Some torsion classes in the Chow ring and cohomology of BPGLn

In the integral cohomology ring of the classifying space of the projective linear group PGLn (over C ), we find a collection of p ‐torsion classes yp,k of degree 2(pk+1+1) for any odd prime divisor p of n , and k⩾0 . If, in addition, p2∤n , there are p ‐torsion classes ρp,k of degree pk+1+1 in the Chow ring of the classifying stack of PGLn , such that the cycle class map takes ρp,k to yp,k . We present an application of the above classes regarding Chern subrings.


Introduction
Let G be an algebraic (resp.topological) group and BG be the classifying stack (resp.space) of G.We follow Vistolis's notations ( [30], 2007) and let A * G (resp.H * G ) denote the Chow ring (resp.integral cohomology ring) of BG.
The Chow ring of BG is first introduced by Totaro ( [27], 1999), as an algebraic analog of the integer cohomology of the classifying space of a topological group.Based on this work, Edidin and Graham ( [9], 1997) developed equivariant intersection theory, of which the main object of interests is the equivariant Chow ring, an algebraic analog of Borel's equivariant cohomology theory.
The Chow rings A * G has been computed by Totaro ([27], 1999) for G = GL n , SL n , Sp n .For G = SO 2n by Field ( [11], 2004), for G = O n and G = SO 2n+1 by Totaro and Pandharipande ([27], 1999 and [21], 1998), for G = Spin 8 by Rojas ( [22], 2006), for the semisimple simply connected group of type G 2 by Yagita ( [33], 2005).The case that G is finite has been considered by Guillot ([16] and [17], 2004).In [23] (2006), Rojas and Vistoli provided a unified approach to the known computations of A * G for the classical groups G = GL n , SL n , Sp n , O n , SO n .Let P GL n be the quotient group of GL n over C, modulo scalars.The case G = P GL n appears considerably more difficult.The mod p cohomology for some special choices of n is considered in [18] (Kameko and Yagita, 2008), [20] (Kono and Mimura, 1975), [28] (Vavpetic and Viruel, 2004) and [19] (Kameko, 2016).In [15] (Gu, 2016), the author computed the integral cohomology of P GL n for all n > 1 in degree less then or equal to 10 and found a family of distinguished elements y p,0 of dimension 2(p + 1) for any prime divisor p of n and any k ≥ 0. (In [15] the author actually considered the projective unitary group P U n , of which the classifying space is homotopy equivalent to that of GL n .)In [30], Vistoli determined the ring structures of A * P GLp and H * P GLp where p is an odd prime.Classes in the rings A * P GLn (resp.H * P GLn ) are important invariants for sheaves of Azumaya algebras of degree n (resp.principal P GL n -bundles.)For instance, the ring H * P GLn plays a key role in the topological period-index problem, considered by Antieau and Williams ( [3] and [4], 2014), Gu ([15], 2017) and Crowley and Grant ([8], 2018).This sugests that the A * P GLn could be of use in the study of the (algebraic) period-index problem ( [7], J.-L.Colliot-Thélène, 2002).The cohomology ring H * P GLn also appears in the study of Dai-Freed Anamolies in particle physics ( [13], García-Etxebarria and Montero, 2018).
In this paper we study the rings A * P GLn and H * P GLn .One may easily find H i P GLn = 0 for i = 1, 2 and H 3 P GLn ∼ = Z/n.Therefore we have a map representing the "canonical" (in the sense to be explained in Section 3) generator x 1 of H 3 P GLn , where K(Z, 3) denotes the Eilenberg-Mac Lane spaces with the 3rd homotopy group Z.
Let p be a prime number.It may be deduced from [6] (Cartan andSerre, 1954-1955) and reviewed in Section 2 that the p-local cohomology ring of K(Z, 3) is generated by the fundamental class x 1 and p-torsion elements of the form y p,I , where I = (i m , • • • , i 1 ) is an ordered sequence of positive integers i Here m is called the length of I.For I = (k), we simply write y p,k for y p,I .The degree of y p,I is deg(y p,I ) = m j=1 (2p ij +1 + 1) + 1.
In [15] the author showed that the images of y p,0 in H 2(p+1) P GLn via X are nontrivial p-torsions, for all prime divisors p of n.In this paper we generalize this to y p,k for k > 0 and (partially) to Chow rings.For simplicity we omit the notation X * and let y p,I denote X * (y p,I ) whenever there is no risk of ambiguity.
(1) In H 2(p k+1 +1) P GLn , we have p-torsions y p,k = 0 for all odd prime divisors p of n and k ≥ 0.
(2) If, in addition, p and n satisfy then there are n-torsion elements ρ p,k ∈ A p k+1 +1 P GLn which maps to y p,k via the cycle class map.Remark 1.1.1.The following are a few cases such that the condition (1.1) is satisfied.
(2) Using Mathematica, the author was able to find, that out of the first 10, 000 odd prime numbers p there are 5, 016 such that for all n satisfying p|n, the condition (1.1) holds.These odd primes starts with 3,7,11,19,23,31  One the other hand, the classes y p,I for I of length greater then 1 seem hard to capture, and so are their counterparts in Chow rings (if any).Nonetheless we are able to verify the following Theorem 1.2.Let p be a (not necessarily odd) prime divisor of n.Suppose that I is of length greater than 1.Then in H * P GLn , we have y p,I = 0 if p 2 ∤ n.In Section 2 we recall the cohomology of the Eilenberg-Mac Lane space K(Z, 3).In particular, we consider the actions of the Steenrod reduced power operations.
In Section 3 we recall a homotopy fiber sequence and the associated Serre spectral sequence converging to H * P GLn , which is the main object of interests in [15] (Gu, 2016).Moreover, we prove Theorem 1.2 in this section.
Section 4 is a recollection of elements in equivatiant intersection theory related to the topic of this paper.In Section 5 we review work of Vezzosi and Vistoli on the cohomology and Chow ring of BP GL p where p is an odd prime.Moreover, we bring the Steenrod reduced power operations into the picture.We prove part (1) of Theorem 1.1 in this section.
In Section 6 and Section 7 we construct the torsion classes ρ p,k in the Chow ring of BP GL n for p|n satisfying (1.1), completing the proof of Theorem 1.1.
The appendix concerns odd primes p such that for all n divisible by p, the condition (1.1) holds.

The Cohomology of the Eilenberg-Mac Lane space K(Z, 3)
The cohomology of the Eilenberg-Mac Lane space K(Z, 3) can be deduced from the notes [6] by Cartan and Serre, and described by the author in [15] in full details.Proposition 2.1 ([15], Proposition 2. 16,2016).The graded ring structure of H * (K(Z, 3); Z) is described in terms of generators of and relations in C(3) as follows: where p runs over all prime numbers, and p H * (K(Z, 3); Z) is a free graded-commutative Z/p-algebra without unit, generated by the elements y p,I where and the degree of y p,I is m j=1 (2p ij +1 + 1) + 1.In particular, when taking I = (i), we have y p,i := y p,I of degree 2p i+1 + 2.
Here x 1 is the fundamental class, i.e, the class represented by the identity map of K(Z, 3).Proposition 2.1 implies, in particular, that any torsion element in H * (K(Z, 3); Z) is of order p for some prime number p.This means that the torsion elements of H * (K(Z, 3); Z), namely all elements above degree 3, are in the image of the Bockstein homomorphism So we to present the classes y p,k in such a way.
For a fixed odd prime p, let P k be the kth Steenrod reduced power operation and let β be the Bockstein homomorphism H * (−; Z/p) → H * (−; Z/p) associated to the short exact sequence For an axiomatic descreibtion of the Steenrod reduced power operations, see [25] ( Steenrod and Epstein, 1962).
To compute the cohomology ring H * (K(Z, 3); Z/p)), it suffices to consider the mod p cohomological Serre spectral sequence associated to the homotpy fiber sequence K(Z, 2) → P K(Z, 3) → K(Z, 3) where P K(Z, 3) is the contractible space of paths in P K(Z, 3).Indeed, an inductive argument on the transgressive elements (Section 6.2 of [24], McCleary, 2001) yields the following Proposition 2.2.For any odd prime p, we have where Λ Z/p [x p,0 , xp,0 , • • • ] is the exterior algebra over Z/p over elements xp,k := P p k P p k− Remark 2.4.The alert reader may argue that Proposition 2.2 merely indicates for some λ ∈ (Z/p) × .However, notice that Proposition 2.1 determines y p,k only up to a scalar multiplication.Hence we may as well choose y p,k such that Proposition 2.3 holds.Proposition 2.3 has the following variation.Proposition 2.5.For k ≥ 1, we have ȳp,k = P p k (ȳ p,k−1 ).
Proof.Recall that for positive integers a, b such that a ≤ pb, we have the Adem relation (Adem, [1]) (2.1) For k > 0, let a = p k and b = p k−1 .Then the only choice of i to offer something nontrivial on the right hand side of (2.1) is i = p k−1 , and (2.1) becomes (2.2) Then it follows by induction that we have Since all classes y p,k are of order p, as stated in Proposition 2.3, we conclude the proof.
3. A Serre Spectral Sequence, Proof of Theorem 1.2 In the introduction we mentioned the map representing the "canonical" class in H 3 P GLn .It is easy to show, for example, in the introduction of [15] (Gu, 2016), that the homotopy fiber is BGL n and there is a homotopy fiber sequence where the first arrow is the obvious projection.Indeed, this may be obtained from the more obvious homotopy fiber sequence by de-looping the first term BC × ≃ K(Z, 2).Indeed, the author used this delooping as the definition of X, and the class x 1 is "canonical" in this sense.
For positive integers r > 1 and s we have ∆ : BP GL r → BP GL rs the obvious diagonal map.The fiber sequences (3.1) and (3.2) then shows the following Lemma 3.1.The following diagram commutes up to homotopy.
As mentioned in the introduction, we omit the notation X * and let y p,I denotes their pull-backs in H * P GLn .If we temporarily denote y p,I by y p,I (r) and y p,I (rs), respectively in the cases n = r and n = rs, then Lemma 3.1 indicates ∆ * (y p,I (rs)) = y p,I (r).In view of this, we do not make the effort of distinguishing y p,I (r) and y p,I (rs), and denote both by y p,I .
Consider the homotopy commutative diagram where P K(Z, 3) is the pointed path space of K(Z, 3), which is contractible, and ϕ is the diagonal map.Moreover, all rows are homotopy fiber sequences and we let Φ and Ψ in the diagram denote morphisms of homotopy fiber sequence.
Remark 3.3.As mentioned in the introduction, the diagram (3.3) is indeed a variation of the one considered in [15], where the groups U n , P U n and the unit circle take the places of GL n , P GL n and C × , respectively.
We take notes of the following notations: where v is of degree 2, and where each v i is of degree 2, and for all i we have Moreover, we have where c i , the ith universal Chern class, is of degree 2i, and Bψ * takes c i to the ith elementary symmetric polynomial in v i 's.
As in [15], we let . This spectral sequence is the main object of interest in [15] (Gu, 2016).In principle, using the homological algebra of differential graded algebras, we are able to determine all the differentials of K E via the following Proposition 3.4 (Gu, Proposition 3.8, [15], 2016).Restricted to symmetric poly- In other words, for each bi-degree (s, t), the first nontrivial differential landing in U E s,t * is determined by restricting T d * , * * to U E * , * * .It is known (Theorem 1.2, [15]) that for a prime p, the class y p,0 ∈ H 2(p+1) P GLn is a nontrivial p-torsion if p|n and is 0 otherwise.Since y p,0 is the p-torsion in H * (K(Z, 3); Z) of the lowest degree (2(p + 1)), and generates the unique p-primary subgroup of 2(p + 1).We have the following Proposition 3.5.If p|n, then the class y p,0 is the nontrivial p-torsion in H * P GLn of the lowest degree, and generates the unique p-primary subgroup of 2(p + 1).If p ∤ n the class y p,0 is trivial in H * P GLn .Proof.The class y p,0 being (non)trivial is just Theorem 1.2 of [15].The uniqueness assertion follows by looking at We proceed to consider the classes y p,I for I of length greater than 1, and prove Theorem 1.2.It follows from Corollary 2.18 of [15] that for each . Now we have all the necessary ingredient for the following We denote is in the image of U d * , * r for some r < 2p im + 1, then there is nothing left to prove.Otherwise, let It follows from Proposition 3.4 and (3.7) that we have Since p 2 ∤ n, we have n p ≡ 0 (mod p) and we conclude.

Equivariant Intersection Theory
Equivatiant intersection theory concerns algebraic spaces with an action of an algebraic group.The main object of interests is the equivariant Chow ring A * G (X) for an algebraic space X with an action of an algebraic group G.This is an analog of Borel's equivariant cohomology theory for algebraic spaces with actions of algebraic groups, of which the construction was made possible in [27] (Totaro, 1999), where he defined the Chow ring A * G of the classifying stacks of an algebraic group G, which is isomorphic to the equivariant Chow ring A * G (spec K), where K is the underlying field, which we assume to be C for the rest of this paper.This ring plays the role of the coefficient ring in Borel's equivariant cohomology theory.See [9] (Edidin and Graham, 1996) for a more comprehensive survey of equivariant intersection theory.
Most of the constructions and properties of equivariant Chow rings needed in this paper can be found in Section 4 of [30] (Vistoli, 2007).We briefly recollect them here.From now on we work with a fixed base field K and the reader is free to assume K = C.
Below an arbitrarily large degree k, the equivariant Chow ring A ≤k G (X) is (by definition) isomorphic to A ≤k (X ′ ) for another variety X ′ .Therefore, just as in the non-equivariant Chow theory, we have the following two properties.(See, for example, [12], Fulton, 1998.)Proposition 4.1.(Homotopy Invariance) Let X → B be a G-equivariant vector bundle over an algebraic space B. Then the induced homomorphism Let f : Y → X be a morphism of G-schemes, then we have the pull-back In the particular clase where X = V is a G-representation of dimension n, and Y = {0} ⊂ V .Then we have the following Proposition 4.3.We have the following exact sequence ) → 0 where the first arrow is the multiplication of the Chern class c n (V ).In particular, it follows from Proposition 4.1 that we have This is no longer a ring homomorphism, but a homomorphism of A * G -modules, in the sense that we have the following projection formula: where the righthand side is the index of H in G. Therefore, we have Similar to the Cartan-Eilenberg double coset formula ([2], Adem and Milgram, 1994), we have the Mackey's formula concerning the transfer and the restriction described above.Once again we adopt the setup appeared in Vistoli's work [30]: Let G be an algebraic group and H, K be algebraic subgroups of G such that H has finite index over G.We will also assume that the quotient G/H is reduced, and a disjoint union of copies of spec K (this is automatically verified when K is algebraically closed of characteristic 0).Furthermore, we assume that every element of (K\G/H)(K) is in the image of some element of G(K).
Let C be a set of representatives of classes of the double quotient K\G/H(K).For each s ∈ C , let There is another way to relate equivariant Chow rings over different algebraic groups.Suppose we have a monomorphism of algebraic groups H → G. Let H act on a scheme X.Then we have the G equivaiant algebraic space G × H X, which is the quotient G × X/ ∼ where the equivalence relation "∼" is defined by (g, hx) ∼ (gh, x) for all h ∈ H.
We proceed to discuss the relation of an algebraic group G (over C) and its maximal torus T (G).Let W be the Weyl group of T (G) in G.
Notice that the analog of Proposition 4.6 for integral cohomology holds as well.(And perhaps better known.) sends torsion-free elements to torsion-free elements.
Let N G be the normalizer of the maximal torus T (G).It is well-know (Gottlieb, [14], 1975) that the restriction homomorphism H * G → H * NG is injective.The analog conclusion holds for Chow rings.We proceed to discuss the cycle class map and Steenrod reduced power operations on Chow rings.Recall that for an algebraic variety X over C we have the cycle class map functorial in X.For a prime number p, it reduces to Fix an odd prime p and a base field K with character different from p.In [5] (2003), Brosnan constructed operations functorial in X, satisfying the Adem relations similar to the Steenrod reduced power operations P i .Moreover, when K = C, the operation S i agrees with P i : Proposition 4.9 (Brosnan, Corollary 9.12, [5], 2003).Let K = C and X be an algebraic variety over C, we have the following commutative diagram: Voevodsky ( [31], 2003) has defined similar operations for motivic cohomology theory, which restricts to Chow rings in the sense that the kth Chow ring functor A k (−) coincides with the motivic cohomology functor H 2k,k mot (−; Z) (Voevodsky, [32], 2002).

The Chow Ring and Cohomology of BP GL p
This section is a recollection of works of Vezzosi and Vistoli ([29], [30]) on the Chow ring and integral cohomology of BP GL p for p an odd prime, together with a few further observations.
Recall that the Weyl group of T (P GL p ), the maximal torus of P GL p , is the permutation group S p , which acts on T (P GL p ) by permuting the diagonal entries.Elements in A * P GLp therefore restrict to (A * T (P GLp) ) Sp , the subgroup of classes fixed by S p .In Vistoli's paper [30], the Chow ring A * P GLp is given an (A * T (P GLp) ) Spalgebra structure via a splitting injection Vistoli fully determined A * P GLp in terms of the above (A * T (P GLp) ) Sp -algebra structure.We partially state his result as follows: Theorem 5.1 (Vistoli, [30], 2007).The (A * T (P GLp) ) Sp -algebra A * P GLp is generated by an element ρ p ∈ A p+1 P GLp of additive order p.
Remark 5.2.See Section 3 of [30] for the complete version of the theorem.
To prove Theorem 5.1, Vistoli considered two elements of P GL p , represented respectively by the matrices where ω is a pth root of unity.They generate two subgroups of P GL p , both cyclic of order p, which we denote by C p and µ p , respectively.Furthermore, the two matrices commute up to a scalar ω, which means they commute in P GL p .
Therefore we obtain an inclusion of algebraic groups C p × µ p ֒→ P GL p , which factors as where the two terms in the middle are the obvious semi-direct products.
Remark 5.3.As pointed out by Vistoli (Remark 11.4, [30]), the element ρ p depends on the choice of the pth root of unity ω.Indeed, one readily verifies that, for a given choice of ω and the corresponding ρ p , other choices of ω, say ω ′ , corresponds to λρ p for λ running over (Z/p) × .
One readily verifies (5.2) where ξ and η, both of degree 1, are respectively the restrictions of the canonical generators of A * Cp and A * µp via the projections.Vistoli showed the following Proposition 5.4 (Vistoli, Proposition 5.4, [30], 2007).The Weyl group of C p × µ p in P GL p is isomorphic to SL 2 (Z/p), of which the action on A * Cp×µp is determined by its action on linear combinations of ξ and η by linear transformations.Furthermore, the ring of invariants (A * Cp×µp ) SL2(Z/p) is generated by Vistoli constructed the class ρ p by lifting r successively via the restrictions associated to the chian of inclusions (5.1).In particular, ρ p restricts to r.Here we present two steps in the lifting process: is an isomorphism.Here W is the Weyl group of C p ⋉ T (P GL p ) in S p ⋉ T (P GL p ).
The integer cohomology of C p × µ p also plays an important role.
Proposition 5.7.We have where ξ and η are of degree 2, and are the images of elements in A * Cp×µp denoted by the same letters, via the cycle class map, whereas x 1 is of degree 3. Furthermore, for k ≥ 0 we have where B is the connecting homomorphism H * (−; Z/p) → H * +1 (−; Z) and ζ is the mod p reduction of ζ, and an integral cohomology class with an overhead bar denotes its mod p reduction.
Proof.It is standard homological algebra that we have in A * Cp×µp ⊗ Z/p.Another essential ingredient of Vistoli's work is the stratification method, which appears in Vezzosi's work [29] that studies A * P GL3 .Following Vistoli ([30]), we adopt the following notation of a stratification: Given an algebraic group G and a (complex If we can obtain generators for some A * G (V ≤i ), then by induction on i and using the localization sequence One of the advantages of working with stratifications is that it may enable us to simplify the group G.For example, for any integer n > 1, consider the P GL nrepresentation sl n of trace-zero n × n matrices on which P GL n acts by conjugation.Similarly, we have a representation D n of the group Γ n := S n ⋉ T (P GL n ), defined as the n × n diagonal trace-zero matrices, on which Γ n acts by conjugation.Let sl * n (resp.D * n ) be the open subvariety of sl n (resp.D * n ) of matrices with n distinct eigenvalues.Then we have the following Proposition 5.9 (Vezzosi, Proposition 3.1, [29], 2000).The composite of natural maps Remark 5.10.Vezzosi stated this proposition only for n = 3, but his proof works for any n > 1.Indeed, his proof is essentially an application of Proposition 4.5, taking G = P GL n , H = Γ n and X = D * n .Based on Proposition 5.9, Vistoli proved the following more refined result when n is an odd prime p.

Indeed, via the localization sequences
we make the following identifications: (5.3) Vistoli then showed the following Lemma 5.12 (Vistoli, within the proof of Proposition 10.1, [30], 2007).All arrows (obvious restriction maps) in the following diagram are ring isomorphisms: The following lemma is also due to Vistoli, though not explicitly stated.
s are all distinct, and each L i \{0} is contained in U , and (b) U contains a point that is fixed under T (P GL p ).

Then the restriction homomorphism
Remark 5.14.Vistoli stated in Lemma 10.2, [30], that under these conditions, the restriction homomorphism A * Cp⋉T (P GLp) (W \{0}) → A * Cp⋉T (P GLp) (U ) is an isomorphism.Lemma 5.13 has the following consequence that will be needed later.Recall the Γ n -representation D n and its open subvariety D * n .When p|n, consider D n as a C p × µ p -representation via the composition where ∆ is the diagonal homomorphism.

On Permutation Groups S p × S n−p and S n
This section is a technical prerequisite for the construction of the class ρ p,0 ∈ A p+1 P GLn , to be presented in Section 7.
Let p denote an odd prime as usual, while n is a positive integer such that p|n and p < n.The canonical actions of S p on the set {1, 2, • • • , p} and S n−p on {p + 1, p + 2 • • • , n} identifies S p,n−p := S p × S n−p as a subgroup of S n .Remark 6.1.In the context of this paper, particularly this and the next section, it is sometimes helpful to regard the permutation group S n as the subgroup of GL n of permutation matrices acting on column vectors e i that forms the canonical basis of C: from the left.(Here "T" means transverse, of course.)More precisely, If s ∈ S n , then as a permutation matrix, s satisfies (6.1) se j = e s(j) .
We will freely let elements in S n acts either on the vectors e i 's, or numbers 1 ≤ i ≤ p, without further explanation.For example, the subgroup S p,n−p of S n consists of matrices of the form where A and B are permutation matrices of dimensions n−p, respectively.For instance, when n = 2p, the subgroup S 2 ⋉ S p,p is the union of 2p × 2p permutation matrices of the form A 0 0 B and 0 A B 0 where A and B are p × p permutation matrices.
We proceed to consider the double quotient S p,n−p \S n /S p,n−p .Lemma 6.2.The left quotient set S n /S p,n−p is in 1-1 correspondence to linear subspaces of dimension p in C n spanned by vectors e i .In particular, we have the index [S n : S p,n−p ] = n p .Moreover, the canonical action of S p,n−p on the left quotient set S n /S p,n−p is trivial.In particular we have the following isomorphism of sets: Proof.In view of Remark 6.1, the left quotient S n /S p,n−p is very similar to the construction of Grassmannians in, for example, Chapter 4 of [26] (Switzer, 1975).Indeed, one readily verifies that two permutation matrices A, B ∈ S n represents the same coset in S n /S p,n−p if and only if the first p rows of A and the first p rows of B span the same linear subspace of C n , and it follows that the left quotient set S n /S p,n−p is in 1-1 correspondence to p-dimensional linear subspaces in C n spanned by vectors of the form e i .
For the assertion on the double quotient, consider a matrix in S p,n−p of the form A 0 0 B as described in Remark 6.1.Then it acts on a coset [M ] ∈ S n /S p,n−p (represented by some M ∈ S n ) by left multiplication of A with the first p rows of M .This does not change the linear subspace generated by these rows, and we conclude that the action is trivial.
To apply Mackey's formula (Proposition 4.4) in Section 7, we need to consider the subgroups of the form sS p,n−p s −1 S p,n−p where s ∈ S n runs over a set of representatives of the double quotient S p,n−p \S n /S p,n−p .
In view of Lemma 6.2, we identify S p,n−p \S n /S p,n−p with S n /S p,n−p , and furthermore, identify S n /S p,n−p with p-linear subspaces of C n spanned by e i 's.Lemma 6.3.Let s ∈ S n represent a left coset of S n /S p,n−p corresponding to a p-subspace F , in the sense of Lemma 6.2 Then we have where is the subgroup preserving F .Here we regard vectors in F as column vectors so any s ∈ S n acts upon them by left multiplication.In particular, if F contains exactly k vectors of the form e i for i > p, then we have the index Proof.Suppose that the p-subspace F corresponds to a coset in S n /S p,n−p represented by s ∈ S n .It follows form Lemma 6.2 that F is determined by the first p rows of s, namely e T s −1 (1) ,• • • ,e T s −1 (p) (because of (6.1)).We proceed to show the "⊆" part of (6.2).Suppose t ∈ sS p,n−p s −1 S p,n−p , i.e., t ∈ S p,n−p , and t = st 0 s −1 for some s ∈ S n and t 0 ∈ S p,n−p .Then for any 1 ≤ j ≤ p, we have st 0 s −1 (j) = k, 1 ≤ k ≤ p, or equivalently, t 0 (s −1 (j)) ∈ {s −1 (k)} k≤p , for any j ≤ p.In other words, the permutation t 0 preserves F .On the other hand, t 0 ∈ S p,n−p , and we conclude t 0 ∈ H F .Hence "⊆" holds.
To show the "⊇" part of (6.2), suppose t = st 0 s −1 with s ∈ S n and t 0 ∈ H F .Then for 1 ≤ j ≤ p, we have 1 ≤ k ≤ p such that or equivalently st 0 s −1 (j) = k.In other words, we have t = st 0 s −1 ∈ S p,n−p .Since H F is a subgroup of S p,n−p , we have t 0 ∈ S p,n−p and therefore t ∈ sS p,n−p s −1 , and we deduce "⊇".
For the assertion on the index[S p,n−p : sS p,n−p s −1 S p,n−p ], we first compute the order of the group H F .If F is represented by s such that k members of {s −1 (j)} j≤p is greater then p, then an element of H F acts separately on 4 subsets of {1, • • • , n}:  We are finally prepared to construct the promised torsion classes ρ p.k ∈ A * P GLn .Throughout this section, p will be an odd prime and n a positive integer such that p|n.
As in [30], we first take Γ n = S n ⋉ T (P GL n ) as an avatar of P GL n and then apply the localization sequence of Chow groups to obtain the desired result for P GL n .Since we consider Γ n as a subgroup of P GL n in the obvious way, elements in it are represented by n × n matrices such that in each row and column there is exactly one nonzero entry.
Recall the subgroup S p,n−p of S p,n−p defined in Section 6.In the obvious sense we take the subgroup Γ p,n−p := S p,n−p ⋉ T (P GL n ) of Γ n .Then elements in Γ p,n−p are represented by block matrices of the form (7.1) A 0 0 B with A, B square matrices of dimensions p and n − p, respectively.We have the following group homomorphisms where A, B are as in (7.1).One readily observes that the composite of the homomorphisms above gives the identity of Γ p .In particular, there is a p-torsion and we conclude.
It follows from Proposition 7.1 that Γn that restricts to ρ ′ p ∈ A p+1 Γp via the diagonal inclusion.In particular, the composition of inclusions We proceed to construct the promised classes ρ p,k ∈ A p+1 P GLn for k ≥ 0. Recall that we have the diagonal map ∆ : P GL p → P GL n and the cycle class map cl : A * G → H 2 * G for any algebraic group G over C. Lemma 7.2.Under the condition of Proposition 7.1 (i.e, (1.1), there is an ntorsion class ρ p,0 ∈ A p+1 P GLn satisfying ∆ * (ρ p,0 ) = ρ p and cl(ρ p,0 ) = y p,0 .Proof.We will necessarily consider both y p,0 ∈ H * P GLp and y p,0 ∈ H * P GLn .To avoid ambiguity we denote them by y p,0 (p) and y p,0 (n), respectively.
Consider the following commutative diagram (7.6) in which all arrows are the obvious restrictions.The vertical arrows are surjective, due to the localization sequences, and ϕ is an isomorphism, due to Proposition 5.9.Therefore, we choose ρ p,0 to be a class such that This assertion together with Lemma 3.1 and Remark 3.2 indicates that, for the restriction along the diagonal map ∆ * : H * P GLn → H * P GLp , we have (∆ * ) −1 (y p,0 (p)) {torsion classes in H * P GLn } = {y p,0 (n)}.Furthermore, it follows from Proposition 4.6 that ρ p,0 is torsion, since ρ p is so.Hence we have cl(ρ p,0 ) = y p,0 (n).
It remains to show that ρ p,0 is n-torsion, for which we have the following In particular, ρ p,0 is n-torsion.
Finally, the arguments in Theorem 1.1 concerning the Chow ring follows from the following Lemma 7.4.For k ≥ 0, under the condition (1.1), there are n-torsion elements ρ p,k ∈ A p k+1 +1 satisfying cl(ρ p,k ) = y p,k .
Proof.For k = 0, this is Lemma 7.2.Inductively, let ρp,k−1 ∈ A * P GLn ⊗ Z/p be the mod p reduction of ρ p,k−1 and we take ρ p,k ∈ A * P GLn to be a lift of S p k (ρ p,k−1 ) ∈ A * P GLn ⊗ Z/p.Notice that, by the definition of A * G (Totaro, [27], 1999), for any k there is a smooth variety X satisfying A k G ∼ = A k (X).Therefore, we can apply Proposition 4.9.It then follows from Corollary 2.5, Proposition 4.9 and Proposition 5.7 that we have cl(ρ p,k ) = y p,k .Finally, Proposition 4.6 indicates that ρ p,k are torsion elements, and by Proposition 7.3, they are n-torsion.
We proceed to give a corollary of Theorem 1.1.In particular, the image of the sets of elements {ρ p,k } k≥0 and {y p,k } k≥0 both generate subalgebras of transcendental degrees 2, after passing to mod p cohomology and Chow ring.
Proof.Consider the first two paragraphs of the corollary.By construction (as describe earlier in this section) in the case of Chow ring, the class ρ p restricts to r.The case of integral cohomology then follows by considering the cycle class map.
The third paragraph then follows from Proposition 5.5.( * t h e t o t a l number o f q u a l i f i e d o nes among t h e f i r s t m pr imes .* ) P r i n t [ " In t h e f i r s t " , m, " odd primes , t h e t o t a l number o f q u a l i f i e d o nes i s " , t o t a l , " ." ] ; P r i n t [ " The f i r s t row o f t h e f o l l o w i n g t a b l e i s t h e f i r s t " , k , " odd pr imes " ." A ' Yes ' benea th an odd pr imes means i t i s q u a l i f i e d , and a

Proposition 4 . 2 .
(Localization Sequences) Let f : Y → X be a proper morphism of G-schemes.Then we have an exact sequence as follows:

Lemma 5 .
13 (Vistoli, within the proof of Lemma 10.2, [30], 2007).Suppose that W is a representation of C p ⋉ T (P GL p ), and U an open subset of W \{0}. Assume that (a) the restriction of W to C p × µ p splits as a direct sum of 1-dimensional representations of which the cardinalities are p − k, k, n − p − k, and k, respectively.Hence we have#(H F ) = (p − k)!k!(n − p − k)!k!,and we compute the index [S p,n−p : sS p,n−p s −1 S p,n−p

l a b e l s = Table [ 1 ,
{ i , m} ] ; ( * a l i s t o f 1 and 0 r e c o r d i n g whether t h e i −th odd prime w i t h i n t h e f i r s t m o nes i s q u a l i f i e d .* ) w r d l a b e l s = Table [ " Yes " , { i , k } ] ;( * a l i s t o f " Yes " and "No" r e c o r d i n g whether t h e i −th odd prime w i t h i n t h e f i r s t k o nes i s q u a l i f i e d .* )oddprimes = Table [ Prime [ i + 1 ] , { i , m} ] ; ( * t h e l i s t o f t h e f i r s t m odd pr imes * ) For [ i = 1 , i < m + 1 , i ++,For [ j = I n t e g e r P a r t [ S q r t [ oddprimes [[ i ] ] − 1 ] ] , j < oddprimes [ [ i ] ] , j ++, I f [ Mod[ 1 + j ˆ2 , oddprimes [ [ i ] ] ] == 0 , l a b e l s [ [ i ] ] = 0 ; Break [ ] ]] ] ; ( * The i −th e n t r y i s now 1 i f t h e i −th odd prime i s q u a l i f i e d , and 0 o t h e r w i s e .This do uble l o o p i s j u s t i f i e d by P r o p o s i t i o n A . 2 .* ) For [ i = 1 , i < k + 1 , i ++, I f [ l a b e l s [ [ i ] ] == 0 , w r d l a b e l s [ [ i ] ] = "No " ] ] ; t o t a l = To ta l [ l a b e l s ] ; 'No ' means o t h e r w i s e ." ] ; P r i n t [ Grid [ { Part [ oddprimes , 1 ; ; k ] , w r d l a b e l s } ] ] It turns out, as mentioned in Remark 1.1.1,that 5, 016 out of the first 10, 000 primes are qualified.
• • •.Remark 1.1.2.In this paper we are unable to determine the precise orders of ρ p,k , though it is reasonable to conjecture that they are of order p.Indeed, Vezzosi has shown ([29], Corollary 2.4, 2000) that all torsion classes in A * P GLn are n-torsions.The same is true and more obvious for H * P GLn .(See, for example, Lemma 4.4 of [15]).Remark 1.1.3.The sets of elements {ρ p,k } k≥0 and {y p,k } k≥0 are not algebraically independent in general.Indeed, when p = n, they both generate subalgebras of transcendental dimension 2, in a suitable sense (See Corollary 7.5).
and there is an embedding K s → H defined by k → sks −1 .
Proposition 4.4 (Vistoli, Proposition 4.4, [30], 2007).(Mackey's formula) k .Corollary 6.4.Let N Sn S p,n−p be the normalizer of S p,n−p in S n .ThenN Sn S p,n−p = S p,n−p , if n = 2p, S 2 ⋉ S p,p , if n = 2p,where the semidirect product is obtained from the action of S 2 on the set {1, ...2p} by permuting the subsets {1, • • • , p} and {p+1, • • • , 2p}.In other words, S 2 = {1, τ } is considered as a subgroup of S n via the action Proof.It follows from Lemma 6.3 that the equation [sS p,n−p s −1 : sS p,n−p s −1 S p,n−p ] = 1 holds for n = 2p if and only if s is the coset represented by either 1 or 0 I p I p 0 , and it holds for n = 2p if and only if s is the coset of 1. 7. Some Torsion Classes in the Chow Ring of BP GL n (u) for each s ∈ C .Suppose s represents a class in the double quotient corresponding to a p-subspace generated by {e i1 , • • • , e ip } such that i j > p for k of the e ′ ij s.It then follows from the projection formula (4.2) and Lemma 6.3 that for such an s and any u ∈ A * subspaces spanned by{e i1 , • • • , e ip }|i j > p for k of the e ij 's} s∈C tr Γp,n−p,s Γp,n−p • res Γp,n−p Γp,n−p,s (u)where Γ p,n−p,s := sΓ p,n−p s −1 Γ p,n−p , and C is a collection of representatives of the double quotient Γ p,n−p \Γ n /Γ p,n−p , which is identified as S p,n−p \S n /S p,n−p , which is then identified as S n /S p,n−p , by Proposition 6.2.Accordingly, we replace each s ∈ C by its coset in S n = Γ n /T (P GL n ).