for a ij Ω . Then a ij y i is an Ω -module generator in Res Ω (X ). Hence there is a non zero element c ij in Ω (X ) Ω Z (p) = CH (X ) such that res Ω (c ij )= a ij y i in grΩ ( ¯ X ). 2000 Mathematics Subject Classification. 55N20, 55R12, 55R40. Key words and phrases. Algebraic cobordism, Rost motive, flag manifold. 1

Transcript of arXiv:1510.08135v2 [math.AT] 4 Apr 2016 › pdf › 1510.08135.pdf · 2018-10-30 ·...

Page 1: arXiv:1510.08135v2 [math.AT] 4 Apr 2016 › pdf › 1510.08135.pdf · 2018-10-30 · arXiv:1510.08135v2 [math.AT] 4 Apr 2016 ALGEBRAIC COBORDISM AND FLAG VARIETIES N.YAGITA Abstract.

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ALGEBRAIC COBORDISM AND FLAG VARIETIES

N.YAGITA

Abstract. Let X be an algebraic variety over k such that X = X ⊗k k iscellular. We study torsion elements in the Chow ring CH∗(X) which corre-sponds to viy in the algebraic cobordism Ω∗(X) where 0 6= y ∈ CH∗(X)/pand vi is the generator of BP ∗ with |vi| = −2(pi − 1). In particular, we tryto compute CH∗(X) from Ω∗(X) when X are twisted complete flag varieties.

1. Introduction

Let X be a smooth algebraic variety over a field k of ch(k) = 0 such thatX = X ⊗k k is cellular. For a fixed prime p, let CH∗(X) = CH∗(X)(p) be theChow ring generated by algebraic cycles modulo rational equivalence, localized atp. Let us write

Ω∗(X) = MGL2∗,∗(X)⊗MU∗ BP ∗

the BP ∗-version of the algebraic cobordism defined by Voevodsky ([Vo1], [Le-Mo1,2], [Ya2,6]) with the coefficient ring Ω∗ = BP ∗ ∼= Z(p)[v1, v2, ...], where |vn| =−2(pn − 1). The relation between these theories are given ([Vo1],[Le-Mo1,2])

CH∗(X) ∼= Ω∗(X)⊗Ω∗ Z(p).

In this paper, we study CH∗(X) by the restriction map resΩ : Ω∗(X)→ Ω∗(X).Let (yi) be a Z(p)-base of CH∗(X) with the degree |yi| ≤ |yi+1| so that

CH∗(X) ∼= ⊕iZ(p)yi, Ω∗(X) ∼= ⊕iΩ∗yi.

(Here Ayi is the A-free module generated by yi.) Define the filtration F i =Ω∗yj|j ≥ i ⊂ Ω∗(X) and the associated graded algebra

gr∗Ω(X) = ⊕iFi/F i+1 ∼= Ω∗ ⊗ CH∗(X).

Let us write ResΩ(X) = gr∗(Im(resΩ) ⊂ gr∗(Ω∗(X)).An ideal J ⊂ BP ∗ is called invariant if r(J) ⊂ J for all (Landweber-Novikov)

cohomology operations r in Ω∗(X) theory. By using the Cartan formula for theLandweber-Novikov operation, it is almost immediate ;

Lemma 1.1. Let (yi) be a Z(p)-base of CH∗(X). Then there are invariant idealsJ(yi) in Ω∗ such that

ResΩ(X) ∼= ⊕iJ(yi)yi ⊂ Ω∗ ⊗ CH∗(X).

In the above lemma, let us write J(yi) = (ai1 , ..., ais) for aij ∈ Ω∗. Then aijyiis an Ω∗-module generator in ResΩ(X). Hence there is a non zero element cij in

Ω∗(X)⊗Ω∗ Z(p)∼= CH∗(X) such that resΩ(cij ) = aijyi in grΩ∗(X).

2000 Mathematics Subject Classification. 55N20, 55R12, 55R40.Key words and phrases. Algebraic cobordism, Rost motive, flag manifold.

1

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2 N.YAGITA

Corollary 1.2. For a base (yi) of CH∗(X) and the invariant ideals J(yi) =(ai1 , ..., ais), there is a surjective map q : CH∗(X)/p ։ Q(X) such that

Q(X) = ⊕iC(yi), where C(yi) = Z/pci1 , ..., cis, |aijyi| = |cij |.

Conjecture 1.3. The above map q is isomorphic.

However, it seems difficult to compute Q(X) directly. Hence we consider moreeasy version. The prime invariant ideals are written as Ω∗ or In = (p, v1, ...., vn−1)for 0 ≤ n ≤ ∞. We consider modulo (In+1

∞ ) version

ResΩ,n(X) ⊂ (gr∗Ω∗(X))/In+1∞∼= Ω∗/In+1

∞ ⊗ CH∗(X),

and let us write by Pn(X) the corresponding Q(X) so that there are surjectivemaps

CH∗(X)/p։Q(X) = P∞(X)։...։P2(X)։P1(X).

Lemma 1.4. Let y 6= 0 ∈ CH∗(X)/p with J(y) = In. Then we have

C(y) ∼= Mn(y) = Z/pc0(y), c1(y), ..., cn−1(y),

with degree |ck(y)| = −2pk + 2 + |y|.

If viy ∈ ResΩ(X), then we see that vjy ∈ ResΩ(X) for all j ≤ i. So definedegv(y) = n if vn−1y ∈ ResΩ(X) but vny 6∈ ResΩ(X). (Let degv(y) = 0 if y ∈ResΩ(X) and degv(y) = −1 if py 6∈ ResΩ(X), that meansM0 = Z/p andM−1 = 0.)Then we can write

Corollary 1.5. Let CH∗(X) ∼= ⊕iZ(p)yi. Then

P1(X) ∼= ⊕iMdi(yi) where di = degv(yi).

In general, to compute degv(y) is not so easy problem. However there are casesdegv(y) are known or computable.

Given a pure symbol a in the mod p Milnor K-theory KMn+1(k)/p, by Rost, we

can construct the norm variety Va of dim(Va) = pn − 1 (such that Va is cellular).Rost and Voevodsky showed that there is y ∈ CHbn(Va) for bn = (pn − 1)/(p− 1)such that yp−1 is the fundamental class of Va. Moreover, there is an irreducible(Rost) motive Ra ( write it by Rn simply) in the motive M(Va) of Va such thatCH∗(Ra) ∼= Z(p)[y]/(y

p). We see degv(yi) = n for all 1 ≤ i ≤ p− 1, and moreover

we know ([Ro1], [Vo1,4], [Me-Su], [Vi-Ya], [Ya4])

Theorem 1.6. We have CH∗(Rn)/p ∼= P1(Rn) and

P1(Rn) ∼= M01 ⊕ ⊕p−1i=1Mn(y

i) ∼= Z/p1 ⊕Mn(y)1, y, ..., yp−2.

We consider the K-theory version of Pi(X). Let k∗ = Z(p)[v1]. Take J(y)′ =J(y) ⊗BP∗ k∗ instead of J(y) (so that I ′n = I2 = (p, v1) for n ≥ 2). Let us writethe corresponding Pi(X) by P (1)i(X). For example, when n ≥ 2,

P (1)1(Rn) ∼= Z/p1 ⊕M2(y)1, y, ..., yp−2.

Recall that gr∗geo(X) is the graded ring associated with the geometric filtration of

the algebraic K-theory K0(X) ([To3], [Ya5]).

Lemma 1.7. If K0(X) ∼= K0(X), then we have gr∗geo(X)/p ∼= P (1)∞(X).

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ALGEBRAIC COBORDISM 3

Another examples which we can see degv(y) are the cases that X = G/Bk

twisted complete versal flag varieties. The degree degv(y) are computed by usingthe Milnor operations Qi.

Let G be a simply connected compact Lie group and T its maximal torus. ByBorel ([Bo], [Mi-Tod]), its mod(p) cohomology is written as

(∗) grH∗(G;Z/p) ∼= P (y)/p⊗ Λ(x1, ..., xℓ), ℓ = rank(G)

with P (y) = Z(p)[y1, ..., yk]/(ypr1

1 , ..., yprk

k )

where the degree |yi| of yi is even and |xj | is odd. (Here Λ(a, ..., b) is the mod(p)exterior algebra generated by a, ..., b.)

Let BT be the classifying space of T . We consider the fibering ([Tod], [Mi-Ni])

Gπ→ G/T

i→ BT and the induced spectral sequence

(∗∗) E∗,∗′

2 = H∗(BT ;H∗′

(G;Z/p)) =⇒ H∗(G/T ;Z/p).

The cohomology of BT is given by H∗(BT ) ∼= S(t) = Z[t1, ..., tℓ] with |ti| = 2. It

is well known that yi are permanent cycles (i.e., exist in E0,∗′

∞ ) and that there is aregular sequence ([Tod], [Mi-Ni]) (b1, ..., bℓ) in H∗(BT )/(p) such that d|xi|+1(xi) =

bi. (These bi are called the transgressive elements.) Thus we get

E∗,∗′

∞∼= grH∗(G/T ;Z/p) ∼= P (y)/p⊗ S(t)/(b1, ..., bℓ).

Moreover we know that G/T is a manifold such that H∗(G/T ) is torsion free, and

grH∗(G/T ) ∼= P (y)⊗ S(t)/(b) for S(t)/(b) = S(t)/(b1, ..., bℓ)

where bi = bi mod(p). Since H∗(G/T ) is torsion free, the Atiyah-Hirzebruch spec-tral sequence (AHss) collapses. Hence we also know

grBP ∗(G/T ) ∼= BP ∗ ⊗ grH∗(G/T ).

Lemma 1.8. In the spectral sequence (∗∗), let d|xi|+1(xi) = bi 6= 0. Then we have

the relation in BP ∗(G/T )/I2∞ such that

bi = py(0) + v1y(2) + ...+ vny(n) + ...

where y(k) ∈ H∗(G/T ;Z/p) with π∗y(k) = Qkxj for π : G→ G/T .

Let Gk be the split reductive algebraic group corresponding to G, and Tk bethe split maximal torus corresponding to T . Let Bk be the Borel subgroup withTk ⊂ Bk. Note that Gk/Bk is cellular, and CH∗(Gk/Tk) ∼= CH∗(Gk/Bk). Hencewe have CH∗(Gk/Bk) ∼= H2∗(G/T ) and CH∗(BBk) ∼= H2∗(BT ).

Let G be a nontrivial Gk-torsor. We will study some motive in the twisted flagvariety F = G/Bk. Moreover let G be versal (For the definition and properties ofversal, see §10 below or [Ga-Me-Se], [To2], [Me-Ne-Za], [Ka1].)

The versal case of the main result in Petrov-Semenov-Zainoulline [Pe-Se-Za] isgiven as follows ; Let G be a versal Gk-torsor, and F = G/Bk. Then there is ap-localized motive X = R(G) such that

CH∗(F) ∼= CH∗(R(G))⊗ S(t)/(b), CH∗(R(G)) ∼= P (y).

Moreover, each element in CH∗(R(G)) is written as a sum of products b′is, sincethe map CH∗(BBk)→ CH∗(F) is surjective when G is versal ([Ka1], [Me-Ne-Za]).

From the preceding corollary, we have

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Lemma 1.9. Suppose degv(y) > 0 for y ∈ P (y)/p and d|x|+1(xj) 6= 0. If inH∗(G;Z/p),

Qn(xj) = y but Qk(xj) = 0 for 0 ≤ k < n

then degv(y) ≥ n+ 1.

In the last sections in this paper, by using the above lemma, we try to compute

gr∗geo(R(G))/p ∼= P (1)∗∞(R(G))

for (G, p) = (Spin(13), E7; p = 2) and (E8; p = 3).Example. Let G = Spin(11) (see §12). Then

grH∗(G;Z/2) ∼= P (y)⊗ Λ(x3, x5, x7, x9, z15)

with P (y) = Z(2)[y6, y10]/(y26 , y

210). Here the suffix means its degree. We denote

ci = bi−1 = d(x2i−1) for 2 ≤ i ≤ 5, and e8 = b5 = d(z15). The Qi-actions are wellknown e.g., Q0(x3) = 0, Q1(x3) = Q0(x5) = y6. From the preceding lemma andLemma 1.8, the restriction map resΩ is given

c2 7→ v1y6, c3 7→ 2y6, c4 7→ v1y10, c5 7→ 2y10,

c2c4 7→ v21y6y10, e8 7→ 2y6y10.

Hence we see J(y6) = J(y10) = (2, v1). Moreover we can show J(y6y10) = (2, v21).Thus we can see

ResΩ(R(G)) ∼= BP ∗1 ⊕ (2, v1)y6, y10 ⊕ (2, v21)y6y10

in grΩ∗(R(G)) ∼= Ω∗ ⊗ P (y). Then C(y6) ∼= Z/2c3, c2 for example. Hence

gr∗geo(R(G))/2 ∼= P (1)∗2(R(G)) ∼= C(1)⊕ C(y6)⊕ C(y10)⊕ C(y6y10)

∼= Z/21, c2, c3, c4, c5, c2c4, e8.

Moreover, Karpenko ([Ka2]) proves gr∗geo(R(G)) ∼= CH∗(R(G)).This paper is organized as follows. In §2 we recall the motivic BP -theory using

the Atiyah-Hirzebruch spectral sequence. In §3, we define degv(y) and study thecase J(y) = In. We generalize it to general invariant ideals in §4. In §5, we studythe graded ring gr∗geo(X) from K-theory. In §6, 7, the norm variety and the Dicksoninvariants are studied, considering relations to the operation Qi. In §8, we studydegv(y) when X is an anisotropic quadric. In §9, we recall the cohomology andBP ∗-theory of topological flag manifolds X = G/T . In §10, we recall the resultsby Petrov, Semenov and Zainoulline, for the motves R(G) of flag varieties G/Bk.In §11, §12, we study ResΩ(R(G)) when G = SO(odd) and G = Spin(odd). Westudy gr∗geo(R(G))/2 for groups for Spin(7), Spin(9), Spin(11) in §13. The groupSpin(13) is studied in §14. The exceptional group E7, E8; p = 2 is studied in§15, §16, and E8, p = 3 is studied in §17.

2. algebraic BP theories

Recall that MU∗(−) is the complex cobordism theory defined in the usual (topo-logical) spaces and

MU∗ = MU∗(pt.) ∼= Z[x1, x2, ...] |xi| = −2i.

Here each xi is represented by a sum of hypersurfaces of dim(xi) = 2i in some

product of complex projective spaces ([Ha],[Ra]). Let MGL∗,∗′

(−) be the mo-tivic cobordism theory defined by Voevodsky [Vo1]. Let us write by AMU the

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ALGEBRAIC COBORDISM 5

spectrum MGL(p) in the stable A1-category representing this motivic cobordismtheory (localized at p), i.e.,

MGL∗,∗′

(−)(p) = AMU∗,∗′

(−).

Here note that AMU2∗,∗(pt.) ∼= MU2∗(p). It is not isomorphic to AMU∗,∗′

(pt) in

general, while AMU∗.∗′

(X) is an MU∗(p)-algebra.

Given a regular sequence Sn = (s1, ..., sn) with si ∈ MU∗(p), we can inductively

construct the AMU -module spectrum by the cofibering of spectra ([Ya2,4], [Ra])

(2.1) T−1/2|si| ∧ AMU(Si−1)

×si−→ AMU(Si−1)→ AMU(Si)

where T = A/(A − 0) is the Tate object. For the realization map tC inducedfrom k ⊂ C, it is also immediate that tC(AMU(Sn)) ∼= MU(Sn) with MU(Sn)

∗ =MU∗/(Sn).

Recall that the Brown-Peterson cohomology theory BP ∗(−) with the coefficientBP ∗ ∼= Z(p)[v1, v2...] by identifying vi = xpi−1. (So |vi| = −2(p

i − 1).) We can

construct spectra (in the stable A1-homotopy category)

ABP = AMU(xi|i 6= pj − 1)

such that tC(ABP ) ∼= BP . For S = (vi1 , ..., vin), let us write

ABP (S) = AMU(S ∪ xi|i 6= pj − 1)

so that tC(ABP (S)) = BP (S) with BP (S)∗ = BP ∗/(S).In particular, let AHZ = ABP (v1, v2, ...) so that AHZ2∗,∗(pt.) ∼= Z(p). In the

A1-stable homotopy category, Hopkins-Morel showed that

AHZ ∼= HZ , i.e., AHZ∗,∗′

(X) ∼= H∗,∗′

(X,Z(p))

the (usual) motivic cohomology. Using this result, we can construct the motivicAtiyah-Hirzebruch spectral sequence.

Theorem 2.1. ([Ya2,6]) Let Ah = ABP (S) for S = (vi1 , vi2 , ...). Then there isAHss (the Atiyah-Hirzebruch spectral sequence)

E(Ah)(∗,∗′,2∗′′)2 = H∗,∗′

(X ;h2∗′′

) =⇒ Ah∗+2∗′′,∗′+∗′′

(X)

with the differential d2r+1 : E(∗,∗′,2∗′′)2r+1 → E

(∗+2r+1,∗′−r,2∗′′−2r)2r+1 .

Note that the cohomology Hm,n(X,h2n′

) here is the usual motivic cohomol-

ogy with (constant) coefficients in the abelian group h2n′

. Here we recall someimportant properties of the motivic cohomology. When X is smooth, we know

CH∗(X) ∼= H2∗,∗(X ;Z(p)), H∗,∗′

(X ;Z(p)) ∼= 0 for 2∗′ < ∗.

Hence if X is smooth, then Em,n,2n′

r∼= 0 for m > 2n. The convergence in AHss

means that there is the filtration by the third degree (by v′is)

Ah∗,∗′

(X) = F ∗,∗′

0 ⊃ F ∗,∗′

1 ⊃ F ∗,∗′

2 ⊃ ...

such that F ∗,∗′

i /F ∗,∗′

i+1∼= E∗+2i,∗′+i,−2i

∞ .Let S ⊂ R = (vj1 , ...). Then the induced map ABP (S) → ABP (R) of spectra

induces the BP ∗(p)-module map of AHss : E(ABP (S))∗,∗

′,∗′′

r → E(ABP (R))∗,∗′,∗′′

r .

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6 N.YAGITA

In general, ABP (S)∗,∗′

(X) 6∼= ABP ∗,∗′

(X)/(S). However, from the above mapsand dimensional reason, we see for a smooth X ,

(2.3) ABP (S)2∗,∗(X) ∼= ABP 2∗,∗(X)/(S),

(2.4) ABP (S)2∗,∗(X)⊗BP∗ Z(p)∼= H2∗,∗(X) ∼= CH∗(X).

In this paper, a connected oriented theory h2∗(X) means ABP (S)2∗,∗(X) asabove. We mainly consider the connected oriented theory ABP 2∗,∗(X). We writeit simply

Ω2∗(X) = ABP 2∗,∗(X) ∼= MGL2∗,∗(X)⊗MU∗ BP ∗.

Hence from (2.4), Ω∗(X)⊗Ω∗ Z(p)∼= CH∗(X) for a smooth X .

Recall the filtration F ∗,∗′

i and the associated graded ring

(2.5) grΩ2∗(X) ∼= ⊕iE2(∗+i),∗+i,−2i∞

∼= ⊕iE2(∗+i),∗+i,−2i2 /Im(d)

∼= ⊕i(Ω−i ⊗ CH∗+i(X))/Im(d)

where Im(d) = ∪Im(dr) ⊂ Ω<0 ⊗ CH∗(X) ∼= E2∗,∗,<02

for the differential in the AHss converging to ABP ∗,∗′

(X). Note that drE2∗,∗,0r = 0

and each element in E2∗,∗,02 is permanent.

When X is cellular, it is known the motivic cohomology is

H∗,∗′

(X) ∼= ⊕∗H2∗,∗(X).

Since the degree r of nonzero differential dr is odd (in fact r = |v|+1 for v ∈ BP<0).Hence all dr = 0 and Im(d) = 0. Thus the AHss collapses

grΩ∗(X) ∼= Ω∗ ⊗ CH∗(X) for X : cellular.

Let yi generate (as an Ω∗-module) Ω∗(X) (hence yi 6= 0 ∈ CH∗(X)/p). Thenfrom (2.5), we can write

grΩ∗(X) ∼= ⊕iFi/F i+1, for F i =

|yj|≥i

Ω∗(yj).

Here, we used the notation that Ω∗(y) means the Ω∗-submodule generated by y inΩ∗(X), (while Ω∗y means the Ω∗-free module generated by y.)

3. Image Res of the restriction

Let us write X = X⊗ k for the algebraically closure k of k. Let resΩ : Ω∗(X)→Ω∗(X) be the restriction map. Recall that grΩ∗(X) ∼= Ω∗ ⊗ CH∗(X) since X iscellular.

Let (yi) be a Z(p)-base of CH∗(X) with the degree |yi| ≤ |yi+1| so that

CH∗(X) ∼= ⊕iZ(p)yi, Ω∗(X) ∼= ⊕iΩ∗yi.

(Here Ayi is an A-free module generated by yi.) Define the filtration F i =Ω∗yj|j ≥ i ⊂ Ω∗(X) and the associated grade algebra

gr∗Ω(X) = ⊕iFi/F i+1 ∼= Ω∗ ⊗ CH∗(X).

(This filtration F i is finer than that in the previous section.)Les us write the image of the restriction map

ResΩ(X) = gr∗(Im(resΩ))

= ⊕i (Im(resΩ) ∩ F i)/(Im(resΩ) ∩ F i+1) ⊂ grΩ∗(X).

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ALGEBRAIC COBORDISM 7

Recall that In = (p = v0, v1, ..., vn−1) and I∞ = (v0, v1, ...) are the prime invari-ant ideals (under Landweber-Novikov operations) of BP ∗ = Ω∗.

Lemma 3.1. Let y ∈ Ω∗(X) be an Ω∗-module generator. If vny ∈ ResΩ(X), thenvjy ∈ ResΩ(X) for all 0 ≤ j ≤ n, namely,

In+1y ⊂ ResΩ(X) ⊂ grΩ∗(X).

Proof. Recall the Quillen (Landweber-Novikov) operation rα in the Ω∗(−) theory([Ha], [Ra], [Ya2,4,7]) so that

rpi∆n−i(vn) = vi mod(I2i ).

Let α = pi∆n−i. Then by the Cartan formula, we have

rα(vnx) = rα(vn) · x+∑

α′+α′′=α, |α′′|>0

rα′(vn)rα′′ (x).

Here rα′′ (x) ∈ F |rα′′ (x)| ⊂ F |x|+2. Hence in grΩ∗(X) we get rα(vnx) = (vi + ai)xwith ai ∈ I2i . When i = 0, 1, it is immediate we can take ai = 0. By induction oni, we see if vnx ∈ ResΩ(X) ⊂ grΩ∗(X), then so is vix for 0 ≤ i ≤ n.

Definition. For a smooth X and an Ω∗-module generator y ∈ Ω∗(X) withy 6∈ ResΩ(X), we say that y has v-degree n

degv(y) = n if vn−1y ∈ ResΩ(X) but vny 6∈ ResΩ(X).

When y ∈ ResΩ(X), let degv(y) = 0 and when py 6∈ ResCH let degv(y) = −1.

Lemma 3.2. If M is an irreducible Ω∗-module, then M is contained in one irre-ducible motive in Ω∗(X).

Proof. Each map from Ω∗(X) to Ω∗(Y ) of motives is represented by a compositionmap f∗g

∗ of the induced map g∗ (of smooth varieties) and the Gysin map f∗. Theboth maps are Ω∗ maps. Hence a decomposition of an Ω∗-motive gives that of anΩ∗-module.

Hereafter in this section, we assume the following condition for y and X suchthat we can define degv(y).

(1) Ω∗(X) is an Ω∗-free module.(2) y ∈ Ω∗(X) with 0 6= y ∈ CH∗(X)/p.

Lemma 3.3. If degvy = n > 0, then there are ci ∈ CH∗(X) for 0 ≤ i ≤ n − 1with |ci| = |y|−2(pi−1) such that the following Mn(y)

′ is contained (as a quotient)in CH∗(X),

CH∗(X) ։ Mn(y)′ = Z(p)c0 ⊕ Z/pc1, ...., cn−1.

Proof. We recall that

CH∗(X) ∼= Ω∗(X)⊗Ω∗ Z(p)∼= Ω∗(X)/(Ω<0Ω∗(X))

where Ω<0 = BP<0 = (v1, ..., vn, ...) is the ideal of Ω∗ generated by negativedegree elements. The assumption degv(y) = n implies Iny ⊂ Ω∗y ∩ResΩ(X).Moreover since vjy 6∈ ResΩ(X) for all j ≥ n, we see

Iny = Ω∗y ∩ResΩ(X) mod ((BP<0)2Ω∗y).

Here we note

In/((BP<0) · In + (BP<0)2) ∼= (p, ..., vn−1)/((BP<0) · (p, ..., vn−1))

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8 N.YAGITA

∼= Z(p)p ⊕ Z/pv1, ..., vn−1 ∼= Mn(y)′.

Here pci = 0 for i 6= 0, since p · vi = vi · p ∈ BP<0In. Hence we see

CH∗(X)(p)։(ResΩ(X) ∩ Ω∗y)/(BP<0ResΩ(X)) ∩ Ω∗y) ∼= Mn(y)′.

Here we use the projection grΩ∗(X)։Ω∗y since grΩ∗(X) is Ω∗-free.

Lemma 3.4. Let M be an irreducible motive in the motive M(X) of smooth X,and y ∈ CH∗(M). If degv(y) = n > 0, then Mn(y)

′ is contained (as a quotient )in CH∗(M).

Recall Mn(y) = Mn(y)′/p is defined for n > 0. Moreover, let M0(y)

′ = Z(p)yand M−1(y)

′ = 0 (when py 6∈ ResΩ(X)).

Corollary 3.5. Let CH∗(X) ∼= ⊕iZ(p)yi. Then CH∗(X)/p has the quotientmodule P1(X) such that

CH∗(X)/p ։ P1(X) = ⊕iMdi(yi) where di = degv(yi).

The surjection of the above corollary is an isomorphic when X is the Rost motivedefined by the norm variety. But it is far from an isomorphism for general cases.

Let us write

PN1 (X) = ⊕|yi|≤NMdi(yi).

Lemma 3.6. For n ≥ 0, let f : X → Y be a map of smooth varieties such that

f∗ : CH∗(Y )/p→ CH∗(X)/p is injective for ∗ ≤ N,

and that degv(f∗y) ≥ 1 for all 0 6= y ∈ CH∗(Y )/p such that degv(y) ≥ 1 and

|y| ≤ N . Then degv(y) ≤ degv(f∗(y)) and

f∗ : PN1 (Y )→ P ∗

1 (X) is injective.

Proof. If vjyi ∈ ResΩ∗ , then f∗(vjyi) = vjf∗(yi) ∈ ResΩ∗ . Hence degv(yi) ≤

degv(f∗(yi)). For d ≤ d′, it is immediate Md ⊂Md′ by the definition.

4. Another Invariant ideals

In this section, we generalize arguments for the module Mn(y)′/p = Mn(y) in

the preceding section as stated in the introduction.Recall that ResΩ(X) = gr∗(Im(resΩ)) ⊂ gr∗(Ω∗(X)). An ideal J ⊂ BP ∗ is

called invariant if r(J) ⊂ J for all (Landweber-Novikov) cohomology operations r inΩ∗(X) theory. By using the Cartan formula for the Landweber-Novikov operationas the proof of Lemma 3.1, we see

Lemma 4.1. Let (yi) be a Z(p)-base of CH∗(X), i.e., CH∗(X) ∼= ⊕iZ(p)yi.Then there are invariant ideals J(yi) in Ω∗ such that

ResΩ(X) ∼= ⊕iJ(yi)yi ⊂ Ω∗ ⊗ CH∗(X).

Proof. Let us write J(yi) = (ai1 , ..., ais) for aij ∈ Ω∗. Then using the Cartanformula as the proof of Lemma 3.1, we have

rα(aijy) = rα(aij)y +∑

|α′′|>0

rα′(aij)rα′′(y) in F |y| ⊂ Ω∗(X),

which is rα(aij)y in F |y|/F |y|+1 ⊂ grΩ∗(X). Hence if aij (y) ∈ ResΩ(X), then so isrα(aij)(y).

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ALGEBRAIC COBORDISM 9

Hence we see J(yi) is an invariant ideal beecause the Quillen operations rαgenerate the Landwber-Novikov operations, and all stable BP ∗-module operationsin ABP ∗,∗′

(X) theory [Ya2,4].

In the above lemma, let us write J(yi) = (ai1 , ..., ais). Then aij (yi) is a Ω∗-module generator in ResΩ(X). Hence there is a non zero element cij in Ω∗(X)⊗Ω∗

Z(p)∼= CH∗(X) such that resΩ(cij ) = aijyi in grΩ∗(X).

Corollary 4.2. For a base (yi) of CH∗(X) and the invariant ideals J(yi) =(ai1 , ..., ais), there is a surjective map q : CH∗(X)/p ։ Q(X) such that

Q(X) ∼= ⊕iC(yi) with C(yi) = Z/pci1 , ..., cis

where the degree is given |aijyi| = |cij |.

Conjecture 4.3. The above map q is isomorphic.

Example. When J(y) = In = (p, ...vn−1), we see

C(y) = Z/pc0, ..., cn−1 ∼= Mn(y) with |ci| = |viy|.

Of course there are many other invariant ideals. For examples the ideal J =(p2, pv1, v

21) is invariant but J

′ = (p2, v21) is not for p 6= 2. In fact,

r∆1(v21) = 2pv1, r2∆1

(v21) = p2

where we used r∆1(v1) = p and rα(v1) = 0 mod(I2∞) for α 6= ∆1. (See [Ha] for

detail examples of invariant ideals.) If J = J(y), then

C(y) ∼= Z/2c0, c1, c2 with |c0| = |y|, |c1| = |v1y|, |c2| = |v21y|.

However, it seems difficult to compute Q(X) directly. Hence we consider moreeasy version. The prime invariant ideals are written as Ω∗ or In = (p, v1, ...., vn−1)for 0 ≤ n ≤ ∞. We consider modulo (In+1

∞ ) version

ResΩ,n = ResΩ(X)/(ResΩ(X) ∩ In+1∞ grΩ∗(X))

⊂ (gr∗Ω∗(X))/In+1∞∼= Ω∗/In+1

∞ ⊗ CH∗(X),

and let us write by Pn(X) the corresponding Q(X) so that there are sujective maps

CH∗(X)/p։Q(X) = P∞(X)։...։P2(X)։P1(X).

Hence from Corollary 3.5, we see P1(X) ∼= ⊕iMdeg(yi)(yi).

If J(y) = (p2, pv1, v21), then C(y) is contained in P2(X). In particular, if J(y) =

(pn), then C(y) = Z/pc0 is contained in Pm(X) for m ≥ n.

5. relation to K-theories

Recall the (topological) Morava K-theory

K(n)∗(X) = BP (p, ..., vn, ...)∗(X)[v−1

n ]

is the generalized cohomology theory with the coefficient ringK(n)∗ = Z/p[vn, v−1n ].

Note that when n = 1, this MoravaK-theory is essentially isomorphic to the usualmod(p) K-theory. Hence when we want to know mod(p) K-theory K∗(X ;Z/p), weonly need to know this Morava K-theory K(1)∗(X).

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10 N.YAGITA

For an algebraic space X , we consider the integral algebraic Morava K-theoryAK(n)∗,∗

(X) and its connected theory Ak(n)∗,∗′

(X) with the coefficeint rings

Ak(n)2∗,∗ ∼= Z(p)[vn], and AK(n)2∗,∗ ∼= Z(p)[vn.v−1n ] so that

Ak(n)∗,∗′

(X)[v−1n ] ∼= AK(n)∗,∗

(X).

We consider the AHss given in Section 2

(5.1) E∗,∗′,∗′′

2 (X) ∼= H∗,∗′

(X)⊗ K(n)∗′′

=⇒ AK(n)∗,∗′

(X).

The associated graded ring is defined as the infinite term ([Ya5,6])

gr(n)∗geo(X) = E2∗,∗,0∞ (X).

Lemma 5.1. We have CH∗(X)/I(n) ∼= gr(n)∗(X), where I(n) is the ideal gener-

ated by vn-torsions in E2∗,∗,0∞ for Ak(n)∗,∗

(X)

Proof. Recall CH∗(X) ∼= H2∗,∗(X) and dr(H2∗,∗(X)) = 0 for smooth X . By the

Ak(n)∗,∗′

(X) version of (5.1), we have

grAk2∗,∗(X) ∼= E2∗,∗,∗′

∞∼= (k(n)2∗,∗ ⊗ CH∗(X))/Im(d).

Since k(n)2∗,∗ ∼= Z(p)[vn], an element in Im(d) is written as dr(x) = vsnx′ for s ≥ 1

and x′ ∈ CH∗(X).

In the AHss for Ak(n)∗,∗′

(X), the element x′ ∈ E2∗,∗,0∞ is a (higher) vn-torsion

element, indeed, vsnx′ = 0. The AHss (5.1) is the localization by vn of the AHss for

Ak(n)∗,∗′

(X). So x′ = 0 ∈ E2∗,∗,0∞ in the AHss converging AK∗,∗′

(X) (5.1).

Recall the definition of Q(X), Pi(X) in the introduction (or the preceding sec-

tion). We consider their k(n) version. Let k(n)∗ = k(n)2∗,∗ ∼= Z(p)[vn]. Let uswrite

J ′(yi) = J(yi)⊗Ω∗ k(n)∗ ∼= J(yi)⊗BP∗ Z(p)[vn]

(while it is not invariant ideal in general). For example, I ′m = (p, vn) for m > n.

When J ′(yi) = (a′i1 , ..., a′ir ) ⊂ k(n)∗, we take c′i1 , ..., c

′ir in CH∗(X)/p. Then

define

P (n)∞(X) = ⊕iZ/pc′i1 , ..., c

′ir.

Similarly we can define P (n)j(X) by considering J ′(yi)/Ij+1∞ so that

CH∗(X)/p։P (n)∞(X)։...։P (n)2(X)։P (n)1(X).

Lemma 5.2. The first surjection of the above is decomposed

CH∗(X)/p ։ gr(n)∗geo(X)/p ։ P (n)∞(X).

Proof. We only need to show I(n) = 0 in P (n)∞(X). Let vsnx = 0 in E2∗,∗,∗′′

∞ for

x ∈ E2∗,∗,0∞ in the AHss converging to Ak(n)∗,∗

(X). Then we have

vsnx = vrnx′ in Ak(n)2∗,∗(X) for s < r.

Hence resk(n)(vs1(x − vr−s

1 x′)) = 0. Since Ak(n)2∗,∗(X) ∼= k(n)∗ ⊗ CH∗(X)

which is k(n)∗-free. Hence x = vr−sn x′ which is not a generator of Resk(n), and

this means x = 0 in P (n)∞(X).

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ALGEBRAIC COBORDISM 11

For ease of notations (and we only consider algebraic spaces), let us write(throughout this section)

K(n)∗(X) = AK(n)2∗,∗(X), h∗(X) = Ak(n)2∗,∗(X) h∗ = Z(p)[vn].

From (2.4), we see h∗(X)⊗h∗ Z(p)∼= CH∗(X). From Lemma 5.1, we see

gr(n)∗(X)/p ∼= CH∗(X)/(p, I(n))

∼= (h∗(X)⊗h∗ Z/p)/I(n) ∼= h∗(X)/(I(n), I∞).

Hence, for a ∈ hm(X), the following conditions are equivalent

a = 0 ∈ gr(n)m(X)/p ⇐⇒ a ∈ (I∞, I(n))h∗(X).

Lemma 5.3. Let a 6= 0 ∈ K(n)0(X). Then there is s ≥ 0 and a′ ∈ h∗(X) suchthat

a = vs1a′ and a′ 6∈ Im(I∞h∗(X)).

Hence 0 6= a′ ∈ gr(n)2s(X).

Proof. Suppose that a 6= 0 ∈ K(n)0(X). Since K(n)∗(X) ∼= h∗(X)[v−1n ], there is s

such that

a = vsna′ for some a′ ∈ h2s(X).

Let us take the largest such s. Then a′ is a h∗/p = Z/p[vn] module generator ofh∗(X)/p and |a| = 0 but |a′| ≥ 0. So s must be nonnegative.

Lemma 5.4. Suppose K(n)0(X) ∼= K(n)0(X). Then we have

gr(n)∗(X)/p ∼= P (n)∞(X).

Proof. Let 0 6= x ∈ gr(n)∗(X) ∼= h∗(X)/(I∞, I(n)). We consider the restrictionresh : h∗(X)→ h∗(X).

Suppose that resh(x) ∈ I∞res(h∗(X)) (note I(n) = 0 ∈ h∗(X), since it isvn-free). Then take x′ ∈ h∗(X) such that

resh(x − vsnx′) ∈ F |x|+1 ⊂ h∗(X).

Continuing this argument we see res(x − a) = 0 with a ∈ I∞h∗(X).Since K0(X) ∼= K0(X), the restriction resK is isomorphic. Hence we see x = a

in h∗(X)/(I(n)). This means x ∈ I∞h∗(X) and x = 0 ∈ gr(n)∗(X), and this is acontradiction. Thus we can take x so that res(x) is a generator of Resh(X). Thisimplies x 6= 0 in P (n)∞(X) and from Lemma 5.2, we have this lemma.

6. Motivic cohomology of the norm variety

Let X be a smooth (quasi projective) variety. Recall that H∗,∗′

(X ;Z/p) is themod(p) motivic cohomology defined by Voevodsky and Suslin. Rost and Voevodskysolved the Beilinson-Lichtenbaum (Bloch-Kato) conjecture ([Vo2.4], [Su-Jo], [Ro2])

H∗,∗′

(X ;Z/p) ∼= H∗et(X ;µ⊗∗′

p ) for ∗ ≤ ∗′.

In this paper, we always assume that k has a primitive p-th root of unity (soµp∼= Z/p). Then we have the isomorphism Hm

et (X ;µ⊗np ) ∼= Hm

et (X ;Z/p). Let

τ be a generator of H0,1(Spec(k);Z/p) ∼= Z/p, so that colimiτiH∗,∗′

(X ;Z/p) ∼=H∗

et(X ;Z/p).

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12 N.YAGITA

Let χX be the Cech complex, which is defined as (χX)n = Xn+1 (see details[Vo1,2,4]). Let χX be defined as a cofiber

χX → χX → Spec(k)

in the stable A1-homotopy category. Voevodsky defined the motivic cohomologyH∗,∗′

(χ;Z/p) for all object χ in the stable A1-homotopy category. Morover, he

showed that there exists the Milnor operation in the motivic cohomology ([Vo1,3])

Qi : H∗,∗′

(χ;Z/p)→ H∗+2pi−1,∗′+pi−1(χ;Z/p)

which is compatible with the usual Milnor operation on H∗(tC(χ);Z/p) for therealization map tC. (Here the topological operation is defined Q0 = β Bockstein

operation and Qi+1 = QiPpi

− P pi

Qi.) This operation Qi can be extended on

H∗,∗′

(M ;Z/p) for a motive M in M(X) (Lemma 7.1 in [Ya4]).

For 0 6= x ∈ H∗,∗′

(X ;Z/p) (or cohomology operation), let us write ∗ = |x|, w(x) =2 ∗′−∗, d(x) = ∗− ∗′ so that 0 ≤ d(x) ≤ dim(X) and w(x) ≥ 0 when X is smooth.We also note

|τ | = 0, w(τ) = 2, |Qi| = 2pi − 1, w(Qi) = −1.

Let fX ∈ CHdim(X)(X) be the fundamental class of X. Suppose degv(fX) = n.It is known that the ideal

I(X) = π∗(X) ⊂ BP ∗(pt.) = BP ∗ for π : X → pt.

is an invariant ideal (e.g. Lemma 5.3 in [Ya4]). Since π∗(fX) = 1, we getπ∗(vn−1fX) = vn−1, and this means I(X) ⊃ In. Thus we have

Lemma 6.1. Let f ∈ CH∗(X) be a fundamental class of X. If degv(f) = n, thenIn ⊂ I(X).

Note that when X = V is the norm variety so that tC(V ) = vn, the idealI(V ) = In+1, infact, π∗(1) = vn. But degv(fV ) = n e.g., vnfV 6∈ ResΩ(X). (SeeCorollary 6.8 below.)

The following two lemmas are known.

Lemma 6.2. (Lemma 6.4,6.5 in [Ya4]) If In ⊂ I(X), then ABP 2∗,∗(χX) is In-

torsion, and H∗,∗′

(χX ;Z/p) is Λ(Q0, ..., Qn−1)-free.

Lemma 6.3. (Corollary 3.4 in [Ya]) Let x ∈ CH∗(X) ∼= E2∗,∗,0∞ and vsx = 0 in

E2∗,∗,∗′

∞ for the AHss converging to ABP ∗,∗′

(X). Then there is b ∈ H∗,∗′

(X ;Z/p)with Qs(b) = x and is a relation in ABP 2∗,∗(X)

vsx+ vs+1xs+1 + ...+ vkxk + ... = 0 mod(I2∞)

with xk = Qk(b) in H2∗,∗(X ;Z/p) for all k ≥ s.

We consider the following assumption for an existence of the motive M in X .

Assumption 6.4. There is a motive M in M(X) such that

(1) there is y ∈ CH∗(M) with degv(y) ≥ 1,

(2) H∗,∗′

(M ;Z/p) ∼= H∗,∗′

(χX ;Z/p) for 0 < ∗′ < ∗ ≤ |y|.

(3) the cohomology H∗,∗′

(χX ;Z/p) for 0 < ∗′ < ∗ ≤ |y| is generated by onlyone element a′ as the K∗

M (k)⊗ Λ(Q0, ..., Qn−1)-modules.

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ALGEBRAIC COBORDISM 13

Theorem 6.5. Assume degv(y) = n and (1)− (3) as above. Then we have

ci(y) = Q0...Qi...Qn−1(a′).

Moreover CH∗(M)/p ∼= C(y) ∼= Mn(y) for 0 < ∗ ≤ |y|.

Proof. Since degv(y) = n, there is cn−1(y) ∈ H2∗,∗(M ;Z/p) which correspondto vn−1y. Since |Qicn−1| ≤ |y| for i ≤ n − 2, we see Qi(cn−1) = 0 from (2)

since w(Qicn−1) = −1. Since I(X) ⊃ In, H∗,∗′

(χX ;Z/p) is Λ(Q0, ..., Qn−1)-freefrom Lemma 4.5. Therefore cn−1(y) is in the Qi-image. Hence there is a′′ ∈

H∗,∗′

(χX ;Z/p) such that

Q0...Qn−2(a′′) = cn−1(y).

Hence a′′ = a′ or a′′ = Qn−1a′. If the second case holds, then cn−1(y) =

Q0...Qn−1(a′), and there can not exist ci(y) for i < n− 1, and a contradiction. So

a′′ = a′.Since cn−1(y) ∈ Im(Q0), it is a p-torsion in CH∗(X), and this means

pcn−1(y) + v1c′n−2 + ...+ vn−1c

′0 = 0 in Ω∗(X)/I2∞

for some c′i ∈ Ω∗(X). In particular, we see that

resΩ(vn−1c′0) = pvn−1y ∈ Ω∗(X)/(resΩ(I

2∞)) ∼= Ω∗(X)/I3∞

from resΩ(pcn−1(y)) = pvn−1y. (Note for i > 0, resΩ(c′i) = 0 mod(I2∞) by dimen-

sional reason.) So resΩ(c′0) = py. Thus we can take c0(y) = −c

′0. (Note that ci(y)

are only decided in Ω∗(X) modulo (Ker(resΩ), I∞).)

Then from Lemma 6.5, there is b ∈ H∗,∗′

(X ;Z/p) such that

Q0(b) = cn−1(y), Qn−1(b) = c0(y).

From (3), this b is determined uniquely, that is b = Q1...Qn−2(a′). Hence we have

c0(y) = Qn−1(b) = Qn−1Qn−2...Q1(a′).

We also get ci(y) = Q0...Qi...Qn−1(a′) by the same arguments, using the follow-

ing relation

vicn−1(y) = −vn−1ci(y) in Ω∗(X)/(I2∞) for 0 < i < n− 1.

Now we recall the norm variety. Given a pure symbol a in the mod p MilnorK-theory KM

n+1(k)/p, by Rost, we can construct the norm variety Va such that

π∗([Va]) = vn, a|k(Va) = 0 ∈ KMn+1(k(Va))/p

where [Va] = 1 ∈ Ω0(Va), π : Va → pt. is the projection and k(Va) is the functionfield of Va over k. Note I(Va) = In+1. Rost and Voevodsky showed that there isy ∈ CHbn(Va) for bn = (pn − 1)/(p− 1) such that yp−1 is the fundamental class ofVa, that is

Z(p)yp−1 ∼= CHpn−1(Va) ∼= CH0(Va) ∼= Z(p).

Then deg(Va) = p and yp−1 6∈ ResCH , but pyp−1 ∈ ResCH. Moreover the samefacts hold for yi with 1 ≤ i ≤ p − 1. Hence each yi satisfies (1),(2) (and somemodified (3)) in Assumption 6.4, in particular degv(y

i) > 0. More strongly, Rostand Voevodsky show the following theorem.

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14 N.YAGITA

Theorem 6.6. ([Ro1,2], [Vo1,4]) For a nonzero symbol a ∈ KMn+1(k)/p, let M(Va)

be the motive of the norm variety Va. Then there is an irreducible motive Ra (writeRn also) in M(Va) such that CH∗(Ra) ∼= Z(p)[y]/(y

p) and

CH∗(Ra) ∼= Z(p) ⊕Mn(y)′[y]/(yp−1).

We can also prove

Theorem 6.7. ([Vi-Ya], [Ya6]) The map resΩ : Ω∗(Ra) → Ω∗(Ra) is injectiveand

ResΩ(X) = Ω∗1 ⊕ Iny, ..., yp−1 ⊂ Ω∗[y]/(yp) ∼= Ω∗(Rn).

Corollary 6.8. For the Rost motive Rn and 1 ≤ i ≤ p− 1, we have degv(yi) = n

and CH∗(Rn)/p ∼= P1(Rn).

By Rost and Voevodsky, it is known that ([Ro1,2], [Vo1-3], [Su-Jo])

H∗,∗′

(χa;Z/p) ∼= H∗,∗′

(Ra;Z/p)

for all ∗ ≤ 2bn, ∗′ ≤ ∗, and there is a short exact sequence [Or-Vi-Vo]

0→ H∗+1,∗(χa;Z/p)τ→ KM

∗+1(k)/p→ KM∗+1(k(Va))/p

where we identify H∗+1,∗+1(χa;Z/p) ∼= KM∗+1(k)/p. Hence there exists a′ =

τ−1(a) ∈ H∗+1,∗(Ra;Z/p).We can define y ∈ CH∗(Ra) by

y = p−1resΩ(Q1...Qn−1(a′)) in grΩ∗(Rn) ∼= Ω∗ ⊗ CH∗(Rn).

We can prove Theorem 6.6 by using Theorem 6.5, 6.7, while to show Assumption6.4 is rather difficult, and we still need Voevodsky’ s original proof ([Vo1,2,4]).

7. Dickson invariant

The contents in this section are not used in other sections but arguments arevery similar to those given in the preceding section. Throughout this section, weassume that k contains ξp2 a primitive p2-root of the unity. Since

Q0(τ) = ξp = 0 in k∗/(k∗)p ∼= H1,1(pt.;Z/p),

we see Qi commutes with τ , i.e., Qi(τ · z) = τ ·Qi(z) for z ∈ H∗,∗′

(X ;Z/p).We consider the case X = B(Z/p)n (while X is of course not cellular) with

H∗,∗′

(X ;Z/p) ∼= KM∗ (k)⊗ Z/p[τ, y1, ..., yn]⊗ Λ(x1, ..., xn)

for p odd prime. Then x = x1...xn is invariant under the SLn(Z/p)-action. Soeach

qi(x) = Q0...Qi...Qn(x) x = x1...xn

is SLn(Z/p)- invariant. The invariant ring on the polynomial ring is well knownby Dickson

Z/p[y1, ..., yn]GLn(Z/p) ∼= Z[cn,0, ...., cn,n−1],

Z/p[y1, ..., yn]SLn(Z/p) ∼= Z[en, cn,1, ...., cn,n−1],

where ep−1n = cn,0 and each cn,i is defined by

Πy∈Z/py1,...,yn(t+ y) = tpn

+∑

tpi

cn,i.

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ALGEBRAIC COBORDISM 15

These cn,i is also represented by Chern classes. Let reg : (Z/p)n → Upn be the(complex) regular representation. Then

cn,i = cpn−pi(reg) = reg∗(cpn−pi).

It is known by Mimura-Kameko [Mi-Kam]

encn,i = Q0...Qi...Qn(x1...xn).

Hence the element encn,i is just qi(x). Hence we get

qi(x) = cn,ien = reg∗(cpn−1)1/(p−1)reg∗(cpn−pi).

Each element a ∈ H1,1(X ;Z/p) ∼= H1et(X ;Z/p) can be represented by A1-

homotopy map X → BZ/p. So we can write

a = a1...an = i∗a(x1...xn) for ia : X → (BZ/p)n.

Theorem 7.1. Let p be odd. For an element a = a1...an ∈ Hn,n(X ;Z/p), we have

qi(a) = Q0...Qi...Qn(a)

= i∗a(encn,i) = i∗a((reg∗(cpn−1))

1/(p−1)reg∗(cpn−pi))

in H2∗,∗(X ;Z/p). Moreover we have pqi(x) = viq0(x) in BP ∗(X)/(I2∞, vj |j > i).

It is known ([Vo1,2]) that

H∗,∗′

(B(Z/2)n;Z/2) ∼= KM∗ (k)⊗ Z/2[τ, y1, ..., yn, x1, ..., xn]/(Re.)

where Re. = (x2i = τyi | 1 ≤ i ≤ n).

Giving grading by multiplying τ , we have

(∗) grH∗,∗′

(B(Z/2)n;Z/2) ∼= KM∗ (k)⊗ Z/2[τ, y1, ..., yn]⊗ Λ(x1, ..., xn).

The above theorem also holds for p = 2, considering this graded ringgrH∗,∗′

(B(Z/2)n;Z/2) instead of H∗,∗′

(B(Z/2)n;Z/2).Here, however, we give an another argument for p = 2, which is quite similar to

that in the previous section. Define dn,i ∈ Z/2[x1, ..., xn] by

Πx∈Z/2x1,...,xn(t+ x) = t2n

+∑

t2i

dn,i

so that d2n,i = cn,i and we have the isomorphism

Z/2[x1, ..., xn]GLn(Z/2) ∼= Z/2[dn,0, ..., dn,n−1].

Moreover for the real regular representation regR : (Z/2)n → BO(2n), we have

τ−2n−1+2i−1+1reg∗R(w2n−2i) = dn,i

where we identify the element dn,i with a homogeneous element of w(dn,i) = 2. Weknow from Lemma 5.7 in [Kam-Te-Ya]

dn,i = Q0...Qi...Qn−2(x1...xn) for i ≤ n− 2

where dn,i is considered in (∗) (see for details §5 in [Kam-Te-Ya]).

Theorem 7.2. Let a = a1...an ∈ Hn,n(X ;Z/2) and τa′ = a. Then we have

τ · qi(a′) = τ ·Q0...Qi...Qn−2(a

′) = Q0...Qi...Qn−2(a)

= i∗a(dn,i) = i∗a(τ−2n−1+2i−1+1reg∗R(w2n−2i)).

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16 N.YAGITA

Remark. Note that when a ∈ KM∗ (k)/2 ∼= H∗,∗(pt.;Z/2) and X = Rn, the

element qi(a′) = ci(y) in the previous section. However, the value of the above

equation is zero (indeed Qia = 0). Hence the above theorem may be seen anextension of the fact τci(y) = 0.

Remark. Smirnov and Vishik constructed the (generalized) Arason map en :In/In+1 → KM

∗ (k)/2 where I is the fundamental ideal of the Witt ring W ∗(k)generated by even dimensional quadratic forms. They used the Nisnevich classify-ing space BO(n)Nis. In this paper we write the etale classifying space by BO(n)etor simply BO(n). The motivic cohomology is given as (Theorem 3.1.1 in [Sm-Vi])

H∗,∗′

(BO(n)Nis;Z/2) ∼= H∗,∗′

(pt. : Z/2)⊗ Z/2[u1, ..., un]

where ui = τ−[(i+1)/2]wi identifying wi ∈ Hi,i(BO(n)et;Z/2). These elements arecalled as subtle characteristic classes in the paper [Sm-Vi].

Let Xq be the quadric for a quadratic form q, and χq = χXq . Smirnov-Vishikproved (Theorem 3.2.34, Remark 3.2.35 in [Sm-Vi]) that when q ∈ In, there is amap fq : χq → BO(N)Nis (with N = dim(q)), and the Arason map can be definedas (for 1 ≤ i ≤ n− 2)

en(q) = τ(Q0...Qi...Qn−2)−1f∗

q (u2n−2i).

We consider the following diagram (not assumed commutative) for ∗′′ = 2n − 2i

H∗,∗′

(χq;Z/2)f∗

q←−−−− H∗,∗′

(BO(N)Nis;Z/2)τ−∗

′′/2j∗

←−−−−−− H∗,∗′

(BO(N)et;Z/2)

y

y

H∗,∗′

(Xq : Z/2)i∗a←−−−− H∗,∗′

(B(Z/2)n)et;Z/2)τ−∗

′′/2+1j∗reg∗

←−−−−−−−−−− H∗,∗′

(BO(2n)et;Z/2)

where N ≥ max(dim(q), 2n), and j : BO(N)Nis → BO(N)et are natural inducedmaps, and i∗a is the induced map from a = a1...an ∈ Hn

et(Xq;Z/2). Hence we havetwo elements

u = f∗q (u∗′′) = f∗

q (τ−∗′′/2 · j∗(w∗′′ ))

v = i∗en(q)dn,i = i∗en(q)(τ−∗′′/2+1 · reg∗R(w∗′′ )).

Then we see τu = v, but it is only zero. Here there is x′ ∈ H1,0((BZ/2)Nis;Z/2)such that τx′ = x. hence τ−1dn.i exists in H∗(B(Z/2)nNis;Z/2). However we cannot see a good map Xq → B(Z/2)nNis.

8. quadrics and p = 2

In this section, we rewrite arguments of quadrics by using degv(y), Mn(y) andP1(X). Let h ∈ CH1(X) be the hyperplane section. Let X be a quadric ofdim(X) = 2ℓ− 1. (The case dim(X) = 2ℓ is given with some modification of oddcase.) From Rost and Toda-Watanabe ([Ro1], [Tod-Wa])

CH∗(X) ∼= Z(2)[h, y]/(hℓ = 2y, y2)

∼= Z(2)1, h, ..., hℓ−1 ⊕ Z(2)y, ..., h

ℓ−1y.

Hence h ∈ ResΩ(X) and degv(h) = 0.Throughout this section, we assume that X is anisotropic.

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ALGEBRAIC COBORDISM 17

This means degv(hℓ−1y) ≥ 1 since hℓ−1y is the fundamental class of the (2ℓ−1)-

dimensional manifold X . Thus we get

degv(hi) = 0, degv(h

iy) ≥ 1 for all 0 ≤ i ≤ ℓ− 1.

Lemma 8.1. Let X be an (anisotropic) quadric of dim(X) = (2ℓ − 1). Let uswrite di = degv(h

iy). Then d0 ≤ d1 ≤ ... ≤ dℓ−1 and

P1(X) ∼= ⊕0≤i≤ℓ−1(M0(hi)⊕Mdi(h

iy)).

Proof. If vnhiy ∈ ResΩ(X), then so is h · vnh

iy from h ∈ ResΩ(X). This impliesdi ≤ di+1. Since J(hiy) ∼= Idi , we have C(hiy) ∼= Mdi(h

iy).

We recall that

M0(hi)⊕Mdi(h

iy) ∼= Z/2hi, c0(hiy) ⊕ Z/2c1(h

iy), ..., cdi−1(hiy).

We note c0(hiy) = 2hiy = hℓ+i. Hence we can write

(∗) P1(X) ∼= Z/2[h]/(h2ℓ)⊕⊕0≤i≤ℓ−1Z/2c1(hiy), ..., cdi−1(h

iy).

Note that the Rost motive is isomorphic to

CH∗−ki(Rdi)/2∼= Z/2hki, c0(h

iy) ⊕ Z/2c1(hiy), ..., cdi−1(h

iy)

where ki = 1/2(|hiy| − |Rdi |) = i+ (2ℓ − 2di). Therefore we have

Corollary 8.2. If the motive M(X) is a sum of Rost motives (e.g., an excellentquadric), then CH∗(X)/2 ∼= P1(X) as above. In particular the set k0, ..., kℓ−1 =0, 1, ..., ℓ− 1.

By using (∗) as above, we can prove

Corollary 8.3. (Theorem 4.7 in [Ya3]) The Chow ring CH∗(X)/2 has the quotientring

P1(X) ∼= Z/2[h]/(h2ℓ)⊕sj=0 Z/2[h]/(h

ℓ−fj)uj

where fj is the smallest f such that vjhfy ∈ ResΩ(X) (i.e., degv(h

fy) = j + 1),and uj = cj(h

fjy) = vjhfjy, and so s = degv(h

ℓ−1y)− 1.

Proof. The result follows from changing order of sums such that

ResΩ(X) ⊃ ⊕Idihiy = ⊕i,j(vjh

iy) = ⊕hi−fj (vjhfj ) = ⊕(hi−fjuj).

(For details see the proof of Theorm 4.7 in [Ya3].)

Let a = a0...an 6= 0 ∈ KMn+1(k)/2 and φn+1 = 〈〈a0, ..., an〉〉 be the (n + 1)-th

Pfister form associated to a. Let q′ is the subform of codimension 1, that is, q′ isthe maximal neighbor of the (n+ 1)-th Pfister form φn+1.

Example 1. ([Ya3]) Let X = Xq′ be the maximal neighbor of the (n+1) Pfisterquadric (ℓ = 2n − 1). Then degv(yh

i) = n for all 0 ≤ i ≤ ℓ− 1. Hence we have thering isomorphism

CH∗(Xq′)/2 ∼= Z[h]/(h2n+1−2)⊕ Z/2[h]/(h2n−1)u1, ..., un−1

where ui = ci(y) = viy ∈ Ω∗(X) so that uiuj = 0.Example 2. ([Ya3]) Let Xq′′ be the minimal neighbor of the (n + 1) Pfis-

ter quadric (ℓ = 2n−1), namely, the norm variety. Then degv(vhℓ−1) = n and

degv(yhi) = n− 1 for all 0 ≤ i ≤ ℓ− 2. Hence we have the ring isomorphisms

CH∗(Xq′′ )/2 ∼= Z/2[h]/(h2n)⊕ Z/2[h]/(h2n−1

)u′1, ..., u

′n−2 ⊕ Z/2u′

n−1

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18 N.YAGITA

where u′i = ci(y) for 1 ≤ i ≤ n− 2 and u′

n−1 = cn−1(h2n−1−1y).

Let f : X → Y be a map of anisotropic quadrics. Suppose f∗(hY ) 6= 0 ∈CH2(X)/2 (i.e. f∗(hY ) = hX). Let yY be the ring generator of CH∗(Y ) suchthat 2yY = hℓY for ℓ = 1/2(dim(Y ) − 1). Similarly define the ring generator yXin CH∗(X)/2. Then f∗(yY ) = hℓY −ℓXyX (from 2yX = hℓX ), and so dim(X) ≤dim(Y ). Thus we see

(1) f∗ : CH∗(Y )→ CH∗(X) is injective for ∗ ≤ dim(X).(2) each hjyX 6∈ ResCH for 0 ≤ j ≤ ℓX − 1.Recall PN

1 (X) = ⊕|yi|≤NMdi(yi). From Lemma 3.6, we have

Theorem 8.4. Let f : X → Y be a map of anisotropic quadrics such that f∗ :CH2(Y )/2→ CH2(X)/2 is non zero. Then the induced map

Pdim(X)1 (Y )→ P1(X) is injective.

Example 3. Let Xmin (resp. Xmax) be the minimal (resp. maximal) neighborof the (n + 1)-Pfister quadric and e : Xmin ⊂ Xmax be the embedding. Notedim(Xmin) = 2n − 1 and we see

P 2n−11 (Xmax) ∼= Z[h]/(h2n)⊕ Z/2u1, ..., un−1

from Example 1. Hence we see that it injects into P1(Xmin), by e∗(ui) = h2n−1−1u′i

for i ≤ n− 2 and e∗(un−1) = u′n−1.

Corollary 8.5. Let Y be an anisotropic quadric and X = Y ∩ H2d for 2d-dimensional plane H2d, and e : X ⊂ Y be the induced embedding. Then theGysin map e∗ : P1(X) → P1(Y ) is injective. Let us write by fj(X) (resp. fj(Y ))the smallest number f such that vjh

fyX ∈ ResΩ(X) (resp. vjhfyY ∈ ResΩ(Y )).

Then we have

fj(Y )− d ≤ fj(X) ≤ fj(Y ) + d.

Proof. First note that e∗(yY ) = hdyX . Note e∗(1) = h2d and e∗(yX) = hdyY . Letf = fj(Y ). Then vjh

fyY ∈ ResΩ(Y ) and hence

e∗(vjhfyY ) = vjh

f+dyX ∈ ResΩ(X).

This means fj(X) ≤ fj(Y ) + d. If vjhgyX ∈ ResΩ(X) then e∗(vjh

gyX) =vjh

g+dyY ∈ ResΩ(Y ). This fact implies fj(Y ) ≤ fj(X) + d.

9. Lie groups G and the flag manifolds G/T

Let Y be a simply connected H-space of finite type (e.g. compact Lie group).By Borel, its mod p-cohomology is (for p odd)

H∗(Y ;Z/p) ∼= P (y)/p⊗ Λ(x1, ..., xℓ), P (y) = ⊗siZ(p)[yi]/(y

pri

i )

where |yi| is even and |xj | is odd. When p = 2, a graded ring grH∗(Y ;Z/2) isisomorphic to the right hand side ring, e.g. x2

j = yij for some yij .For ease of arguments, we assume p odd in this section (however the similar

results also hold for p=2). Let us write by QH∗(−) the indecomposable module

QH∗(Y ;Z/p) = H∗(Y ;Z/p)/(H+(Y ;Z/p))2 ∼= Z/py1, ..., ys, x1, ..., xℓ.

Then it is known by Lin, Kane [Ka]

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ALGEBRAIC COBORDISM 19

Lemma 9.1. (J.Lin, §35 in [Ka]) Let p be an odd prime and Y be a finite simplyconnected H-space. Then

QevenH∗(Y ;Z/p) =∑

n≥0

βPnQoddH∗(Y ;Z/p).

For connected but non simply connected space, the above theorem holds for∗ ≥ 4. Moreover

Lemma 9.2. (§36 in [Ka]) By the same assumption as the preceding lemma, wesee

Q2nH∗(Y ;Z/p) = 0 unless n = pk + ...+ pi + ...+ 1.

For example, the possibility of even degree of ring generators are

(p+ 1), (p+ 1, p2 + 1), (p2 + p+ 1, p3 + p+ 1, p3 + p2 + 1), ...

Moreover, it is known that each generator is combined by reduced power operations.The first degree type |y| = 2(p + 1) appears for p = 3, exceptional Lie groups

F4, E6, E7, and for p = 5, E8 (and for p = 2, G2, F4, E6). The second type appearsfor p = 3, E8. However it seems that there is no example for other types for p odd.

So we may assume the first or second types above. Let us say that (simplyconnected) simple Lie groups are of type (I) and (II) respectively. When type (I),there are x1, x2, y in H∗(Y ;Z/p) such that

(∗) x2 = P 1(x1), y = Q1(x1) = Q0(x2)

with |x1| = 3, |x2| = 2p + 1 and |y| = 2p + 2. The existence of the above y alsoholds for p = 2. For type (II), there are more elements x3, x4 and y′ such that

(∗∗) y′ = P p(y) = Q2(x1) = Q1(x3) = Q0(x4),

where |y′| = 2(p2 + 1), |x3| = 2p2 − 2p + 3, |x4| = 2p2 + 1. For p = 2, the othertypes appear but we see ;

Lemma 9.3. Let G be a simply connected Lie group such that H∗(G;Z) has p-torsion. Then there are x1, x2, y satisfying (∗).

Next we recall the cohomology of flag manifolds G/T . Let T be the maximaltorus of a simply connected compact Lie group G and BT the classifying space ofT . We consider the fibering

(8.1) Gπ→ G/T

i→ BT

and the induced spectral sequence

E∗,∗2 = H∗(BT ;H∗(G;Z/p)) =⇒ H∗(G/T ;Z/p).

The cohomology of the classifying space of the torus is given by

H∗(BT ) ∼= S(t) = Z[t1, ..., tℓ] with |ti| = 2.

where ℓ is also the number of the odd degree generators xi in H∗(G;Z/p). It isknown that yi are permanent cycles and that there is a regular sequence ([Tod],[Mi-Ni]) (b1, ..., bℓ) in H∗(BT )/(p) such that d|xi|+1(xi) = bi. Thus we get

E∗,∗′

∞∼= grH∗(G/T ;Z/p) ∼= P (y)/p⊗ S(t)/(b1, ..., bℓ).

Moreover we know that G/T is a manifold of torsion free, and

(8.2) H∗(G/T )(p) ∼= Z(p)[y1, .., yk]⊗ S(t)/(f1, ..., fk, b1, ..., bℓ)

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20 N.YAGITA

where bi = bi mod(p) and fi = ypri

i mod(t1, ..., tℓ). Since H∗(G/T ) is torsion free,we also know

(8.3) BP ∗(G/T ) ∼= BP ∗[y1, ..., yk]⊗ S(t)/(f1, ..., fk, b1, ..., bℓ)

where bi = bi mod(BP<0) and fi = fi mod(BP<0).Here we will study a relation between Qi-actions on H∗(G;Z/p) and vi-module

structure of BP ∗(G/T ). Recall that k(n)∗(X) is the connected Morava K-theorywith the coefficients ring k(n)∗ ∼= Z/p[vn] and ρ : k(n)∗(X) → H∗(X ;Z/p) isthe natural (Thom) map. Recall that there is an exact sequence (Sullivan exactsequence [Ra], [Ya2]) induced from (the topological version of) (2.1)

...→ k(n)∗+2(pn−1)(X)vn→ k(n)∗(X)

ρ→ H∗(X ;Z/p)

δ→ ...

Here it is known that ρ · δ(x) = Qn(x) the Milnor operation.We consider the Serre spectral sequence

E∗,∗′

2∼= H∗(B;H∗′

(F ;Z/p)) =⇒ H∗(E;Z/p),

induced from the fibering Fi→ E

π→ B with H∗(B) ∼= Heven(B).

Lemma 9.4. (Lemma 4.3 in [Ya1]) In the spectral sequence E∗,∗′

r above, supposethat there is x ∈ H∗(F ;Z/p) such that

(∗) y = Qn(x) 6= 0 and b = d|x|+1(x) 6= 0 ∈ E∗,0|x|+1.

Moreover suppose that E0,|x||x|+1

∼= Z/px ∼= Z/p. Then there are y′ ∈ k(n)∗(E) and

b′ ∈ k(n)∗(B) such that i∗(y′) = y, ρ(b′) = b and that in k(n)∗(E),

(∗∗) vny′ = λπ∗(b′) for λ 6= 0 ∈ Z/p.

Conversely if (∗∗) holds in k(n)∗(E) for y = i∗(y′) 6= 0 and b = ρ(b′) 6= 0, thenthere is x ∈ H∗(F ;Z/p) such that (∗) holds.

Corollary 9.5. In the spectral sequence converging to H∗(G/T ;Z/p), let b 6= 0 bethe transgression image of x, i.e. d|x|+1(x) = b. Then we have its lift b ∈ BP ∗(BT )

such that in BP ∗(G/T )/I2∞

b = py(0) + v1y(1) + ...+ viy(i) + ...

where y(i) ∈ H∗(G/T ;Z/p) with π∗y(i) = Qix.

10. versal flag varieties

Let Gk be the split reductive algebraic group corresponding to G, and Tk bethe split maximal torus corresponding to T . Let Bk be the Borel subgroup withTk ⊂ Bk. Note that Gk/Bk is cellular, and CH∗(Gk/Tk) ∼= CH∗(Gk/Bk). Hencewe have CH∗(Gk/Bk) ∼= H2∗(G/T ) and CH∗(BBk) ∼= H2∗(BT ).

Recall the algebraic cobordism Ω∗(X). There is a natural (realization) mapΩ∗(X) → BP ∗(X(C)). In particular, we have Ω∗(Gk/Bk) ∼= BP ∗(G/T ). RecallIn = (p, v1, ..., vn−1) and we also note

Ω∗(Gk/Bk)/I∞ ∼= CH∗(Gk/Bk)/p ∼= H∗(G/T )/p.

Let G be a nontrivial Gk-torsor. We can construct a twisted form of Gk/Bk by(G×Gk/Bk)/Gk

∼= G/Bk. We will study the twisted flag variety F = G/Bk.Let us consider an embedding of Gk into the general linear group GLN for some

large N . This makes GLN a Gk-torsor over the quotient variety S = GLN/Gk. Let

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ALGEBRAIC COBORDISM 21

F be the function field k(S) and define the versal Gk-torsor E to be the Gk-torsorover F given by the generic fiber of GLN → S. (For details, see [Ga-Me-Se], [To2],[Me-Ne-Za], [Ka1].)

The corresponding flag variety E/Bk is called the versal flag variety, whichis considered as the most complicated twisted flag variety (for given Gk). It isknown that the Chow ring CH∗(E/Bk) is not dependent to the choice of genericGk-torsors E (Remark 2.3 in [Ka1]).

Note. Hereafter in this paper, we always assume that G (and hence F) meansversal. Hence CH∗(F) means the Chow ring defined over the field k(S) above (butnot over k).

Karpenko proves the following result for a versal flag variety.

Theorem 10.1. (Karpenko Lemma 2.1 in [Ka1], [Me-Ne-Za]) Let h∗(−) be an ori-ented cohomology theory e.g., h∗(X) = CH∗(C),K∗(X),Ω∗(X). Then the naturalmap h∗(BBk)→ h∗(G/Bk) is surjective.

Corollary 10.2. The cohomology h∗(F) = h∗(G/Bk) is multiplicatively generatedby elements ti in S(t).

The versal case of the main result in Petrov-Semenov-Zainoulline [Pe-Se-Za] isgiven as

Theorem 10.3. (Theorem 5.13 in [Pe-Se-Za]) There is a p-localized motive R(G)such that the (p-localized) motive M(F) of the variety F is decomposed

M(F)(p) ∼= R(G)⊗ T with T = (⊕uT⊗u).

Here T⊗u are Tate motives with CH∗(T )/p ∼= S(t)/(p, b) = S(t)/(p, b1, ..., bℓ).Hence, we have additively

CH∗(F)/p ∼= CH∗(R(G))⊗ S(t)/(p, b).

For R(G) = R(G)⊗ k, we have CH∗(R(G))/p ∼= P (y)/p.

Hence we have surjections for the variety F

CH∗(BBk) ։ CH∗(F)pr.։ CH∗(R(G)).

We study in [Ya6] what elements in CH∗(BBk) generate CH∗(R(G)).For ease of notations, let us write A(b) = Z/p[b1, ..., bℓ]. By giving the filtration

on S(t) by bi, we can write grS(t)/p ∼= A(b)⊗S(t)/(b). In particular, we have maps

A(b)iA→ CH∗(F)/p→ CH∗(R(G))/p. We also see that the above composition map

is surjective (see also Lemma 10.5 below).

Lemma 10.4. Let pr : CH∗(F)/p→ CH∗(R(G))/p, and 0 6= x ∈ Ker(pr). Thenx =

b′t′ with b′ ∈ A(b), 0 6= t′ ∈ S(t)+/(p, b) i.e., |t′| > 0.

Let us write

ytop = Πsi=1y

pri−1i (resp. ttop)

the generator of the highest degree in P (y) (resp. S(t)/(b)) so that f = ytopttop isthe fundamental class in H2d(G/T ) for 2d = dimR(G/T ). For N > 0, let us write

A(b)N = Z/pbi1 ...bik ||bi1 |+ ...+ |bik | ≤ N ⊂ A(b).

Lemma 10.5. For N = |ytop|, the map A(b)N → CH∗(R(G))/p is surjective.

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22 N.YAGITA

Proof. In the preceding lemma, AN ⊗ t′ for |t′| > 0 maps zero in CH∗(R(G))/p.Since each element in S(t) is written by an element in AN ⊗ S(t)/(b), we have thelemma.

Corollary 10.6. If bi 6= 0 in CH∗(X)/p, then so in CH∗(R(Gk))/p.

Proof. Let pr(bi) = 0. From Lemma 10.4, bi =∑

b′t′ for |t′| > 0, and henceb′ ∈ Ideal(b1, ..., bi−1). This contradict to that (b1, ..., bℓ) is regular.

Now we consider the torsion index t(G). Let dimR(G/T ) = 2d. Then the torsionindex is defined as

t(G) = |H2d(G/T ;Z)/i∗H2d(BT ;Z)| where i : G/T → BT.

Let n(G) be the greatest common divisor of the degrees of all finite field extensionk′ of k such that G becomes trivial over k′. Then by Grothendieck [Gr], it isknown that n(G) divides t(G). Moreover, if G is versal, then n(G) = t(G) ([To2],[Me-Ne-Za], [Ka1]), which implies that each element in t(G)P (y) is represented byelement in CH∗(BBk).

It is well known that if H∗(G) has a p-torsion, then p divides the torsion in-dex t(G). Torsion index for simply connected compact Lie groups are completelydetermined by Totaro [To1], [To2].

Lemma 10.7. ([Ya6]) Let b = bi1 ...bik in S(t) such that in H∗(G/T )

b = ps(ytop +∑

yt), |t| > 0

for some y ∈ P (y) and t ∈ S(t)+. Then t(G)(p) ≤ ps.

From Lemma 9.5, we have

Lemma 10.8. Suppose that degv(y) > 0 for y ∈ P (y)/p and d|xj |+1(xj) 6= 0.(Hence dk(xj) = 0 for all k ≤ |xj |.) Then the degree degv(y) is the max of thenumber n+ 1 such that in H∗(G;Z/p)

Qn(xj) = y but Qk(xj) = 0 for 0 ≤ k < n.

Proof. From Lemma 9.5, we have Ω∗(R(G))/(I2∞)

d|xj |+1(xj) = bj = py(1) + ...+ vn−1y(n− 1) + vny(n) + ...

with π∗(y(i)) = Qi(xj).Suppose that the condition for Qi is satisfied. For k < n, we assumed Qk(x) = 0.

Since π∗(y(k)) = 0, we see y(k) =∑

yb for y ∈ P (y) and b ∈ A(b)+. So y(k) ∈I∞k(n)∗(G/T ) and vky(k) = 0 in mod(I2∞). Here we have π∗(y(n)) = y, and hencebj is written

(∗) vny + vn+1y(n+ 1) + ... ∈ ResΩ(X).

Hence degv(y) ≥ n+ 1.Conversely, suppose (∗) with mod(I2∞). Here we note that we can write b = bk

since bi ∈ I∞k(n)∗(G/T ). Then vny = b = 0 mod(I2∞) = mod(v2n) in k(n)∗(G/T ).So there is y′ ∈ k(n)∗(G) with vny

′ = 0 and y = y′ mod(vn). By the Sullivan exactsequence, there is x ∈ H∗(G;Z/p) such that Qn(x) = y and d|x|+1(x) = b.

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ALGEBRAIC COBORDISM 23

11. The orthogonal group SO(2ℓ+ 1) and p = 2

At first we consider the orthogonal groups G = SO(m) and p = 2 , while itis not simply connected. The mod(2)-cohomology is written as ( see for example[Mi-Tod], [Ni])

grH∗(SO(m);Z/2) ∼= Λ(x1, x2, ..., xm−1)

where |xi| = i, and the multiplications are given by x2s = x2s.

For ease of argument, we only consider the case m = 2ℓ + 1. Hereafter thissection, we assume (G, p) = (SO(2ℓ + 1), 2).

H∗(G;Z/2) ∼= P (y)⊗ Λ(x1, x3, ..., x2ℓ−1)

grP (y)/2 ∼= Λ(y2, ..., y2ℓ), letting y2i = x2i (hence y4i = y22i).

The Steenrod operation is given as Sqk(xi) =(

ik

)

(xi+k). The Qi-operations aregiven by Nishimoto [Ni]

Qnx2i−1 = y2i+2n+1−2, Qny2i = 0.

It is well known that the transgression bi = d2i(x2i−1) = ci is the i-th elementarysymmetric function on S(t). Moreover we see that Q0(x2i−1) = y2i in H∗(G;Z/2).From Corollary 8.3, we have

Corollary 11.1. In BP ∗(G/T )/I2∞, we have

ci = 2y2i +∑

n≥1

vny(2i+ 2n+1 − 2)

for some y(j) with π∗(y(j)) = yj.

We have c2i = 0 in CH∗(F)/2 from the natural inclusion SO(2ℓ+1)→ Sp(2ℓ+1)(see [Pe], [Ya6]) for the symplectic group Sp(2ℓ+ 1).

Let us write by ΛZ(a1, ..., as) the Z(p) -free module such that ΛZ(a1, ..., as)/p ∼=Λ(a1, ..., as). Then we have

Theorem 11.2. ([Pe], [Ya6]) Let (G, p) = (SO(2ℓ+ 1), 2). Then

CH∗(R(G)) ∼= ΛZ(c1, ..., cℓ).

Corollary 11.3. We have

J(y2i1 ...y2ir ) = (2r) ⊂ BP ∗ for 1 ≤ i1 < ... < ir ≤ ℓ.

Hence CH∗(R(G))/2 ∼= Pℓ(R(G)), and P1(R(G)) ∼= Z/21, c1, ..., cℓ.

Proof. Since 2y2i = ci mod(I2∞), we have

2ry2i1 ...y2ir = ci1 ...cir mod(Ir+1∞ )

which is in ResΩ(X). Hence C(y2i1 ...y2ir ) ⊃ Z/2ci1 ...cis.On the other hand, from the preceding theorem, we see

⊕(i1<...<ir)Z/2ci1...cir∼= CH∗(R(G))/2.

So we have Pℓ(R(G)) ∼= ⊕(i1<...<ir)C(y2i1 ...y2ir )∼= CH∗(R(G))/2.

The above example may not be so interesting since

degv(y2i) = 1, and degv(y2i1 ...y2ir ) = −1 for r ≥ 2.

However we consider an example degv(y) = n for each n, in an extension over thefield k, We study CH∗(X |K)/2 for some interesting extension K over k. Let K be

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24 N.YAGITA

an extension of k such that X does not split over K but splits over an extensionover K of degree 2a, (a, 2) = 1. Suppose that

(∗) y2ℓ 6∈ ResΩ(X |K) but y2i ∈ ResΩ(X |K) for 1 ≤ i ≤ ℓ− 1.

Lemma 11.4. Suppose (∗). Then ℓ = 2n − 1 for n > 0.

Proof. We can see that if ℓ 6= 2n − 1, then each y2ℓ is a target of the Steenrodoperation Sq2k of a sum of products of y2i for i < ℓ.

Lemma 11.5. Suppose (∗) and ℓ = 2n − 1. Then vn−1y2ℓ ∈ ResΩ(X |K), i.e.,degv(y2ℓ) = n for X |K.

Proof. From Corollary 11.1, we see

cℓ−2j+1 = 2y(2(ℓ− 2j + 20)) + v1y(2(ℓ− 2j + 21)) + ...+ vj(y(2ℓ))

= vj(y2ℓ) mod (y2, y4, ..., y2ℓ−2).

Hence we have resΩ(cℓ−(2 j−1 )) = vj (y2ℓ) mod (y2 , y4 , ..., y2ℓ−2 ).

Thus we have

Corollary 11.6. Suppose (∗) and ℓ = 2n − 1. Then we have the surjection

CH∗(R(G)|K)/2 ։ Λ(y2, ..., y2ℓ−2)⊗ (M0(1)⊕Mn(y2ℓ)).

At last of this section, we consider the case X(C) = G/P with

G = SO(2ℓ+ 1) and P = U(ℓ).

Let us write this X by Y , i.e. Y = G/Pk. From the fibering SO(2ℓ+1)→ Y (C)→BU(ℓ), we have the spectral sequence

E∗,∗′

2∼= H∗(SO(2ℓ+ 1);Z/2)⊗H∗′

(BU(ℓ))

∼= P (y)⊗ Λ(x1, ..., x2ℓ−1)⊗ Z/2[c1, ..., cℓ] =⇒ H∗(Y (C);Z/2).

Here the differential is given as d2i(x2i−1) = ci. Hence

CH∗(Y )/2 ∼= H∗(Y (C);Z/2) ∼= P (y)/2.

This case is studied by Vishik [Vi] and Petrov [Pe] as maximal orthogonal (orquadratic) grassmannian. (see Theorem 5.1 in [Vi]). From Theorem 11.2, we have

Theorem 11.7. ([Vi],[Pe]) Let Y = Gk/U(ℓ)k. Then

CH∗(Y )/2 ∼= CH∗(R(G))/2 ∼= Λ(c1, ..., cℓ).

In [Vi], Vishik originally defined the J-invariant J(q) of a quadratic form q whichcorresponds to the quadratic grassmannian (see Definition 5.11, Corollary 5.10 in[Vi]) by

J(q) = ik|y2ik ∈ ResΩ(X) = ik|degv(yik) = 0 ⊂ 0, ..., ℓ.

Let I be the fundamental ideal of the Witt ringW (k) so that grW (k) = ⊕nIn/In+1 ∼=

KM∗ (k)/2 where KM

∗ (k) is the Milnor K-theory of k. Smirnov and Vishik (Propo-sition 3.2.31 in [Sm-Vi]) prove that

q ∈ In if and only if 0, ..., 2n−1 − 2 ⊂ J (q).

Hence the condition (∗) in Coeollary 11.6 is equivalent to q ∈ In for the quadraticform q (with ℓ = 2n − 1) corresponding to Y |K .

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ALGEBRAIC COBORDISM 25

Theorem 11.8. Let q ∈ In be the quadratic corresponding Y |K . Then there ism ≥ n, and the surjection for ℓ = 2m − 1 such that

CH∗(R(G)|K)/2։Λ(y2, ..., y2ℓ−2)⊗ (M0(1)⊕Mm(y2ℓ)).

Corollary 11.9. Let Xq be the quadric for an anisotropic quadratic form q, and

f ∈ CHdim(Xq)(Xq) be a fundamental class. If q ∈ In, then degv(f) ≥ n.

Proof. For each generator 0 6= y ∈ CH∗(Xq)/2, we see degv(y) ≤ degv(f) fromLemma 8.1. The result follows from m = degv(y2ℓ) ≤ degv(f).

.

12. Spin(2ℓ+ 1) and p = 2

Throughout this section, let p = 2, G = SO(2ℓ + 1) and G′ = Spin(2ℓ + 1).It is well known that G/T ∼= G′/T ′ for the maximal torus T ′ of the spin group.By definition, we have the 2 covering π : G′ → G. It is well known that π∗ :H∗(G/T ) ∼= H∗(G′/T ′).

Let 2t ≤ ℓ < 2t+1, i.e. t = [log2ℓ]. The mod 2 cohomology is

H∗(G′;Z/2) ∼=∼= P (y)′ ⊗ Λ(x3, x5, ..., x2ℓ−1)⊗ Λ(z), |z| = 2t+2 − 1

where P (y)′ ∼= P (y)/(y2). (Here z is defined by d2t+2(z) = y2t+1

for 0 6= y ∈H2(BZ/2;Z/2) in the spectral sequence induced from the fibering G′ → G →BZ/2.) Hence

grP (y)′ ∼= ⊗2i6=2jΛ(y2i) ∼= Λ(y6, y10, y12, ..., y2ℓ)

where ℓ = ℓ− 1 if ℓ = 2j for some j, and ℓ = ℓ otherwise.The Qi operation for z is given also by Nishimoto [Ni]

Q0(z) =∑

i+j=2t+1,i<j

y2iy2j , Qn(z) =∑

i+j=2t+1+2n+1−2,i<j

y2iy2j for n ≥ 1.

We know that

grH∗(G′/T ′)/2 ∼= P (y)′ ⊗ S(t′)/(2, c′2, ....., c′ℓ, c

2t+1

1 ).

Here c′i = π∗(ci) and d2t+2(z) = c2t+1

1 in the spectral sequence convergingH∗(G′/T ′).Take k such that G is a versal Gk-torsor so that G′

k is also a versal G′k-torsor.

Let us write F = G/Bk and F′ = G

′/B′k. Then

CH∗(R(G′))/2 ∼= P (y)′/2, and CH∗(R(G))/2 ∼= P (y)/2.

The Chow ring CH∗(R(G′))/2 is not computed yet for ℓ ≥ 6, while we have thefollowing lemmas.

Lemma 12.1. Let 2t ≤ ℓ < 2t+1. Then there is a surjection

Λ(c′2, ...c′ℓ)⊗ Z/2[c2

t+1

1 ] ։ CH∗(R(G′))/2.

Lemma 12.2. We have

P1(R(G′)) ∼=

Z/21, c′2, c′3 for ℓ = 3, 4

Z/21, c′2, ..., c′ℓ, c2

t+1

1 for ℓ ≥ 5.

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26 N.YAGITA

Proof. By Nishimoto, for ℓ > 4, Q0(z) = y 6= 0 ∈ P (y) and 2y = c2t+1

1 is anΩ∗-module generator of ResΩ(X). So it is nonzero in P1(R(G)). For ℓ = 3, 4, we

see Q0(z) = 0 and c2t+1

1 = 0 in P1(R(G)).Let ℓ > 4. We note 4y2iy2j , 2v1y2iy2k+2 are in ResΩ(X), Since 4, 2v1 = 0

mod(I2∞), the element C(y2iy2j) does not appear in P1(R(G)). By the Qi-actions,we can see

ResΩ(X) ∼= (2)y2i|i 6= 2j, 2j + 1 ⊕ (2, v1)y2j+2 ⊕ (2)Q0(z)

as a submodule of grΩ∗ ((G))/(I2∞). In fact if v2y2i ∈ ResΩ(X), then there isx ∈ H∗(G;Z/2) such that Q0(x) = Q1(x) = 0 but Q2(x) = y2i. But this is not thecase from the Qi-action by Nishimoto.

The restrictions y2i 7→ ci′ for i 6= 2j , v1y2j+2 7→ c′2i , c

2t

1 7→ 2Q0(z) implies thelemma.

13. Spin(7), Spin(9), Spin(11)

From this section to the next section, we only consider G = Spin(2ℓ+ 1). Thespace we consider ResΩ(X) is X = R(G) the generalized Rost motive.

Hereafter in this paper, we change notations as follows. Let us write the elementcj1 in the preceding section, by ej, and take off the dash of c′i, i.e.

e2t+1 = c2t+1

1 , c2 = c′2, c3 = c′3, ...., cℓ = c′ℓ.

Now we consider examples. The groups Spin(7), Spin(9) are type (I) of the rankℓ = 3, 4 respectively. Hence R(G) ∼= R2 the original Rost motive. The cohomologyis given

grH∗(G;Z/2)) ∼=

Z/2[y6]/(y26)⊗ Λ(x3, x5, z7) G = Spin(7),

Z/2[y6]/(y26)⊗ Λ(x3, x5, x7, z15) G = Spin(9).

The cohomology operations are Q1(x3) = Q0(x5) = y6. Hence P (y) ∼= Z(2)1, y6,and resΩ(c2) = v1y6, resΩ(c3) = 2y6.

Theorem 13.1. ([Ya6]) Let G = Spin(7) or Spin(9). Then

ResΩ(R(G)) ∼= BP ∗1⊕(2, v1)y6, CH∗(R(G)/2 ∼= P1(R(G)) ∼= Z/21, c2, c3.

Next, we consider G = Spin(11). The cohomology is written as

H∗(G;Z/2) ∼= Z/2[y6, y10]/(y26 , y

210)⊗ Λ(x3, x5, x7, x9, z15).

By Nishimoto, we know Q0(z15) = y6y10. It implies 2y6y10 = d16(z15) = e8. Sinceytop = y6y10, we have t(G) = 2.

Let us write k∗(X) = Ak(1)2∗,∗(X). (It is written in §5 as h∗(X) for n = 1. Weuse Lemma 5.4 for n = 1.)

From Corollary 2.1, we can take in k∗(F) such that c2 = v1y6, c4 = v1y10.Moreover in k∗(F)/(I2∞) we have c3 = 2y6, c5 = 2y10.

Theorem 13.2. (Karpenko [Ka2], [Ya6], [Ya5]) Let G = Spin(11). Then we have

CH∗(R(G))/2 ∼= grgeo(R(G))/2 ∼= Z/21, c2, c3, c4, c5, c2c4, e8,

where c2c4 = pr(c2 · c4) for the cup product · in gr∗geo(F).

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ALGEBRAIC COBORDISM 27

Corollary 13.3. Let G = Spin(11). Then the restriction map resΩ is injective

ResΩ(R(G)) = BP ∗1 ⊕ (2, v1)y6, y10 ⊕ (2, v21)y6y10.

That means CH∗(R(G))/2 ∼= P2(R(G)).

Proof. The right hand side module is contained in ResΩ(X). Its number of theBP ∗-module generators is just 7 = rank2(CH∗(R(G))/2). Hence it is ResΩ(X).

We proved the above corollary from the preceding theorem. However, we seethe theorem from the above corollary, as stated in the introduction.

14. Spin(13).

Throughout this section. let G = Spin(13) so that ℓ = 6. Then we have

grP (y) ∼= Λ(y6, y10, y12).

Hence ytop = y6y10y12. Since 2t ≤ ℓ < 2t+1, so t = 2. Hence e8 exists inCH∗(R(G)). Note e28 = pr(e8 · e8) = 0 in CH∗(R(G)) since |e28| > |ytop|. Wealso note t(G) = 22 = 4 by Totaro, in fact e8c6 = 4ytop.

Note. From Corollary 5.5 in [Ya7], the projection pr(ci1 ...cis) is uniquely de-termined in k∗(R(G))/(Is+1

∞ ) ∼= k∗ ⊗ P (y)/(Is+1∞ ).

We take y6, y10, y12 such that

c2 = v1y6, c4 = v1y10, y12 = y26 in k∗(F).

(See [Ya7] for details.) Recall that the invariant ideal I∞ in k∗-theory is I∞ =(2, v1) ⊂ k∗. Note that pr(I∞) ⊂ I∞. Then we can take ci ∈ k∗(R(G)) such thatin k∗(R(G))/(I3∞)

c3 = 2y6, c5 = 2y10 + v1y12, c6 = 2y12,

e8 = 2y6y10 + v1y6y12.

Here we used P (y)∗ = 0 for ∗ = 8, 14, 20, and Q1x9 = y12, Q1z15 = y6y12.

Theorem 14.1. ([Ya7]) We have

grgeo(R(G)) ∼= P (b)/(c2c5, c2c4c5)⊕ Z/2c4c3, e8c4, c2c6, c4c6

⊕Z(2)1, c3, c6, e8.c3c6, c5c6, e8c6

where P (b) = ΛZ(c2, c4, c5)/(2c2, 2c4)

Corollary 14.2. We have the additive isomorphism

grgeo(R(G)) ∼= A⊗ (Z(2)1, c6 ⊕ Z/2c4)⊕ Z/2c6c4

where A = Z(2)1, c3, c5, e8 ⊕ Z/2c2.

Lemma 14.3. ([Ya7]) The restriction image ResΩ(X) ⊗Ω∗ k∗ for X = R(G) isgiven by

ResΩ(R(G′))⊗Ω∗ k∗ ⊕ (2)y12 ⊕ (4, 2v1, v21)y6y12, y10y12 ⊕ (4, v21)ytop

where G′ = Spin(11). In particular grgeo(X)/2 ∼= P (1)2(X).

Here to prove the above lemma, we used Theorem 14.1 in [Ya7]. However thistheorem is also induced from the above lemma, and it seems more easy. We cansee the above lemma as follows.

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28 N.YAGITA

Proof of Lemma 14.3. The image of the restriction map is written (with mod(I3∞))as follows

c6 7→ 2y12, c6c3 7→ 4y6y12, c6c5 7→ 4y10y12, c6e8 7→ 4ytop

c6c2 7→ 2v1y6y12, c6c4 7→ 2v1y10y12, (e8c5 7→ 4v1ytop)

(c3c4 − v1e8) 7→ v21y6y12, c4c5 7→ v21y10y12, e8c4 7→ v21ytop.

Hence the formula in this lemma is contained in ResΩ(X). (Note that all generatorsc ∈ A(c) in the left hand side in the above appear as c in the preceding theorem.The generators y ∈ BP ∗ ⊗ P (y) in the right hand side appear in ResΩ(X) in thislemma.)

For an invariant ideal J ⊂ BP ∗, the ideal J ′ = J ⊗BP∗ k∗ is written

(2, vi1), (4, 2vi1, vj1), or (4, vj1).

However, we can check vi1y12, 2v1ytop 6∈ ResΩ(X). In fact if resΩ(c) = v21y12, thenby dimensional reason, we see c = c22 = 0, and this is a contradiction. We can see2v1ytop 6∈ ResΩ(X) from c5e8 = (2y10 + v1y12)(2y6y10 + v21y6y12) = 0 mod(I3∞).

By using gr∗geo(R(G))/2 ∼= P (1)∞(R(G)) (Lemma 5.4 ), we have the theoremfrom computing ⊕yC(y).

15. The case G = E7 and p = 2.

Throughout this section, let (G, p) = (E7, 2).

Theorem 15.1. ([Mi-Tod], [Ko-Mi]) The cohomology grH∗(G;Z/2) is given

Z/2[y6, y10, y18]/(y26, y

210, y

218),⊗Λ(z3, z5, z9, z15, z17, z23, z27).

Here the suffix means its degree. We can rewrite

grH∗(G;Z/2) ∼= Z/2[y1, y2, y3]/(y21 , y

22 , y

23)⊗ Λ(x1, ..., x7).

Lemma 15.2. ([Ko-Mi]) The cohomology operations act as

x1 = z3Sq2

−−−−→ x2 = z5Sq4

−−−−→ x3 = z9Sq8

−−−−→ x4 = z17

x5 = z15Sq8

−−−−→ x6 = z23Sq4

−−−−→ x7 = z27

x5 = z15Sq2

−−−−→ x4 = z17

The Bockstein acts Sq1(xi+1) = yi for 1 ≤ i ≤ 3, and

Sq1 : x5 = z15 7→ y1y2, x6 = z23 7→ y1y3, x7 = z27 7→ y2y3.

Lemma 15.3. ([Ya5,6]) In H∗(G/T )/(4), for all monomials u ∈ P (y)+/2, exceptfor ytop = y1y2y3, the elements 2u are written as elements in H∗(BT ). Moreover,in BP ∗(G/T )/I2∞, there are bi ∈ BP ∗(BT ) such that

b2 = 2y1, b3 = 2y2, b4 = 2y3, b6 = 2y1y3, b7 = 2y2y3,

b1 = v1y1 + v2y2 + v3y3, b5 = 2y1y2 + v1y3.

Proof. The last two equations are given by Q1(x1) = y1, Q2(x1) = y2,

Q3(x1) = y3, Q1(x5) = y3, Q0(x5) = y1y2.

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ALGEBRAIC COBORDISM 29

Lemma 15.4. ([To1]) We have t(E7)(2) = 22.

Proof. We get the result from b2b7 = (2y1)(2y2y3) = 22ytop.

By Chevalley’s theorem, resK is surjective (moreover isomorphic). Hence inK∗(R(G))/2, we see

vs1y2 = b ∈ Z/2[b1, ..., bℓ] some s ∈ Z.

The facts that |bi| ≥ 6 and |y2| = 10 imply s = 2.

Lemma 15.5. We can take b2 = 2y1 + v21y2 in k∗(G/T )/(v31).

Proposition 15.6. ([YaG]) There is a filtration whose associated graded ring is

grK∗(R(G)) ∼= K∗ ⊗ (B1 ⊕B2) where

B1 = P (b)/2⊕ Z/2b1b7, where P (b) = ΛZ(b1, b2, b5)

B2 = Z2, 2b1, b3, b4, b1b3, b6, b7, b2b7

Proof. We have isomorphisms

K∗(R(G)) ∼= K∗ ⊗ P (b) ∼= K∗ ⊗ P (y) ∼= K∗(R(G)).

Let us write b2 = b2 − 2v−11 b1 and b5 = b5 − v−1

1 b1b3 so that b2 = v21y2, b5 = v1y3in K∗(R(G)). Moreover let P (b) = Λ(b1, b2, b5).

Let us write gr2K∗(R(G)) = K∗(R(G))/2 ⊕ 2K∗(R(G)), and we will compute

gr(gr2K∗(R(G)) for some filtration of gr2K

∗(R(G)).We have

K∗(R(G)) ⊃ 2K∗ ⊗ P (b) = 2K∗ ⊗ P (b)

= 2K∗1, b1, b2, ..., b2b3, b1b2b3 = 2K∗1, y1, y2, ..., y1y2y3

= K∗2, 2b1, b3, b4, b1b3, b6, b7, b1b7.

Here we used 2v1y1 = 2b1, 2y2 = b3, 2y3 = b4, 2v1y1y2 = b1b3, and

2y1y3 = b6, 2y2y3 = b7, 2v1y1y2y3 = b1b7.

The equation 2b1b7 = 4v1y1y2y3 = v1b2b7 implies that

grK∗(R(G)) = gr2K∗(R(G))/(2b1b7)⊕ Zb2b7,

which gives the graded ring in this lemma.

By dimensional reason we have with mod(I3∞)

(∗) b6 = 2y1y3 + λv21y2y3, for λ = 0 or 1.

However I can not decide this λ here.Suppose that λ = 0. Then we get the restriction map resΩ

(b2, b1)→ (2, v1)y1, (b3)→ (2)y2, (b4, b1b3 − v1b5)→ (2, v31)y3,

(b5, b1b2)→ (2, v31)y1y2, (b6, b1b5)→ (2, v21)y1y3,

(b7, b1b4 − b2b5)→ (2, v31)y2y3, (b2b7, b1b7, b1b2b5)→ (4, 2v1, v41)ytop.

For examples, we can compute mod(I4∞)

b1b2 = v1y1(2y1 + v21y2) = 2µv31y1y2 + v31y1y2 for µ ∈ Z/2,

b1b4 − b2b5 = 2v1y1y3 − (2y1 + v21y2)(2y1y2 + v1y3) = v31y2y3.

Here we used y21 = µv21y1y2 + ... by dimensional reason.

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30 N.YAGITA

Suppose λ = 1. Then we can compute

b1b6 = v1y1(2y1y3 + v21y2y3) = v31ytop.

Hence we only need to change one place

(b2b7, b1b7, b1b6)→ (4, 2v1, v31)ytop.

We can not decide grgeo(R(G))/2 here, while we have

Theorem 15.7. Let B = Z/21, b2, ..., b7. Then we have additively

grgeo(R(G))/2 ∼=

B ⊕Bb1 ⊕ Z/2b2b7

or B ⊕B/(b6)b1 ⊕ Z/2b2b7, b1b2b5.

The image ResΩ(R(G))⊗BP∗ k∗ is isomorphic to the sum of the right hand side(y) ⊂ Ω∗ ⊗ P (y) in the above restriction map. For example if λ = 1, then

ResΩ(R(G))⊗BP∗ k∗ ∼= k∗1 ⊕ (2, v1)y1 ⊕ (2)y2

⊕(2, v31)y3, y1y2, y2y3 ⊕ (2, v21)y1y3 ⊕ (4, 2v1, v31)ytop.

16. The case G = E8 and p = 2

Throughout this section, let G = E8 and p = 2. It is known ([Mi-Tod]) that

grH∗(G;Z/2) ∼= Z/2[y1, y2, y3, y4]/(y81 , y

42 , y

23, y

24)⊗ Λ(x1, ..., x8)

with |x8| = 29, |y4| = 30 so that

grH∗(E7, ;Z/2) ∼= grH∗(G,Z/2)/(y21 , y22 , y4, x8).

Hence we can write

grP (y) ∼= grP (y)′ ⊗ Λ(y21 , y41 , y

22 , y4), grP (y)′ = Λ(y1, y2, y3).

The cohomology operations forH∗(G;Z/2) are almost same as that ofH∗(E7;Z/2),which are given also by Lemma 15.2 (with mod(y21 , y

22 , y4)). Moreover, we have

Sq2(x7) = x8, Sq1(x8) = y4.

Thus we have (see Lemma 15.2)

(∗) Q3(x5) = Q2(x6) = Q1(x7) = Q0(x8) = y4, |y4| = 30.

We can not decide ResΩ(R(G)) here, but only give an example

Lemma 16.1. We have J(y21)∼= (4, 2v1, v

21).

Proof. We can see J(y1) = (2, v1) as the case E7. So J(y21) ⊂ (4, 2v1, v21). But

elements 2, v1 are not contained in J(y21).

Let K be an extension of k such that X does not split over K but splits over anextension over K of degree 2a, (2, a) = 1. Suppose that

(∗) y1, y2, y3 ∈ ResΩ(R(G)|K) but y4 6∈ ResΩ(R(G)|K ).

(Compare the above condition (∗) with the condition (∗) in §10.) That is, theJ-invariant J(GK) = (0, 0, 0, 1) and such K exists (see [Pe-Se-Za], [Se]). Then wehave the following theorem by arguments similar to those to get Theorem 7.12.(The motive R(Gk)|K in the theorem is an example of motives given in Lemma 8.4in [Se].)

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ALGEBRAIC COBORDISM 31

Theorem 16.2. Let X = R(G)|K . Then we have the isorphism and the surjection

ResΩ(X) ∼= Z(2)[y1, y2, y3]/(y81 , y

42 , y

23)⊗ (BP ∗1 ⊕ I4y4),

CH∗(X)/2 ։ Z/2[y1, y2, y3]/(y81 , y

42 , y

23)⊗ CH∗(R4)/2,

where CH∗(R4)/2 ∼= M0(1) ⊕M4(y4) ∼= Z/21, 2y4, v1y4, v2y4, v3y4. The restric-tion map is given as bj 7→ v8−jy4 if 5 ≤ j ≤ 8, and bj 7→ 0 if 1 ≤ j ≤ 4.

Proof. From (∗), we have the relations in Ω∗(X)/I2∞

b5 = 2y1y2 + ...+ v3y8, b6 = 2y1y3 + ...+ v2y8, , ...

Here if y ∈ Λ(y1, y2, y3), then y ∈ ResΩ(X). Hence v3y8 ∈ ResΩ(X), and so in I4.Thus we have J(y8) = I4.

17. The case G = E8 and p = 3

We consider cases H∗(G) has p-torsion for an odd prime p. Then (G, p) is oftype (I) or (II). For G of type (I), by Petrov-Semenov-Zinoulline, the motiveR(G) ∼= R2 and are studied in §4 detailedly. For examples

ResΩ(R2) = BP ∗1 ⊕ (p, v1)y, ..., yp−1 ⊂ Ω∗[y]/(yp),

CH∗(R2)/p ∼= Z/p1 ⊕M2(y)1, ..., yp−2.

In this section, we study the case of type(II), that is (G, p) = (E8, 3). Hereafterthis section, we assume (G, p) = (E8, 3).

The cohomology H∗(G;Z/3) is isomorphic to ([Mi-Tod])

Z/3[y8, y20]/(y38 , y

320)⊗ Λ(z3, z7, z15, z19, z27, z35, z39, z47).

By Kono-Mimura [Ko-Mi] the actions of cohomology operations are also known

Theorem 17.1. ([Ko-Mi]) We have P 3y8 = y20, and

β : z7 7→ y8, z15 7→ y28 , z19 7→ y20, z27 7→ y8y20, z35 7→ y28y20,

z39 7→ y220, z47 7→ y8y220,

P 1 : z3 7→ z7, z15 7→ z19, z35 7→ z39

P 3 : z7 7→ z19, z15 7→ z27 7→ −z39, z35 7→ z47.

We use notations y = y8, y′ = y20, and x1 = z3, ..., x8 = z47. Then we can

rewrite the isomorphisms

H∗(G;Z/3) ∼= Z/3[y, y′]/(y3, (y′)3)⊗ Λ(x1, ..., x8),

grH∗(G/T ;Z/3) ∼= Z/3[y, y′]/(y3, (y′)3)⊗ S(t)/(b1, , ..., b8).

From Corollary 2.2, we have

Corollary 17.2. ([Ya5]) We can take b1 ∈ BP ∗(BT ) such that

v1y + v2y′ = b1 in BP ∗(G/T )/I2∞.

From the preceding theorem, we know that all yi(y′)j except for (i, j) = (0, 0)and (2, 2) are β-image. Hence we have

Corollary 17.3. ([Ya5]) For all nonzero monomials u ∈ P (y)+/3 except for (yy′)2,it holds 3u ∈ S(t). In fact, in BP ∗(G/T )/I2∞

b1 = v1y + v2y′, b2 = 3y, b3 = 3y2 + v1y

′, b4 = 3y′,

b5 = 3yy′, b6 = 3y2y′ + v1(y′)2, b7 = 3(y′)2, b8 = 3y(y′)2.

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32 N.YAGITA

In the paper [Ya5], we compute

gr∗γ(G/T )/3 ∼= gr∗geo(G/Bk)/3.

We first compute some graded ring of K∗(R(G)).

Proposition 17.4. (Proposition 9.7 in [Ya5]) There is a filtration whose associatedgraded ring is

gr′K∗(R(G)) ∼= K∗ ⊗ (B1 ⊕B2) where

B1 = (P (b)/(3)⊕ Z/3b1b2, b1b8), P (b) = Z(3)[b1, b3]/(b31, b

33),

B2 = Z(3)3, b2, b4, b5, b6, b7, b8 ⊕ Z(3)b2b8.

Remark. The arguments in the last lines in the page 151 in [Ya5] were notcorrect. The element b22 is zero in grγ(R(G))/2. Delete b22 in the module inLemma 9.6 and B in Proposition 9.7.

Theorem 17.5. (Theorem 9.12 in [Ya5]) For B1, B2 in Proposition 9.7, we have

grgeo(R(G))/3 ∼= B1/(3, b1b23, b

21b

23)⊕ Z/3b1b6, b

21b6 ⊕B2/3.

Corollary 17.6. We have

grgeo(R(G))/3 ∼= B′1 ⊕B′′

1 ⊕B2

where

B′1∼= Z/3[b1, b3]/(b

31, b

32, b1b

23, b

21b

23)∼= Z/31, b1, b3, b

21, b1b3, b

23, b

21b3,

B′′1∼= Z/3b1b2, b1b8, b1b6, b

21b6,

B2∼= Z/3b2, b4, b5, b6, b7, b8, b2b8.

The restriction maps are given as

(b2, b1)→ (3, v1)y, (b3, b21)→ (3, v21)y

2, (b4, b1b2 − v1b3)→ (3, v21)y′,

(b5, b1b3)→ (3, v21)yy′, (b6, b

21b3)→ (3, v31)y

2y′, (b7, b23−2v1b6)→ (3, v21)(y

′)2,

(b8, b1b6)→ (3, v21)y(y′)2, (b2b8, b1b8, b

21b6)→ (9, 3v1, v

31)(yy

′)2.

Note all elements bi...bj in B′1, B

′′1 , B2 in the above corollary appear in the left hand

side of the above maps.

Lemma 17.7. We have the isomorphism

ResΩ (R(G))⊗P∗ k∗ ∼= k∗1 ⊕ (3, v1)y ⊕ (3, v21)y2, y′, yy′, (y′)2, y(y′)2

⊕(3, v31)y2y′ ⊕ (9, 3v1, v

31)ytop.

Hence grγ(R(G)/3 ∼= P (1)3(R(G)).

Thus we can also prove the above theorem (and corollary) from the above lemmaand the arguments for C(y).

For ease of arguments, we write d(x) = 1/4|x|, e.g.,

d(v1) = −1, d(b1) = 1, d(b2) = 2, d(b3) = 4, d(b4) = 5

d(b5) = 7, d(b6) = 9, d(b7) = 10, d(b8) = 12.

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ALGEBRAIC COBORDISM 33

Proof of Lemma 17.7. Let X = R(G). From the above restriction maps, we seethe right hand side ideals are contained in ResΩ(X). We can see J(y2) = (3, v21).Otherwise J(y2) = (3, v1) and v1y

2 ∈ ResΩ(X). Then there is a ring generator xi ∈H∗(G;Z/3) such that Q0(xi) = 0, Q1(x) = y2 This is a contradiction. Similarly wecan see J = (2, v21) for cases where v21 appears.

Therefore we only need to see

v21y2y′, v21ytop are not in ResΩ(X).

Suppose x = v21y2y′ ∈ ResΩ(X). Since x mod(I3∞) is a sum of products of bi

and bj (since each bi ∈ I∞). Since d(x) = −2 + 4 + 5 = 7, we see x = b2b4 = 9yy′

mod(I3∞). This is a contradiction. Suppose x = v21ytop ∈ ResΩ(X). Then d(x) =−2 + 4 + 10 = 12. Hence x = b4b5 or b2b7. We see b4b5 = 9y(y′)2 = b2b7. But9y(y′)2 = 3b8 and it is not a BP ∗-module generator in ResΩ(X).

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faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

E-mail address: [email protected],