Some torsion classes in the Chow ring and cohomology of BPGLn

Abstract

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

A Serre-type spectral sequence for motivic cohomology

“…Section 7 provides a Postnikov system for the motive of a Severi-Brauer variety (see Proposition 7.1) induced by the one for the map BGL n → BP GL n studied in Section 6. Finally, in Section 8, we find torsion classes in the motivic cohomology of BP GL n generalising some results from [7] and [5] (see Corollary 8.6).…”

“…These important results shed new light on the structure of these rings. In this paper, by following the lead of [7] and [5], we show that there are similar non-trivial torsion classes in the motivic cohomology of the Nisnevich classifying space of P GL n . By exploiting the homomorphism in motivic cohomology induced by the canonical map BP GL n → B ét P GL n (that replaces the cycle class map over general fields), we then generalise Gu's results on the torsion classes in the Chow ring of B ét P GL n to fields of characteristic not dividing n containing a primitive nth root of unity.…”

“…As we have already pointed out, a complete description of the singular cohomology of BP U n seems to be still out of reach. Neverthless, in [7] and [5] Gu has constructed non-trivial torsion classes in the cohomology ring of BP U n and in the Chow ring of B ét P GL n over the complex numbers, by using Steenrod operations and the cycle class map going from Chow groups to singular cohomology. These important results shed new light on the structure of these rings.…”

“…In this section, following [7] and [5], we find torsion classes in the motivic cohomology of BP GL n . This allows also to generalise some results about the Chow groups of B ét P GL n from the complex numbers (see [7, Theorem 1.1] and [5, Theorem 1]) to more general fields.…”

The $p$-primary subgroups of the cohomology of $BPU_n$ in dimensions less than $2p+5$

“…None of the works above dealt with H * (BP U n ; Z), the ordinary cohomology of BP U n with coefficients in Z, for n not a prime number. The first named author considered H * (BP U n ; Z), as well as the Chow ring of BP GL n (C) for an arbitrary n in [10], [12] and [13]. In particular, in [12], the first named author determined the ring structure of H * (BP U n ; Z) in dimensions less than or equal to 10.…”