Мультипликативная структура алгебраических кобордизмов ортогональных групп
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1 Introduction 2
1.1 Overview of results 3
2 Basic definitions 5
2.1 Oriented cohomology theories 5
2.1.1 Formal group laws and free theories 8
2.2 Torsors 10
3 Cohomology of split maximal orthogonal Grassmannian 10
3.1 Construction of multiplicative generators 11
3.2 Chern subalgebra 16
3.2.1 Case of maximal Lagrangian Grassmannian 19
3.3 Relations in connective K-theory and hyperbolic cohomology . 20
3.3.1 Examples 22
4 Karpenko conjecture 25
4.1 Case of special orthogonal groups 26
4.2 Methods of constructing counterexamples 28
4.3 Computation of Tor functors in known cases 29
4.3.1 Spin7 30
4.3.2 Spin9 31
Bibliography
1.1 Overview of results 3
2 Basic definitions 5
2.1 Oriented cohomology theories 5
2.1.1 Formal group laws and free theories 8
2.2 Torsors 10
3 Cohomology of split maximal orthogonal Grassmannian 10
3.1 Construction of multiplicative generators 11
3.2 Chern subalgebra 16
3.2.1 Case of maximal Lagrangian Grassmannian 19
3.3 Relations in connective K-theory and hyperbolic cohomology . 20
3.3.1 Examples 22
4 Karpenko conjecture 25
4.1 Case of special orthogonal groups 26
4.2 Methods of constructing counterexamples 28
4.3 Computation of Tor functors in known cases 29
4.3.1 Spin7 30
4.3.2 Spin9 31
Bibliography
The present thesis is devoted to computations of oriented cohomology theories of maximal orthogonal Grassmannians and applications to corresponding twisted flag varieties.
The concept of complex oriented cohomology theory was introduced in the algebraic topology in the 1960s. It is a generalized cohomology theory that admits Chern classes for complex vector bundles (also, it can be defined in terms of Thom isomorphisms). The analog of this notion in algebraic geometry was introduced by Panin and Smirnov [27]. Nowadays, there are several approaches to this concept. One of them assumes the existence of a long exact sequence of localization and closely related to the Morel-Voevodsky A1 -homotopy theory. It includes such examples as motivic cohomology, higher algebraic K-theory, and the Voevodsky algebraic cobordism. Another one is based on the consideration of ’’small” cohomology theories which usually compose the diagonal part of ”big” ones. This approach was investigated by Levine and Morel [24]. The main examples are the Chow rings, the graded version of the Grothendieck K0 ring, and the algebraic cobordism of Levine- Morel, which is a universal oriented cohomology theory. We will follow the second one.
An arbitrary oriented cohomology theory are extensively studied now [6], [10], [11], [13], [22], [23]. In particular, Vishik used the algebraic cobordism theory in his construction of fields with u-invariant 2n + 1 [34]. The general context of oriented cohomology theories suggested Vishiks’ excellent connections [35] and Petrov-Semenov connections [30].
The Chow groups of maximal orthogonal Grassmannians were computed by Vishik [32] and allowed him to introduce the notion of J-invariant of a quadratic form. This invariant has a large number of applications and results behind it. Later, it was generalized to an arbitrary linear algebraic groups to study twisted forms by Petrov, Semenov, and Zainoulline [31].
The following conjecture describes the structure of the Chow ring of most twisted flag variety corresponding to the given split semisimple linear group and it is the motivation for this work.
Conjecture (Karpenko conjecture). Let B be a Borel subgroup in a split semisimple linear algebraic group G and E be a generic G-torsor. Then epimorphism CH*(E/B) ^ grTK°(E/B) is an isomorphism.
This conjecture was proven for any group of type An, Cn, G2, F4, also for SOm,Elc and Spinn,n < 12. However, in 2018, Yagita found a counterexample, which is the spinor group of rank 17 [38]. Later, it was also disproved for Spin19 and Spin21 [1] (notice, that answers for Spin2n+1 and Spin2n+2 are equivalent).
The concept of complex oriented cohomology theory was introduced in the algebraic topology in the 1960s. It is a generalized cohomology theory that admits Chern classes for complex vector bundles (also, it can be defined in terms of Thom isomorphisms). The analog of this notion in algebraic geometry was introduced by Panin and Smirnov [27]. Nowadays, there are several approaches to this concept. One of them assumes the existence of a long exact sequence of localization and closely related to the Morel-Voevodsky A1 -homotopy theory. It includes such examples as motivic cohomology, higher algebraic K-theory, and the Voevodsky algebraic cobordism. Another one is based on the consideration of ’’small” cohomology theories which usually compose the diagonal part of ”big” ones. This approach was investigated by Levine and Morel [24]. The main examples are the Chow rings, the graded version of the Grothendieck K0 ring, and the algebraic cobordism of Levine- Morel, which is a universal oriented cohomology theory. We will follow the second one.
An arbitrary oriented cohomology theory are extensively studied now [6], [10], [11], [13], [22], [23]. In particular, Vishik used the algebraic cobordism theory in his construction of fields with u-invariant 2n + 1 [34]. The general context of oriented cohomology theories suggested Vishiks’ excellent connections [35] and Petrov-Semenov connections [30].
The Chow groups of maximal orthogonal Grassmannians were computed by Vishik [32] and allowed him to introduce the notion of J-invariant of a quadratic form. This invariant has a large number of applications and results behind it. Later, it was generalized to an arbitrary linear algebraic groups to study twisted forms by Petrov, Semenov, and Zainoulline [31].
The following conjecture describes the structure of the Chow ring of most twisted flag variety corresponding to the given split semisimple linear group and it is the motivation for this work.
Conjecture (Karpenko conjecture). Let B be a Borel subgroup in a split semisimple linear algebraic group G and E be a generic G-torsor. Then epimorphism CH*(E/B) ^ grTK°(E/B) is an isomorphism.
This conjecture was proven for any group of type An, Cn, G2, F4, also for SOm,Elc and Spinn,n < 12. However, in 2018, Yagita found a counterexample, which is the spinor group of rank 17 [38]. Later, it was also disproved for Spin19 and Spin21 [1] (notice, that answers for Spin2n+1 and Spin2n+2 are equivalent).
In this section, we will compute Tor-functors from the propositions above in known cases of Karpenko conjecture for spinor groups of small rank. Let E be a generic Spin2n+i-torsor, and P C G is a parabolic subgroup such that Spin2n+i/P = OGr(n), as always. We will begin with the following lemma.
Lemma 4.3. For a free oriented cohomology theory A* the image of the restriction map Im(A*(E/P) ^ A* (Spin2n+1 /P)) is generated by z1kcf .. .0%, 0 < k < dim(Srpin2n+1/P), ij E {0,1} as A*(pt)-module. Here c, stands for cAE).
Proof. It is known, that the same monomials generate the image of rescH : CH*(E/P) ^ CH*(Spin2n+1/P) (see [17]). Let us consider commutative diagram
A*(pt)N > Im(resA)
ZN » Im(resCH),
where the top arrow sends the standard basis of the free module to the elements zlcf .. .c%, 0 < k < dim(Spin2n+1/P), ij E {0,1}, and the bottom one the standard basis of ZN to the same monomials in e1 and cCH (En) (in fact N = dim(Spin2n+1/P)2n-1). Since the bottom map is surjective, it follows from Lemma 3.2, that the top map is surjective as well. ■
First, consider the case of n =1. Then it is obvious, that A*(OGr(1)) = A*(pt)[z1]/(z‘2), and the restriction map is surjective. Hence, Coker(resA) = 0 and Tor functors from Proposition 4.2 and Proposition 4.3 is zero. For n = 2 by our algorithm CK*(OGr(2)) = Z[t][z1, z2]/(z2 — z2, z2) and the restriction map is again surjective (the connective K-theory of OGr(2) generated by 1, z1, z2, z3). The same holds for SE*(OGr(2)). In fact, in these cases the restriction maps are isomorphisms, since Spin3 and Spin5 are special groups.
4.3.1 Spin7
In this case dimension of Spin7/P = OGr(3) is equal to 6 and by computations from 3.3.1 in CK*(OGr(3)) there are relations z2 = 0, z2 = 2z1 z3+tz2z3 and z1 = z2 + tz3. The generators of Im(resCK) are given in the following table. At the intersection of the column with z1 at the top and the row with c22 c33 from the left is its product (notice, that here c3 = 2z3 without sign).
Proposition 4.4. Let E be a generic Spin7-torsor and P C Spin7 be a parabolic such that Spin7/P = OGr(3). Then
Torf[t](Coker(rescK), Z) = gryK0(E/P)tors = Z/2Z®4.
Proof. Let us consider the short exact sequence
0 ^ Im(resCK) ^ CK*(OGr(3)) ^ Coker (resCK) ^ 0.
Taking tensor product — ®Z[t] Z, we get
0 ^ Tor^ (Coker (resCK), Z) ^ Im(resCK) ®Z[t] Z ^ CH * (OGr(3)).
Hence, Tor can be identified with the kernel of the last map. Therefore, we should find non-rational elements, such that they became rational after multiplying by t. In other words, we need to find all t-torsion in Coker(resCK). Note, that z2 is rational for connective K-theory (it comes from c2 — z2 + tc3 = z2). It follows, that the only possible such elements are z3, ziz3, z2z3, z1z2z3, or their linear combination. It is easy to see, that these elements give us t- torsion (for example, z2 — z2 = tz3, or z3 — z1z2 = tz1z3). Direct computation says, that none of these elements is rational until multiplied by t. Hence, kernel is generated by z3, z1z3, z2z3, z1z2z3. Also, since 2z3 is rational it is obvious, that these elements has order 2. Therefore,
TorZ[t} (Coker (rescK), Z) = Z/2Z®4.
The desired isomorphism for the associated graded ring is known and can be computed directly. ■
Proposition 4.5. In the notations of previous proposition, the following holds
Torf[M1,M2] (Coker (res SE ), Z[t]) = 0.
Proof. By the same argument as in the previous proposition corresponding Tor can be identified with kernel of Im(resSE) CZ[M1,M2] Z[t] ^ CK*(OGr(3)). In other words, we need to show, that there is no non-rational element, which became rational after multiplication by p2. Calculations similar to that given above show this. ■ 4.3.2 Spin9
In this case dimension of Srpin9/P = OGr (4) is equal to 10 and by computations from 3.3.1 in CK*(OGr(4)) there are relations z24 =0, z2 = 2z2z4—tz3z4, z2 = 2z 1z3 — tz2z3 — z4 + 2tz1z4 — t2z2z4 and z2 = z2 + tz3 + t2z4.
Proposition 4.6. Let E be a generic Spin9-torsor and P C Spin7 be a parabolic subgroup such that Spin9/P = OGr(4). Then
Tor?}(Coker(rescK), Z) = gryK0(E/P)tors = Z/2Z®8.
Proof. By the same argument as in the proof of Proposition 4.4, we can identify TorZt(Coker(resCK),Z) with the kernel of Im(resCK) ®Z[t] Z ^ CH*(OGr(4)). First of all, we should notice, that c2 — z2 + tc3 + t2c4 = z2 lies in the image of the restriction map (here we ignore sign of the Chern class c3 for simplicity again, i.e. c3 = 2z3 — tz4). Hence, tz3 and t2z4 are also in the image of resCK because tz3 = 2z2 — c2, t2z4 = z2 — z2 — tz3. Consider
z4 = 2z1z3 — z4 + 2tz1z4 + 3t2z2z4 + t3z3z4 + tz2z3, it is easy to see that all summands except 2z1z3 and z4 are rational by arguments above. Using z1c3 = 2z1z3 — tz1z4 we get that z4 — tz1z4 lies in the image of the restriction map. After multiplication by t it gives us rationality of tz4. Applying this to the expression of z4 we get that z4 is rational as well. It follows, that the only possible elements whose are non-rational, but became rational after multiplying by t are z3, z1z3, z2z3, z3z4, z1z2z3, z1z3z4, z2z3z4, z1z2z3z4 or their linear combination. Direct computation says, that none of these elements is rational until multiplied by t (notice, that even projections of this elements to the Chow groups are non-rational). Since z1, z2, z4 and tz3 are rational it follows that the elements above give us t-torsion. Also, since c3 + tz4 = 2z3 G Im(rescK) all these elements has order 2. Therefore,
TorZ[t] (Coker (rescK), Z) = Z/2Z®8.
The desired isomorphism for the associated graded ring is known and can be computed directly. ■
Using similar computations in SE*(OGr(4)) and method from Proposition 4.5 we get the following proposition.
Proposition 4.7. In the notations of previous proposition, the following holds
TorZ[M1,№] (Coker(ressE), Z[t]) = 0.
Lemma 4.3. For a free oriented cohomology theory A* the image of the restriction map Im(A*(E/P) ^ A* (Spin2n+1 /P)) is generated by z1kcf .. .0%, 0 < k < dim(Srpin2n+1/P), ij E {0,1} as A*(pt)-module. Here c, stands for cAE).
Proof. It is known, that the same monomials generate the image of rescH : CH*(E/P) ^ CH*(Spin2n+1/P) (see [17]). Let us consider commutative diagram
A*(pt)N > Im(resA)
ZN » Im(resCH),
where the top arrow sends the standard basis of the free module to the elements zlcf .. .c%, 0 < k < dim(Spin2n+1/P), ij E {0,1}, and the bottom one the standard basis of ZN to the same monomials in e1 and cCH (En) (in fact N = dim(Spin2n+1/P)2n-1). Since the bottom map is surjective, it follows from Lemma 3.2, that the top map is surjective as well. ■
First, consider the case of n =1. Then it is obvious, that A*(OGr(1)) = A*(pt)[z1]/(z‘2), and the restriction map is surjective. Hence, Coker(resA) = 0 and Tor functors from Proposition 4.2 and Proposition 4.3 is zero. For n = 2 by our algorithm CK*(OGr(2)) = Z[t][z1, z2]/(z2 — z2, z2) and the restriction map is again surjective (the connective K-theory of OGr(2) generated by 1, z1, z2, z3). The same holds for SE*(OGr(2)). In fact, in these cases the restriction maps are isomorphisms, since Spin3 and Spin5 are special groups.
4.3.1 Spin7
In this case dimension of Spin7/P = OGr(3) is equal to 6 and by computations from 3.3.1 in CK*(OGr(3)) there are relations z2 = 0, z2 = 2z1 z3+tz2z3 and z1 = z2 + tz3. The generators of Im(resCK) are given in the following table. At the intersection of the column with z1 at the top and the row with c22 c33 from the left is its product (notice, that here c3 = 2z3 without sign).
Proposition 4.4. Let E be a generic Spin7-torsor and P C Spin7 be a parabolic such that Spin7/P = OGr(3). Then
Torf[t](Coker(rescK), Z) = gryK0(E/P)tors = Z/2Z®4.
Proof. Let us consider the short exact sequence
0 ^ Im(resCK) ^ CK*(OGr(3)) ^ Coker (resCK) ^ 0.
Taking tensor product — ®Z[t] Z, we get
0 ^ Tor^ (Coker (resCK), Z) ^ Im(resCK) ®Z[t] Z ^ CH * (OGr(3)).
Hence, Tor can be identified with the kernel of the last map. Therefore, we should find non-rational elements, such that they became rational after multiplying by t. In other words, we need to find all t-torsion in Coker(resCK). Note, that z2 is rational for connective K-theory (it comes from c2 — z2 + tc3 = z2). It follows, that the only possible such elements are z3, ziz3, z2z3, z1z2z3, or their linear combination. It is easy to see, that these elements give us t- torsion (for example, z2 — z2 = tz3, or z3 — z1z2 = tz1z3). Direct computation says, that none of these elements is rational until multiplied by t. Hence, kernel is generated by z3, z1z3, z2z3, z1z2z3. Also, since 2z3 is rational it is obvious, that these elements has order 2. Therefore,
TorZ[t} (Coker (rescK), Z) = Z/2Z®4.
The desired isomorphism for the associated graded ring is known and can be computed directly. ■
Proposition 4.5. In the notations of previous proposition, the following holds
Torf[M1,M2] (Coker (res SE ), Z[t]) = 0.
Proof. By the same argument as in the previous proposition corresponding Tor can be identified with kernel of Im(resSE) CZ[M1,M2] Z[t] ^ CK*(OGr(3)). In other words, we need to show, that there is no non-rational element, which became rational after multiplication by p2. Calculations similar to that given above show this. ■ 4.3.2 Spin9
In this case dimension of Srpin9/P = OGr (4) is equal to 10 and by computations from 3.3.1 in CK*(OGr(4)) there are relations z24 =0, z2 = 2z2z4—tz3z4, z2 = 2z 1z3 — tz2z3 — z4 + 2tz1z4 — t2z2z4 and z2 = z2 + tz3 + t2z4.
Proposition 4.6. Let E be a generic Spin9-torsor and P C Spin7 be a parabolic subgroup such that Spin9/P = OGr(4). Then
Tor?}(Coker(rescK), Z) = gryK0(E/P)tors = Z/2Z®8.
Proof. By the same argument as in the proof of Proposition 4.4, we can identify TorZt(Coker(resCK),Z) with the kernel of Im(resCK) ®Z[t] Z ^ CH*(OGr(4)). First of all, we should notice, that c2 — z2 + tc3 + t2c4 = z2 lies in the image of the restriction map (here we ignore sign of the Chern class c3 for simplicity again, i.e. c3 = 2z3 — tz4). Hence, tz3 and t2z4 are also in the image of resCK because tz3 = 2z2 — c2, t2z4 = z2 — z2 — tz3. Consider
z4 = 2z1z3 — z4 + 2tz1z4 + 3t2z2z4 + t3z3z4 + tz2z3, it is easy to see that all summands except 2z1z3 and z4 are rational by arguments above. Using z1c3 = 2z1z3 — tz1z4 we get that z4 — tz1z4 lies in the image of the restriction map. After multiplication by t it gives us rationality of tz4. Applying this to the expression of z4 we get that z4 is rational as well. It follows, that the only possible elements whose are non-rational, but became rational after multiplying by t are z3, z1z3, z2z3, z3z4, z1z2z3, z1z3z4, z2z3z4, z1z2z3z4 or their linear combination. Direct computation says, that none of these elements is rational until multiplied by t (notice, that even projections of this elements to the Chow groups are non-rational). Since z1, z2, z4 and tz3 are rational it follows that the elements above give us t-torsion. Also, since c3 + tz4 = 2z3 G Im(rescK) all these elements has order 2. Therefore,
TorZ[t] (Coker (rescK), Z) = Z/2Z®8.
The desired isomorphism for the associated graded ring is known and can be computed directly. ■
Using similar computations in SE*(OGr(4)) and method from Proposition 4.5 we get the following proposition.
Proposition 4.7. In the notations of previous proposition, the following holds
TorZ[M1,№] (Coker(ressE), Z[t]) = 0.
