Chow Filtration on Representation Rings of Algebraic Groups

Abstract

We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on $R(G)$ and show that all three define on $R(G)$ the same topology. For any $n\geq 1$, we compute the Chow filtration on $R(G)$ for the special orthogonal group $G:=O^+(2n+1)$. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of $G$ over any field of characteristic $\ne 2$ is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety $X$ such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of $X$.

A counter-example by Yagita

“…Let us conclude by remark that the conjecture we are discussing has been verified for many groups including the spinor ones up to Spin (12), see [10]. The cases of Spin(13/14) and Spin (15/16) are still open.…”

“…For k ≥ 0, we write c K k ∈ K(X) (k) for the K-theoretic Chern class of the tautological vector bundle on X, defined as in [12,Example 2.3]. For k ≥ 1, we also define the element…”

A remark on connective K-theory

“…The Grothendieck group K(X) is actually also a ring (with multiplication given by tensor product of locally-free sheaves) and it is endowed with the Chow filtration (see [8]), i.e., the filtration by codimension of supports of coherent sheaves:…”

Connective K-theory and Adams operations

“…where CH i (BG) are the equivariant Chow groups (defined by Totaro in [20]). The (topological) filtration on R(G) was defined in [10]. The following example illustrates how the calculation of equivariant connective groups CK i takes c CH i to the class of c K i if i is even and to 0 if i is odd.…”

“…The following example illustrates how the calculation of equivariant connective groups CK i takes c CH i to the class of c K i if i is even and to 0 if i is odd. In particular, Ker(ϕ * ) is generated by c CH i with i ≥ 3 odd (see [10,Example 5.3]). The same reasoning to prove that the ring CH(BG) is generated by Chern classes in [20, §15] can be applied to show that CK(BG) is also generated by CK-theoretic Chern classes c 1 , c 2 , .…”