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]
Summary
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
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