Abstract

In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq 0$. If in addition, $p^2\nmid n$, there are $p$-torsion classes $\rho_{p,k}$ of degree $p^{k+1}+1$ in the Chow ring of the classifying stack of $PGL_n$, such that the cycle class map takes $\rho_{p,k}$ to $y_{p,k}$. We present an application of the above classes regarding Chern subrings.

Highlights

  • Which the main object of interest is the equivariant Chow ring, an algebraic analog of Borel’s equivariant cohomology theory

  • When the algebraic group G is over the base field of complex numbers C, it has an underlying topological group which we denote by G, and there is a cycle class map cl : A∗G → HG∗ from AiG to HG2i, which plays a crucial role in this paper

  • The mod p cohomology for some special choices of n is considered by Toda [40], Kono and Mimura [27], Vavpetic and Viruel [43]

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Summary

Equivariant intersection theory

We refer to Edidin and Graham [11] and Totaro [41] for definitions and basic facts on equivariant intersection theory. A∗G−n −c−n−(V−→) A∗G → A∗G(V \{0}) → 0 is exact, where the first arrow is the multiplication by the Chern class cn(V ) It follows from Proposition 4.1 that we have. H has finite index in G, we have the transfer map trHG : A∗H → A∗G This is no longer a ring homomorphism, but a homomorphism of A∗G-modules, in the sense that we have the following projection formula:. TrKKs · ressKHss−1 ·γs : A∗H → A∗K , s∈C where γs is the restriction associated to the conjugation sHs−1 → H There is another way to relate equivariant Chow rings over different algebraic groups. For G = GLn, SLn or a torus, the cycle class map cl : A∗G → HG∗ is an isomorphism of rings. The inclusion of a maximal torus λ : T (P GLn) → P GLn induces the following isomorphism: λ∗ : A∗P GLn ⊗ Q −∼=→ (A∗T (P GLn))Sn ⊗ Q

The Steenrod reduced power operations for motivic cohomology
The Chow ring and cohomology of BP GLp
The subgroups of P GLn of diagonal block matrices
The permutation groups and their double quotients
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