key: cord-0233009-0tf67ecz
authors: Gu, Xing
title: A distinguished subring of the Chow ring and cohomology of $BPGL_n$
date: 2020-12-01
journal: nan
DOI: nan
sha: ae9cf44f736477c6f8eb81cf258dd4bbb4e40c92
doc_id: 233009
cord_uid: 0tf67ecz

We determine a subring of the Chow ring and the cohomology of $BPGL_n$, the classifying space of the projective linear group of degree $n$ over complex numbers, and explain a way in which this computation might play a role in the period-index problem. In addition, we show that the Chow ring of $BPGL_n$ is not generated by the Chern classes of linear representations of $PGL_n$.

The cohomology of classifying spaces of Lie groups is among the fundamental subjects in topology. A similar role in algebraic geometry is played by the Chow rings of the classifying spaces of algebraic groups over a field, defined by Totaro [40] , which may alternatively be described in terms of motivic cohomology. In this paper we consider the Chow ring and cohomology of the classifying space of the complex projective linear group.

• H s,t M (X; R): the motivic cohomology group of bidegree (s, t) for a motivic space X with coefficients in a commutative unital ring R, where the term "motivic space" is defined in Section 2;

• H ś et (X; F ): theétale cohomology of anétale sheaf F over a scheme X. • H s (Y ; R): the singular cohomology group of degree s for a topological space Y with coefficients in R; • H s,t M (X) = H s,t M (X; Z), H s (Y ) = H s (Y ; Z); • BG: the classifying space of a Lie group G, or the geometric classifying space of an algebraic group G which is discussed Section 2; • CH t (X) := H 2t,t M (X): the Chow group of degree t for X a smooth scheme over C or X = BG for G an algebraic group, or equivalently Totaro's Chow ring of BG defined in [40] .

• cl : H s,t M (X) → H s (X(C)): the (complex) cycle class map for X a smooth scheme over C , and X(C) the manifold of complex points of X, or, in the sense of Totaro [40] , for X = BG where G is an algebraic group over C and X(C) = BG(C) for G(C) the Lie group of complex points of G. This is discussed in Section 2. In the case of Chow rings, we have cl : CH t (X) → H 2t (X(C)) which is the cycle class map in the classical sense.

• GL n := GL n (C) and SL n := SL n (C): the general liniear group and the special linear group of degree n over C; • P GL n := GL n /C × : the projective lienar group of degree n over C, i.e., GL n modulo its center, the subgroup of invertible scalar matrices; • P U n := U n /S 1 : the projective unitary group of order n, i.e., the unitary group U n modulo its center. • K(R, s): the Eilenberg-Mac Lane space representing the cohomology functor H s (−; R) for a commutative unital ring R.

In the case of singular cohomology, we always consider BU n and BP U n instead of BGL n and BP GL n , since U n and P U n are respectively the maximal compact subgroups of GL n and P GL n . Among the Chow rings CH * (BG) and H * (BG), the case G = P GL n (or G = P U n ) is one of the most difficult, as pointed out by Molina Rojas and Vistoli [31] , in which a unified approach is provided to the Chow rings of classifying spaces for many classical groups, not including P GL n .

On the other hand, the case for P GL n is potentially of the richest structure. For instance, the torsion classes in CH * (BP GL n ) and H * (BP U n ) are all n-torsions, by Proposition 2.3 of [42] .

In addition to the significance of BP GL n and BP U n in their own rights, the cohomology of BP U n has applications in the topological period-index problem [4] , [20] and the study of anomalies in physics [8] , [14] .

The cohomology algebra H * (BP U 4n+2 ; F 2 ) is determined by Kono and Mimura [27] and Toda [39] . The cohomology algebra H * (BP U 3 ; F 3 ) is determined by Kono, Mimura, and Shimada [28] . Vavpetič and Viruel [41] show some properties of H * (BP U p ; F p ) for an arbitrary odd prime p.

The Chow ring CH * (BP GL 3 ) is almost determined by Vezzosi [42] , which is subsequently improved by Vistoli [43] , which completes the study of CH * (BP GL 3 ) and determined the additive structure as well as a large part of the ring structures of CH * (BP GL p ) and H * (BP U p ), for p an odd prime. The Brown-Peterson cohomology of BP U p for an odd prime p is determined by Kono and Yagita [29] .

The author [19] determines the ring structure of H * (BP U n ) for any n > 0 in dimensions less than or equal to 10, and obtains partial results on the Chow ring and the Brown-Peterson cohomology of BP GL n in [23] and [21] .

In [19] , the author considers a map in which we have classes y p,k ∈ H 2p k+1 +2 (BP U n ), k ≥ 0 which are nontrivial p-torsion classes for p | n and trivial otherwise. In the case p | n and p 2 ∤ n, the author [23] shows that there are p-torsion classes ρ p,k ∈ CH p k+1 +1 (BP GL n ), k ≥ 0 satisfying cl(ρ p,k ) = y p,k . However, the author [23] does not show anything about CH * (BP GL n ) for n with p-adic valuation greater than 1. Here, the p-adic valuation of n means the greatest integer r satisfying p r | n.

The classes ρ p,k and y p,k are "periodic" in the following sense. Suppose we have p | m | n. Then we have the obvious diagonal homomorphism (1.2) ∆ : P GL m → P GL n ,

which induces a map of classifying spaces B∆ : BP GL m → BP GL n . Then it is shown in [23] that we have B∆ * (y p,k ) = y p,k . If in addition we have p 2 ∤ n, then B∆ * (ρ p,k ) = ρ p,k . Despite the works discussed above, very little has been understood about the role of the p-adic valuation of n in CH * (BP GL n ) and H * (BP U n ). The purpose of this paper is to offer some insight into this, in the form of the following two theorems.

Theorem 1. Let p be an odd prime, and n a positive integer divisible by p. Then there are nontrivial p-torsion classes

for k ≥ 0, such that for p | m | n and ∆ : P GL m → P GL n , we have

Furthermore, suppose r ≥ 1 is the p-adic valuation of n. Then there are injective ring homomorphisms

and

Notice that, away from degree 0, the ring

, with the degree of Y p,k equal to p k+1 + 1 in the case of Chow rings, or 2p k+1 + 2, in the case of singular cohomology. For each n > 1, we define the subrings

Theorem 2. Let p be an odd prime and n > 1 an integer with p-adic valuation r > 0. Then the homomorphisms B∆ * restrict to isomorphisms

In Theorem 1, the condition 0 ≤ k ≤ 2r − 1 in (1.4) and (1.5) is essential at least when n is of p-adic valuation 1, as shown in the following Theorem 3. For p and odd prime, and n > 0 an integer satisfying p | n and p 2 ∤ n, there are nontrivial polynomial relations in the rings R M (n) and R(n) as follows:

p,0 + y p+1 p,1 + y p p,0 y p,2 = 0. Remark 1.1. There are inclusions of p-elementary abelian subgroups θ : V 2r ֒→ P GL p r such that when r = 1, the homomorphism Bθ * is injective when restricted to the torsion subgroup of CH * (BP GL p ) (Lemma 6.4). The polynomial relations (6.4) and (6.4) are therefore detected by the relations in CH * (BV 2 ). For a general r, we hope that similar polynomial relations may be detected by CH * (BV 2r ).

Outline of proofs. The classes y p,k are constructed in [23] , which we recall in this paper. To construct the classes ρ p,k , we define a class ζ 1 ∈ H 3,2 M (BP GL n ) viá etale cohomology and the Beilinson-Lichtenbaum conjecture. The classes ρ p,k are constructed by applying Steenrod reduced power operations to the class ζ 1 .

To verify the injectivity of the homomorphisms (1.4) and (1.5), it suffices to verify the latter, from which the former follows via the cycle class map. We reduce it to the case n = p r and consider an inclusion of a non-toral elementary abelian p-subgroup θ : V 2r → P U p r , and show that the composition

Theorem 3 follows from Vistoli [43] and some additional computation involving the transfer maps tr H G :

The period-index problem. The classical version of the period-index problem ( [15] , [18] ) concerns a field k and the degrees of central simple algebras over k and its Brauer group, or more generally the Brauer group to a scheme and the degrees of Azumaya algebras over it. In [3] , Antieau and Williams initiated the study of a topological analog of the period-index problem, which we call the topological period-index problem.

The cohomology of BP U n plays an important role in the study of the topological period-index problem. In this paper we briefly discuss how CH * (BP GL n ) may play a similar role in the period-index problem for schemes.

The Chern subrings. We have an interesting consequence of Theorem 1, regarding the Chern subrings. Definition 1.2. For G an algebraic group over C, and a commutative unital ring R, the Chern subring of CH * (BG)) ⊗ R is the subring generated by Chern classes of all representations of ϕ : G → GL r for some r, i.e., the image of the pull-back homomorphisms

If the Chern subring is equal to CH * (BG) ⊗ R, then we say that CH * (BG) ⊗ R is generated by Chern classes. The Chern subrings for any generalized cohomology theories of BG are similarly defined.

For an abelian group A, let A (p) denote the localization of A at p, or equivalently, tensor product with Z (p) . Vezzosi [42] shows that CH * (BP GL 3 ) (3) is not generated by Chern classes. The same is shown for CH * (BP GL p ) (p) for all odd primes p independently by Kameko and Yagita [26] , and Targa [37] , and is shown for CH * (BP GL n ) (p) with p | n and p 2 ∤ n by the author [23] . The same result for the Brown-Peterson cohomology BP * (BP GL p ) is proved in Kono and Yagita [29] . It is shown in [23] and [21] , respectively, that H * (BP GL n ) (p) and BP * (BP GL n ) are not generated by Chern classes for p | n. We extend the above mentioned results for CH * (BP GL n ) (p) to the most general case: Theorem 4. Let n > 1 be an integer, and p one of its odd prime divisor. Then the Chow ring CH * (BP GL n ) (p) is not generated by Chern classes. More precisely, the class ρ i p,0 is not in the Chern subring for p − 1 ∤ i. Organization of the paper. Section 2 is a brief review motivic homotopy theory required in the rest of this paper. In Section 3 we recall the definition of the classes y p,k in [23] , and construct the classes ρ p,k . In Section 4 we prove a lemma on the cohomology of an extraspecial p-group, which plays a key role in the construction of the non-toral p-elementary subgroup V 2r of P U p r . Then we study the cohomology of BV 2r in Section 5, where we complete the proof of Theorem 1. In Section 6 we prove Theorem 3. Section 7 is a brief discussion on the period-index problem. In Section 8 we discuss the Chern subrings and prove Theorem 4. In the appendix we discuss a Jocobian criterion for algebraic independence over perfect fields, which is used in Section 5.

Acknowledgement. The author is grateful to Burt Totaro for pointing out an error in an earlier version, and for other helpful conversations.

The author is grateful to Zhiyou Wu for communications on basic knowledge on motivic homotopy theory, to Benjamin Antieau, Diarmuid Crowley, Christian Haesemeyer, and Ben Williams for their continuing interests in the study of BP GL n , to Kasper Anderson, Denis Nardin, Oliver Röndigs, David E Speyer, and Mathias Wendt for discussions via MathOverflow, to the anonymous referees who offered various pieces of advice that improve the paper. Finally the author would like to thank the Max Planck Institute for Mathematics in Bonn for various supports amid the COVID-19 global pandemic.

In this section we review the necessary backgrounds in motivic cohomology and homotopy theory. Let Sm k be the category of smooth schemes over a field k, and We call objects of Mot k • motivic spaces. We consider the pointed motivic homotopy category HMot k • over the base field k, which is the homotopy category of the category of simplicial presheaves ∆ op PShv • (Sm k ), taking Bousfield localization with respect to the Nisnevich hypercovers and the canonical projections X × A 1 → X, where A 1 is the affine line. We also consider the homotopy category of pointed locally contractible topological spaces HTop • .

Remark 2.2. We choose to take ∆ op PShv • (Sm k ) as the ambient category of motivic spaces, instead of ∆ op Shv • (Sm k N is ), where Sm k N is is the Nisnevish site over Sm k , as done by Morel and Voevodsky [32] . The resulting homotopy categories are the same, as explained in [10] , for instance. Our choice of simplicial presheaves makes it slightly easier for arguments on monoidal structures.

The motivic stable homotopy category. In the category HTop • , we have the suspension functors Σ s = S s ∧ −, where S s is the s-dimensional sphere, and ∧ is the smash product.

In the category HMot k • , we have smash products defined by the object-wise smash product of simplicial presheaves. The notion of spheres in HMot k • is slightly complicated. We define the simplicial circle S 1,0 := ∆ 1 /∂∆ 1 , where the simplicial sets are regarded as constant simplicial presheaves, and the Tate circle S 1,1 := G m , where G m is the algebraic group Spec k[x ±1 ]. We therefore have spheres S s,t := (S 1,0 ) ∧s−t ∧ (S 1,1 ) ∧t for s ≥ t, and the bigraded suspension functors (2.1) Σ s,t := S s,t ∧ − : HMot k • → HMot k • . By "formally inverting the suspension fucntors", we obtain the stabilization of HTop • and HMot k • , which we denote by SHT and SHM k , respectively. We call objects of SHT spectra, and objects of SHM k motivic spectra. For the construction of SHM k , see [11] . In both the topological and motivic cases, we have the stabilization functors Σ ∞ :

The motivic Eilenberg-Mac Lane spaces and spectra. For a commutative unital ring R, consider the Eilenberg-Mac Lane motivic spectrum H M R representing H * , * M (−; R), i.e., for a smooth scheme X over k, we have natural isomorphisms of groups

. The left-hand side is canonically an abelian group, as SHM k is a triangulated category. The notation H M R is set to be distinguished from HR, the classical Eilenberg-Mac Lane spectrum in SHT. For s ≥ t ≥ 0, we have motivic Eilenberg-Mac Lane spaces K(R(t), s) which are abelian group objects of HMot k • , representing the motivic cohomology functor H s,t M (−; R), i.e., for a smooth scheme X over C, we have natural isomorphisms

. See [24] for the construction of K(R(t), s). We may extend the definition of motivic cohomology to a functor from HMot k • , by letting X at the left-hand-side of (2.3) be any object in the category HMot k • . The ring structure of motivic cohomology yields a morphism

Passing to the stable homotopy category H M R, we have the following Proposition 2.3. For R a commutative unital ring, H M R is a motivic commutative ring spectra, i.e., we have a unital, commutative, associative morphism

which gives the product of motivic cohomology.

For the short exact sequence of Z-modules Z ×n − − → Z → Z/n, the associated long exact sequence of motivic cohomology groups yields a Puppe sequence

in which every two consecutive arrows yield a fiber sequence of spaces. The last arrow represents the Bockstein homomorphism

Passing to the stable homotopy category, the Bockstein homomorphism above yields a morphism (2.6) δ :

The C-realization functor. Consider the functor

of taking complex points with a disjoint base point. Let ∆ op Sets • be the category of pointed simplicial sets. For a pointed topological space Y , let Sing(Y ) be the pointed simplicial set of singular complexes of Y , i.e., we have

with the obvious face and degeneracy maps, and ∆ n the standard topological simplices. Then we have a functor

We take the left homotopy Kan extension of (2.8) and obtain a functor

which is a left Quillen functor. We denote the total left derived functor by t C : HMot C • → HTop • which we also call the C-realization functor, noticing that the homotopy category of ∆ op Sets • , with the classical model structure, is well known to be equivalent to HTop • ( [35] ). We make the choice of ∆ op Sets • over Top • as the target category since the former is easier for comparison with simplicial R-modules.

Remark 2.4. We may take, for instance, the following model for the left homotopy Kan extension:

is a collection of cosimplicial objects with the obvious co-face and co-degeneracy maps.

As explained in Section 3.3 of [32] , the functor t C takes a presheaf represented by a simplicial smooth scheme X to the geometric realization of X (C), the simplicial topological space of degree-wise complex points of X . Therefore, we have

It is shown in [34] , Theorem A.23, that t C is a strict symmetric monoidal Quillen functor, where the strict symmetric monoidal structure on Top • is given by smash products. It is shown in [24] , Theorem 5.5, that t C (H M R) ∼ = HR. Therefore, we have The Beilinson-Lichtenbaum Conjecture. It is shown in [5] that when the base field k has characteristic prime to n, any locally constant torsionétale abelian sheaf with torsion order n is invariant under base changes along A 1 . As an immediate consequence, we have Proposition 2.7 ([30], Corollary 9.25). For a base field k of characteristic prime to n. Then any locally constant torsionétale abelian sheaf with torsion order n is A 1 -local.

With the Lichtenbaum cohomology H s,t L (−; Z/n)( [45] , Definition 10.1) acting as a bridge, this enables the construction of the "étale cycle class map", i.e., the natural transformation

when the characteristic of the base field k is prime to n.

The following theorem is known as the Beilinson-Lichtenbaum Conjecture:

Theorem 2.8 (Voevodsky, Theorem 6.17, [30] ). For smooth schemes over a field k and n be an integer prime to the characteristic of k, and nonnegative integers s ≤ t, the homomorphism (2.11) is an isomorphism.

When k = C, we have the inclusions Nisnevich covers ⊂étale covers ⊂ local homoemorphisms, which yields the following Proposition 2.9. Let R be a commutative unital ring and X be a complex smooth scheme. The complex cycle class map cl factors, functorial in X, as

is the underlying complex manifold of X. For R = Z/n, the second arrow is identified, via the identification Z/n ∼ = (Z/n) ⊗t and Theorem 2.8, to the usual comparison map H ś et (−; µ ⊗t n ) → H s (−; Z/n). The motivic Steenrod reduced power operations. In [46] , Voevodsky constructs stable operations satisfying a set of axioms and Adem relations similar to those of the Steenrod reduced power operations for singular cohomology.

Let p be an odd prime and F p be the field of order p. Then the motivic Steenrod reduced power operations are:

The reader may refer to [46] for the Adem relations.

Remark 2.10. As in the case of classical Steenrod operations, the operation β is the composition of the Bockstein homomorphism δ and the mod p reduction:

Remark 2.11. The notations above coincide with those of the classical Steenrod operations, which will appear in this paper as well. It will be made clear by the context which is intended.

The motivic Steenrod operations are compatible with the classical ones in the following sense. As pointed out in 3.11 of [44] , for k = C, we have the commutative diagrams

Totaro's Chow rings of classifying spaces. It is well known (Preface of [30] ) that for a smooth scheme X over k we have

. This may extend to X = BG, in which case CH * (BG) is the Chow ring of BG in the sense of [40] and [12] . The definition requires some prerequisite as follows.

Lemma 2.12 (Eddidin-Graham, Lemma 9, [12] ). Let G be an algebraic group. For any i > 0, there is a representation V of G and an open set U ⊂ V such that V − U has codimension more than i and such that a principal bundle quotient U → U/G exists in the category of schemes.

Theorem 2.13 (Totaro, Theorem 1.1, [40] ). Let G be a linear algebraic group over a field k. Let V be any representation of G over k such that G acts freely outside a G-invariant closed subset S ⊂ V of codimension ≥ s. Suppose that the geometric quotient (V − S)/G (in the sense of [33] ) exists as a variety over k. Then the ring CH * ((V − S)/G), restricted to degrees less than s, is independent (in a canonical way) of the representation V and the closed subset S. Now we may present the definition of the Chow ring of a classifying space of an algebraic group. Definition 2.14 (Totaro, Definition 1.2, [40] ). For a linear algebraic group G over a field k, define CH i (BG) to be the group CH i ((V − S)/G) for any (V, S) as in Theorem 2.13 such that S has codimension greater than i in V .

The existence of the co-complete category HMot k • gives the colimit construction above on the level of (homotopy types) of motivic spaces, which is called the geometric classifying space of G and is denoted by BG (4.2, [32] ). More precisely, for any base field k, consider HMot k • , the pointed motivic homotopy category over k. For a faithful representation G × A m → A m , and the associated diagonal representations G × A im → A im . Let U i be the maximal open sub-scheme of A im on which G acts freely, and the geometric quotient

exists as a smooth scheme (Lemma 2.12). Then we have a chain of morphisms · · · → V i → V i+1 → · · · such that its colimit in HMot k • depends on G and is independent of any choice involved.

For G, V , and

. taking homotopy colimits, we have t C (BG) ∼ = B(G(C)), as well as the cycle class map

If there is a compactificationḠ(C) of the Lie group G(C), we may write

instead of (2.16).

To describe the universal property of BG, we need to work in the category HMot k N is , the homotopy category of motivic spaces with respect to the localization with respect to the Nisnevich topology.

With some general model-categorical construction ( [32] , Chapter 4), we obtain an isomorphism of functors

The classes ρ p,k and y p,k Let p be an odd prime, and n a positive integer divisible by p. In this section we recall the p-torsion classes y p,k ∈ H 2p k+1 +2 (BP U n ), and construct p-torsion classes ρ p,k ∈ CH p p+1 +1 (BP GL n ) satisfying cl(ρ p,k ) = y p,k .

In [19] and [23] , the author considered the following construction. By the definition of P U n , we have a short exact sequence

which yields a homotopy fiber sequence

As BS 1 is of the homotopy type of the Eilenberg-Mac Lane space K(Z, 2) ≃ ΩK(Z, 3), we have the Puppe sequence which extends the above to another homotopy fiber sequence

Alternatively, the map χ may be constructed as follows. Consider the short exact sequence 1 → µ n → SU n → P U n → 1, where µ n is the cyclic group of complex nth roots of unity. The sequence yields a Bockstein homomorphism

Lemma 3.1. The map χ : BP U n → K(Z, 3) represents the following composition:

The proof is a routine check. The classes y p,k are defined by means of the map χ and the cohomology of K(Z, 3). In general, the cohomology of the Eilenberg-Mac Lane space K(A, n) for A a finitely generated abelian group can be deduced from [7] . The integral cohomology of K(Z, 3) is described in [19] in terms of Steenrod reduced power operations, resembling the description of the mod p cohomology of K(A, n) by Tamanoi [36] . Instead of repeating the above results, we only presents some particular cohomology classes.

Let

be the Bockstein homomorphism,

the mod p reduction of δ, and P i the ith Steenrod reduced power operation. 

wherex 1 denote the mod p reduction of x 1 .

In [23] , the author shows the following Proposition 3.3 (Theorem 1.1, [23] ). For p | n and k ≥ 0, the classes χ * (y p,k ) ∈ H 2p k+1 +2 (BP U n ) are nontrivial.

For simplicity, we omit the notation χ * and write x 1 ∈ H 3 (BP U n ) and y p,k ∈ H 2p k+1 +2 (BP U n ) instead.

We proceed to construct a motivic counterpart of x 1 . Consider the short exact sequence of algebraic groups 1 → µ n → SL n → P GL n → 1, which induces a morphism in HMot C N is :

On the other hand, we have As shown in [32] , Chapter 4, for any Nisnevich sheaf of groups G over Sm k , there is an isomorphism We then take the following compositon, which is a morphism in HMot C • denoted by (3.9) χ M : BP GL n → K(Z/n(2), 2)

where δ is the Bockstein homomorphism. Let ζ 1 ∈ H 3,2 M (BP GL n ) be the class represented by χ M . Then ζ 1 is an n-torsion class. It is the desired motivic counterpart of x 1 , in the sense of the following

Proof. This follows immediately from Proposition 2.9 and Lemma 3.1.

In what follows, we let overhead bars indicate mod p reductions of integral (motivic and singular) cohomology classes.

Definition 3.5. For p an odd prime, p | n, and k ≥ 0, we define p-torsion classes ρ p,k := δ P p k P p k−1 · · · P p P 1 (ζ 1 )

The classes ρ p,k satisfy the properties given in Theorem 1: Proposition 3.6. For p | n, the classes ρ p,k ∈ CH p k+1 +1 (BP GL n ) satisfy cl(ρ p,k ) = y p,k .

Proof. This follows immediately from Lemma 3.4 and the functorial property of cl, and the compatibility of the Steenrod reduced power operations and cl:

where the horizontal arrows are the operations δ P p k P p k−1 · · · P 1 .

On the extraspecial p-groups p 1+2r + For an odd prime number p, a finite p-group G is called an extraspecial p-group if its center Z(G) is cyclic of order p, and the quotient G/Z(G) is a nontrivial elementary abelian p-group, i.e., an abelian group in which every nontrivial element is of order p. A particular type of extraspecial p-groups play an important role in the construction of non-toral p-elementary subgroups of P U p r .

The complete classification of extraspecial p-groups is known, by a theorem of P. Hall (Theorem 5.4.9, [16] ). In this section, we concern ourselves with only one type of extraspecial p-groups for each odd prime p. The main result of this section is Lemma 4.6.

The cohomology of the extraspecial p-groups are studied in depth by Tezuka and Yagita [38] and Benson and Carlson [6] . In this parer we merely need a partial result, which we deduce independently, for the sake of completeness.

Throughout the rest of this paper, we denote by Z(G) the center of a group G.

The orders of extraspectial p-groups are of the form p 1+2r for r > 0, and conversely, for each r > 0 we have two extra special p-groups of order p 1+2r , one of which is denoted by p 1+2r + . We present p 3 + in terms of generators and relations: (4.1)

is the cyclic group Z/p generated by z, and the quotient group p 3 + /Z(p 3 + ) is isomorphic to (Z/p) 2 , which is commutative. To study the groups p 1+2r + for r > 1, we recall the following Definition 4.1. Let G 1 , G 2 be groups such that there is an isomorphism φ : Z(G 1 ) → Z(G 2 ). The central product of G 1 and G 2 with respect to φ is

We often omit the subscript φ when it is clear from the context. In particular, we write G * G in the case that φ is the identity on Z(G).

Remark 4.2. The central product is associative and we feel free to write G 1 * φ1 G 2 * φ2 · · · * φr−1 G r , and in particular G * G * · · · * G. Definition 4.3. We define the group p 1+2r + := p 3 + * · · · * p 3 + (r-fold central product). The following is well known to group theorists, and its proof is a straightforward computation. In the rest of this paper, we use both V and Z/p as notations for the cyclic group of order p. We use the former if we consider its classifying space, and the latter if we regard it as subgroup or quotient of a (co)homology group or a Chow ring. In particular, we denote by V k the k-fold direct sum of V k .

The following is an immediate consequence of Proposition 4.4

Corollary 4.5. Let V 2r = (Z/p) 2r be the Cartesian product of cyclic groups of order p, with a basis e 1 , · · · , e r , f 1 , · · · , f r . There is a short exact sequence of groups 1 → V → p 1+2r

where Z/p maps onto Z(p 1+2r + ). 

In this section we prove that the ring homomorphisms (1.4) and (1.5) in Theorem 1 are injective, by studying the cohomology of a p-elementary subgroup of P U p r . Since we have the cycle class map cl : CH * (BP GL n ) → H * (BP U n ) with cl(ρ p,k ) = y p,k , the injectivity of (1.4) follows from that of (1.5). Hence, we will focus on the proof of (1.5) in this section.

The non-toral p-elementary subgroups of P U n and their normalizers are studied by Griess [17] (Table II) , where a systematic investigation of elementary p-subgroups of algebraic groups is carried out. Andersen, Grodal, Møller, and Viruel [2] present a more detailed discussion. For the purpose of this section, it suffices to consider the case n = p r for p an odd prime.

In the special case r = 1, much of the constructions presented in this section appears in various works such as [43] , [25] , and [29] .

We present the p-elementary subgroups of P U p r as follows. First we construct monomorphisms of Lie groupsθ : p 1+2r + ֒→ U p r , where p 1+2r + is the extraspecial p-group studied in Section 4. Passing to quotients over centers we obtain monomorphisms of the form θ : V 2r → P U p r , where V 2r = (Z/p) ⊕2r as in Section 4.

We proceed to present the monomorphismsθ : p 1+2r + ֒→ U p r . First we consider r = 1, in which case we have (4.1):

It is straightforward to check that the above indeed gives a monomorphism of Lie groups.

Taking r-fold direct produces, we obtain a homomorphism θ ×r : (p 3 + ) ×r ֒→ U ×r p ֒→ U p r , where the inclusion U ×r p ֒→ U p r is given by the canonical action of U ×r p on the r-fold tensor product of C p with the canonical Hermitian inner product. For z ∈ Z(p 3 + ), let z (i) := (1, · · · , ith z , · · · , 1) ∈ (p 3 + ) ×r . Notice that the elementθ ×r (z (i) ) is independent of i, and the above homomorphism factors through the r-fold central product and we have a homomorphism p 1+2r + ∼ = (p 3 + ) * r ֒→ U p r which is also denoted byθ. Taking the quotient group over the centers on both sides, we obtain a monomorphism

Let N (V 2r ) be the normalizer of V 2r in P U p r , and let W = N (V 2r )/V 2r . Then the group W acts upon the cohomology ring H * (BV 2r ) in such a way that the restriction homomorphism θ * : H * (BP U p r ) → H * (BV 2r ) has image in H * (BV 2r ) W , the subring of H * (BV 2r ) of W -invariants. It is therefore important to study the group W and its action on H * (BV 2r ), for which we introduce a symplectic bilinear form on V 2r .

Recall the generators z, e i , f i , 1 ≤ i ≤ r of p 1+2r + as given in Proposition 4.4. The quotient group V 2r = p 1+2r + /Z(p 1+2r + ) is generated by e i , f i . In the obvious way, we regard V 2r as a F p -vector space of dimension 2r with a basis (5.2) e 1 · · · , e r , f 1 , · · · , f r .

Let −, − be a simplectic bilinear form on V 2r , such that its matrix associated to the basis (5.2) is

The following is a special case of Theorem 8.5 of [2] .

Proposition 5.1 (Andersen-Grodal-Møller-Viruel, Theorem 8.5, [2] ). The normalizer of V 2r in P U p r is Sp r , the symplectic group over F p of order 2r, which acts on V 2r with respect to the symplectic bilinear form −, − .

Consider the cohomology algebra

Here we have a i , b i ∈ H 1 (BV 2r ; F p ), andξ i ,η i are respectively the mod p reductions of the integral cohomology classes ξ i , η i ∈ H 2 (BV 2r ) which satisfy

where δ : H * (−; F p ) → H * +1 (−) denotes the Bockstein homomorphism. In other words, we haveξ i = β(a i ) andη i = β(b i ) where β is the mod p reduction of δ. By Proposition 5.1 we have

For a suitable choice of a i , b i , 1 ≤ i ≤ r as above, and a symplectic bilinear form −, − on the F p -vector space H 1 (BV 2r ; F p ) given by the matrix Ω with respect to the basis a 1 · · · , a r , b 1 , · · · , b r , the Sp r -actions on H * (BV 2r ; F p ) and H * (BV 2r ) are described as follows. Suppose g ∈ Sp r .

• It acts tautologically as the symplectic transformations on the F p -vector space H 1 (BV 2r ; F p ) with respect to the symplectic bilinear form −, − . • For g ∈ Sp r and a ∈ H 1 (BV 2r ; F p ), we have gβ(a) = β(ga).

• For a, b ∈ H * (BV 2r ; F p ), we have g(ab) = (ga)(gb).

• For any ξ ∈ H k (BV 2r ), k > 0, there is a unique a ∈ H k−1 (BV 2r ; F p ) satisfying ξ = δ(a), and we have gξ = δ(ga).

In particular, the Bockstein homomorphism δ is Sp r -equivariant.

be the graded exterior F p -algebra generated by a 1 , · · · , a r , b 1 , · · · , b r , each of which is of degree 1, regarded as an subalgebra of H * (BV 2r ; F p ) in the sense of (5.4).

Then the Sp r -action on H * (BV 2r ; F p ) in Corollary 5.2 restricts to Λ * , and the

Proof. It is straightforward to check that the Sp r -action on H * (BV 2r ; F p ) in Corollary 5.2 restricts to Λ * . An arbitrary element in Λ 2 may be written as

for r ij , s ij , t ij ∈ F p , or more conveniently

where we have a = a 1 · · · a n , b = b 1 · · · b n and R = (r ij ), S = (s ij ), T = (t ij ) ∈ F r×r p . Hence, the class w is Sp r -invariant if and only if for any P ∈ Sp r we have

For (5.7) to hold for all

it is necessary that we have R = T = 0 and S = sI r for some s ∈ F p , which are easily verified also as a sufficient condition for (5.7). Therefore we have

Proposition 5.4. We have the invariant subgroup H 3 (BV 2r ) Sp r ∼ = Z/p, which is generated by the class δ(

Proof. The short exact sequence

induces a long exact sequence

Since the groups H k (BV 2r ) are p-torsion for k > 0, the long exact sequence breaks down to short exact sequences

for k > 0, and in particular, we have an Sp r -equivariant isomorphism induced by δ:

where the left hand side is an F p -vector space with a basis consisting of the conjugate classes of

The proposition now follows from Lemma 5.3.

Proposition 5.5. The homomorphism

is surjective. In other words, we have

for some λ ∈ Z, p ∤ λ .

Proof. By Corollary 4.5 we have the following commutative diagram: Bp 1+2r

where D is the map representing the Bockstein homomorphism

Now we have the following commutative diagram:

Let ( V E * , * * , V d * , * * ) be the integral cohomological Serre spectral sequence associated to the second row of (5.9):

By Lemma 4.6, we have

On the other hand, by the universal coefficient theorem we have

Observe that the only nontrivial entry of V E * , * 2 of total degree 2 is V E 0,2 2 , i.e., we have

By (5.11), (5.12), and (5.13), we have

. It follows from (5.14) that there is no nontrivial differential landing on V E 3,0 * . Therefore, we have /p, 2) ) ∼ = Z/p. In other words, we have a short exact sequence

On the other hand, we have

which follows by studying the differentials of the Serre spectral sequence U E s,t 2 = H s (K(Z, 3); H t (BU p r )) ⇒ H s+t (BP U p r ). For instance, see Corollary 3.4 of [19] .

Comparing (5.10), (5.16), and (5.17), we have

Compare the above and Proposition 5.4, and we conclude.

The following is not required for the proof of Theorem 1, but nontheless interesting. Proof. This follows from Lemma 4.6, (5.12), and (5.14).

Recall the classes ξ i = δ(a i ), η i = δ(b i ) ∈ H 2 (BV 2r ).

Corollary 5.7. There is a λ ∈ Z, p ∤ λ, satisfying

Proof. This is a computation involving Steenrod reduced power operations. Consider the cohomology algebra H * (BV 2r ; F p ) = Λ F/p [a 1 , · · · , a r , b 1 , · · · , b r ] ⊗ F p [ξ 1 · · · ,ξ r ,η 1 , · · · ,η r ], and recall the relations

. We recall the two most relevant of the axioms for the Steenrod reduced power operations:

• Dimension axiom:

In particular, for k ≥ 0, we have

• Cartan formula: P k (x · y) = i+j=k P i (x) · P j (y). The computation is then carried out as follows:

Bθ * (y p,k ) = Bθ * (δ P p k P p k−1 · · · P p P 1 (x 1 )) = δ P p k P p k−1 · · · P p P 1 (λ · β(

Lemma 5.8. In the polynomial algebra F p [ξ 1 , · · · ,ξ r ,η 1 , · · · ,η r ], regarded as an F p -subalgebra of H * (BV 2r ; F p ), the polynomials

are algebraically independent.

Proof. A straightforward computation shows that the Jacobian determinant of the collection of polynomials (5.18) in the variablesξ i ,η i is

which coincides with one of the canonical generators of the Dickson invariant algebra [9] of F p [ξ 1 , · · · ,ξ r ,η 1 , · · · ,η r ]. We have J = 0, since the term r i=1ξ

p r+i i occurs once and once only in its expansion, an observation made at the beginning of Section 3, Chapter III of [1] . It then follows from the partial Jacobian criterion The first part of the lemma then follows. The proof for the second part is similar, using the diagrams above withétale cohomology and motivic cohomology replaced by singular cohomology.

Proof Theorem 1. The classes ρ p,k and y p,k are constructed in Section 3 as classes in the images of

respectively. The periodicity condition (1.3), i.e., B∆ * (ρ p,k ) = ρ p,k and B∆ * (y p,k ) = y p,k for the diagonal homomorphism ∆ : P GL m → P GL n , follows from Lemma 5.9. It remains to show that that the homomorphisms (1.4) and (1.5) are injective. We break the proof into several steps.

Step 1. We prove the injectivity of (1.5) for n = p r . Consider the composite homomorphism

where the last arrow is the mod p reduction. It follows from Corollary 5.7 and Lemma 5.8 that the above homomorphism is injective in degrees above 0, and we conclude.

Step 2. We prove the injectivity of (1.5) for n = p r m, with p ∤ m. By Lemma 5.9, the homomorphism χ : H * (K(Z, 3)) → H * (BP U p r ) factors as

Hence, the homomorphism (1.5)

and we conclude from Step 1.

Step 3. We prove the injectivity of (1.4). This follows from the fact that the homomorphism (1.5) factors as

and we conclude from Step 2.

In this section we prove Theorem 2, which asserts that the subrings

are determined by the p-adic valuation of n, and Theorem 3, which asserts the existence of a nontrivial polynomial relation in ρ p,k ∈ CH * (BP GL n ) (resp. y p,k ∈ H * (BP GL n )) for n of p-adic valuation 1 and k = 0, 1, 2. Theorem 3 tells us that the role of the p-adic valuation of n is essential in Thoerem 1.

For 0 ≤ m ≤ n, we define a subgroup of SL n as follows:

Passing to quotients by centers, we obtain a subgroup P GL m,n−n of P GL n . For the rest of this section, we denote by r the p-adic valuation of n. Then there is a diagonal homomorphism

together with a left inverse, the projection map P GL p r ,n−p r → P GL p r ,

Recall the motivic class ζ 1 ∈ H 3,2 M (BP GL n ), which is represented by χ M : BP GL n → K(Z(2), 3).

Consider the short exact sequence of algebraic groups (6.3) 1 → µ p r → SL p r ,n−p r → P GL p r ,n−p r → 1.

The procedures (3.3), (3.4), and (3.9) that produce ζ 1 viaétale cohomology and the Beilinson-Lichtenbaum conjecture may be applied to P GL p r ,n−p r and yield the following morphism in HMot C • :

We denote the corresponding class by ζ ′ 1 ∈ H 3,2 M (BP GL p r ,n−p r ). 

The desired commutative diagram is obtained once we apply the Bockstein homomorphism to the second row in the diagram above.

For an algebraic group or a compact Lie group G, let T (G) denote a maximal torus of G. Then the normalizers of T (P GL p r ), T (P GL n ), T (P GL p r ,n−p r ) are respectively the inner semi-direct products      Γ p r := S p r ⋉ T (P GL p r ), Γ n := S n ⋉ T (P GL n ), Γ p r ,n−p r := S p r ,n−p r ⋉ T (P GL n ), where S p r ,n−p r = S p r × S p r ,n−p r . Therefore, we have a diagram (6.4) Γ p r Γ p r ,n−p r Γ n P GL p r P GL p r ,n−p r P GL n in which the arrows on the top row are restrictions of the ones on the bottom row. In particular, the straight arrows are inclusions and the bent ones are the projections defined by (6.2). One easily checks that the diagram (6.4), without the bent arrows, is commutative. As there are too many homomorphisms of algebraic/Lie groups in sight, we introduce the following systematic notations. For a homomorphism H → G which is clear from the context, such as one in the diagram (6.4), we write We shall now prove Theorem 2.

Theorem (Theorem 2). Let p be an odd prime and n > 1 an integer with p-adic valuation r > 0. Then the homomorphisms B∆ * restrict to isomorphisms

Proof. We only consider the case for Chow rings. The proof for the case of cohomology is verbatim. Let ρ ∈ R M (n),ρ = res P GLn P GL p r (ρ) ∈ R M (p r ), u = res P GLn Γn (ρ) ∈ CH * (BΓ n ),û = res Γn Γ p r (u) ∈ CH * (BΓ p r ). By Lemma 6.1, we have (6.5) res P GL p r P GL p r ,n−p r (ρ) = res P GLn P GL p r ,n−p r (ρ), res Γ p r Γ p r ,n−p r (û) = res Γn Γ p r ,n−p r (u). Now we have n p r u = [Γ n : Γ p r ,n−p r ](u) = tr Γ p r ,n−p r Γn · res Γn Γ p r ,n−p r (u) (Lemma 6.2) = tr Γ p r ,n−p r Γn · res Γ p r Γ p r ,n−p r (û) (6.5).

Since r is the p-adic valuation, we have n p r ∤ 0 (mod p), and by (6.6), we have u = 0 ifû = 0. The injectivity of B∆ * = res P GLn P GL p r follows from Theorem 6.3.

The following lemma is essentially due to Vistoli [43] .

Lemma 6.4. For p and odd prime, consider the subgroup of CH * (BP GL p ) of torsion classes, which we denote by CH * (BP GL p ) tor . The homomorphism

then restricts to CH * (BP GL p ) tor . The restriction

is injective.

Proof. Consider the inclusion V 2 Bθ − − → P GL p . We have the homomorphisms induced by the inclusions (6.7) res P GLp

Since CH * (BT (P GL p )) is torsion-free, (6.7) restricted to CH * (BP GL p ) tor has the following form:

It follows from Proposition 9.4 of [43] that (6.7) is injective for n = p. Therefore, so is (6.8).

We shall now prove Theorem 3.

Theorem (Theorem 3). For p and odd prime, and n > 0 an integer satisfying p | n and p 2 ∤ n, the classes ρ p,k ∈ CH * (BP GL n ) for k = 0, 1, 2, satisfy a nontrivial polynomial relation (6.9) ρ p 2 +1 p,0 + ρ p+1 p,1 + ρ p p,0 ρ p,2 = 0, and similarly for y p,k ∈ H * (BP U n ), k = 0, 1, 2, we have (6.10) y p 2 +1

p,0 + y p+1 p,1 + y p p,0 y p,2 = 0. Proof. We consider only the case for Chow rings. The case for singular cohomology follows from the existence of the cycle class map.

For n = p, a routine computation yields Bθ * (ρ p 2 +1 p,0 + ρ p+1 p,1 + ρ p p,0 ρ p,2 ) = 0, and the desired result follows from Lemma 6.4. The general case follows from Theorem 2. For n = p, it is shown by Vistoli (Proposition 5.4, [43] ) that the latter is generated, as a ring, by Bθ * (ρ p,0 ) and a class q satisfying Bθ * (ρ p,0 )q = Bθ * (ρ p,1 ).

To conclude this section, we observe that Theorem 3 provides an obstruction to the reduction of principal P GL n -bundles. Recall that by the end of Section 2 we present an isomorphism of functors With the right-hand side of the above passing to HMot C • , we obtain a natural transformation Θ : H 1 et (−; G) → Hom HMot C • (−, BG). Proposition 6.6. Let X be a smooth scheme over C, P anétale principal P GL nbundle over X, with p 2 | n, and f = Θ(P ) : X → BP GL n the associated morphism in HMot C • . Letρ p,k = f * (ρ p,k ) ∈ CH * (X). Let m > 0 be an integer satisfying p | m, p 2 ∤ m and m | n. If theétale principal P GL n -bundle P may be reduced to a principal P GL m -bundle via the diagonal map ∆ * : P GL m → P GL n , then we havẽ

p,0 +ρ p+1 p,1 +ρ p p,0ρ p,2 = 0. A parallel assertion holds for X a CW complex, P a (topological) principal P U nbundle, and the cohomology classesỹ p,k = f * (y p,k ).

Proof. The proofs for the cases of schemes and CW complexes are parallel, and we only present the proof for schemes.

By the functorial property of Θ, theétale principal P GL n -bundle P may be reduced via ∆ : P GL m → P GL n only if f : X → BP GL n factors through the diagonal map ∆ * : P GL m → P GL n as

Therefore, we havẽ

p,0 +ρ p+1 p,1 +ρ p p,0ρ p,2 = g * · B∆ * (ρ p 2 +1 p,0 + ρ p+1 p,1 + ρ p p,0 ρ p,2 ). By Theorem 3, we have B∆ * (ρ p 2 +1 p,0 + ρ p+1 p,1 + ρ p p,0 ρ p,2 ) = 0, and we conclude.

The period-index problem originally concerns the Brauer group of a field k and the degrees of central simple algebras over k, which is then generalized to the Brauer group of a scheme and the degrees of Azumaya algebras over it. For more backgrounds on the period-index problem, see [15] and [18] . Antieau and Williams [3] , [4] are the first to consider the topological version of the period-index problem.

The cohomology of BP U n plays a key role in the study of the topological periodindex problem, as demonstrated in [4] and [20] . We refer the reader to [3] and [4] for the background of the topological period-index problem. In a nutshell, it concerns a finite CW-complex Y equipped with a cohomology class α ∈ H 3 (Y ) and the greatest common divisor of all positive integers n such that there is a homotopy commutative diagram

In this case we say that the principal P U n -bundle P realizes the class α. Notice that such a class α is an n-torsion class, and for this reason we define the topological Brauer group of Y to be the subgroup of torsion classes of H 3 (Y ), and an element in this group a (topological) Brauer class of Y . The torsion order of α ∈ H 3 (Y ) is called the period of α and denoted by per(α). The greatest common divisor of all n such that a homotopy commutative diagram of the form (7.1) exists is called the index of α and denoted by ind(α). Similarly, we may consider the period-index problem for motivic spaces andétale P GL n -torsors. We may call the torsion subgroup of H 3,2 M (X) the motivic Brauer group of X and call an element of the motivic Brauer group of X a motivic Brauer class of X. However, since the natural map H 1 et (X; P GL n ) → Hom HMot C • (X, BP GL n ) is not in general a bijection, the lifting problem in the homotopy category HMot C • :

is not equivalent to the problem of finding a P GL n -torsor over X representing α ′ . Yet in this section we are able prove an interesting result by working only in the A 1 -homotopy category HMot C • . The torsion order of α ′ is called the period of α ′ and denoted by per(α ′ ), and the greatest common divisor of all n such that there is a homotopy commutative diagram of the form (7.2) is called the index of α ′ and denoted by ind(α ′ ).

So far, the main examples for per(α) = ind(α) are 2d-skeletons of the Eilenberg-Mac Lane spaces K(Z/m, 2) with a cell decomposition. See [4] , [20] and [22] . In what follows we suggest an alternative source of examples.

Consider the non-toral p-elementary subgroup V 2r of P U p r and the map θ : V 2r → P U p r defined in (5.1). Recall the generator x 1 of H 3 (BP U p 2 ), and similarly we have the motivic Brauer class of BP GL p 2 ζ 1 ∈ H 3,2 M (BP GL p 2 ). Finally, we define α := Bθ * (x 1 ) ∈ H 3 (BV 4 ), α ′ := Bθ * (ζ 1 ) ∈ H 3,2 M (BV 4 ). For the indices, notice that there is a canonical map BV 4 → BP GL n as a morphism in the category of simplicial presheaves, and we have the pullback of the universaĺ etale principal bundle over BP GL n . Therefore, we have

On the other hand, suppose we have anétale principal P GL n -torsor P over BV 4 representing the class α. Then we have a homotopy commutative diagram BP GL n BV 4 K(Z(2), 3). for p | n and p 2 ∤ n. This implies Θ(P ) * (ρ p 2 +1 p,0 + ρ p+1 p,1 + ρ p p,0 ρ p,2 ) = 0, which is absurd, by Theorem 3. The argument for α ∈ H 3 (BV 4 ) is similar, and we have In this section we prove Theorem 4:

Theorem (Theorem 4). Let n > 1 be an integer, and p one of its odd prime divisor. Then the ring CH * (BP GL n ) (p) is not generated by Chern classes. More precisely, the class ρ i p,0 is not in the Chern subring for p − 1 ∤ i. For a base field F, we have the Jacobian criterion for the algebraic independence of a collection of polynomials {ϕ i } in the polynomial ring F[x 1 , · · · , x n ], which is well known to hold in the case that the base field has characteristic 0, or sufficiently large characteristics relative to the degrees of {ϕ i }. We establish a partial Jacobian criterion in the same vein over perfect fields of positive characteristics, which plays a key role in the proof of Lemma 5.8. The criterion may be deduced from, for example, Corollary 16.17 and Corollary A1.7 of Eisenbud [13] . For completeness and simplicity we present an alternative proof.

Proposition A.1. Consider the polynomial algebra F[x 1 , · · · , x n ], where F is a perfect field of characteristic p > 0. Let ϕ 1 , · · · , ϕ m ∈ F[x 1 , · · · , x n ], m ≤ n be polynomials such that the Jacobian matrix (∂ϕ j /∂x i ) ij is of rank m. Then ϕ 1 , · · · , ϕ m are algebraically independent.

Proof. Suppose ϕ 1 , · · · , ϕ m are algebraically dependent. Let f (y 1 , · · · , y m ) be the nontrivial polynomial of the lowest degree such that we have f (ϕ 1 , · · · , ϕ m ) = 0.

Since the Jocobian matrix is of full rank, we have ∂f /∂ϕ i = 0 for all i. Therefore, we have f (ϕ 1 , · · · , ϕ m ) = g(ϕ p 1 , · · · , ϕ p m ) for some polynomial g(z 1 , · · · , z m ) = i1,··· ,im a i1,···im z i1 1 · · · z im m .

Since F is a perfect field of characteristic p > 0, we have b i1,··· ,im ∈ F satisfying b p i1,··· ,im = a i1,··· ,im .Let g(w 1 , · · · w m ) = i1,··· ,im b i1,···im w i1 1 · · · w im m = 0.

Then we have 0 = f (ϕ 1 , · · · , ϕ m ) = g(ϕ p 1 , · · · , ϕ p m ) = i1,··· ,im (b i1,···im ϕ i1 1 · · · ϕ im m ) p =ḡ(ϕ 1 , · · · , ϕ m ) p .

Therefore,ḡ(ϕ 1 , · · · , ϕ m ) = 0 is a nontrivial polynomial relation for ϕ 1 , · · · , ϕ m , and the polynomialḡ has degree lower than that of f , a contradiction. Therefore, ϕ 1 , · · · , ϕ m are algebraically independent.

Cohomology of finite groups

The classification of p-compact groups for p odd

The period-index problem for twisted topological K-theory

The topological period-index problem over 6-complexes

Séminaire de géométrie algébrique du bois-marie 1963-1964

The cohomology of extraspecial groups

Algebre de Eilenberg-MacLane et homotopie, Secrétariat mathématique

Anomalies in the space of coupling constants and their dynamical applications II

A fundamental system of invariants of the general modular linear group with a solution of the form problem

Universal homotopy theories

Motivic homotopy theory: lectures at a summer school in Nordfjordeid

Equivariant intersection theory (with an appendix by Angelo Vistoli: The Chow ring of M 2 )

Commutative Algebra: with a view toward algebraic geometry

Dai-Freed anomalies in particle physics

Central simple algebras and Galois cohomology

Finite groups

Elementary abelian p-subgroups of algebraic groups

Le groupe de Brauer. I. algebres d'Azumaya et interprétations diverses. Dix exposés sur la cohomologie des schémas

On the cohomology of the classifying spaces of projective unitary groups

The topological period-index problem over 8-complexes, I

On the Brown-Peterson cohomology of BP Un in lower dimensions and the Thom map

The topological period-index problem over 8-complexes

Some torsion classes in the Chow ring and cohomology of BP GLn

From algebraic cobordism to motivic cohomology

On the integral Tate conjecture over finite fields

Chern subrings

On the cohomology of the classifying spaces of P SU (4n + 2) and P O(4n + 2)

Cohomology of classifying spaces of certain associative H-spaces

Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups

Lecture notes on motivic cohomology

On the Chow rings of classifying spaces for classical groups. Rendiconti del Seminario Matematico della Università di Padova

A 1 -homotopy theory of schemes

Geometric invariant theory

On Voevodsky's algebraic K-theory spectrum

Homotopical algebra

Q-subalgebras, Milnor basis, and cohomology of Eilenberg-MacLane spaces

Chern classes are not enough

The varieties of the mod p cohomology rings of extra special p-groups for an odd prime p

Cohomology of classifying spaces

The Chow ring of a classifying space

On the mod p cohomology of BP U (p)

On the Chow ring of the classifying stack of P GL 3

On the cohomology and the Chow ring of the classifying space of P GLp

Voevodsky's Seattle lectures: K-theory and motivic cohomology

Motivic cohomology with Z/2-coefficients

Reduced power operations in motivic cohomology