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- Tags: - #AG - Refs: - #todo/add-references - Links: - computational properties of Chow

Chow ring

• What is the degree of a cycle?

Definitions

See Weil divisor.

Rational equivalence:

Cycles associated to subschemes

Relation to divisor class group:

Paper on Chow Rings

• Reference: a recent result, https://arxiv.org/pdf/math/0505560.pdf

Cohomology for Symmetric algebra on the group of characters.

There is a map from the Chow ring back into cohomology, which in general fails surjectivity and injectivity. Tensoring this map with \({\mathbf{Q}}\) creates an isomorphism, though. In this case, both have the ring structure of invariants under the maximal torus. (Classical result, Leray and Borel.)

An inverse of the cycle class map?

Chow rings have not been computed for \(\operatorname{PGL}_n\). Need to know about Euler class,

\(A_*\) known for all \(O_n\) and \(SO_n\) for \(n\) odd in 80s, general result for \(SO_n\) 2004. \(PGL_n\) case is much harder. Understood for \(n=2\), since \(\operatorname{PGL}_2 \cong SO_3\). Other bits that have been computed: \(H^*({\mathbf{B}}\operatorname{PGL}_3, {\mathbf{Z}}/3), H^*({\mathbf{B}}\operatorname{PGL}_n, {\mathbf{Z}}_2)\) for \(n = 2 \operatorname{mod}4\) in 70s/80s, incomplete results for \(H^*({\mathbf{B}}\operatorname{PGL}_p, {\mathbf{Z}}_p)\) in 2003.

Relation to K-theory:

\begin{align*}{\mathsf{K}}_{0}(X) \otimes_{\mathbb{Z}} \mathbb{Q} \cong \prod_{i} {\operatorname{CH}}^{i}(X) \otimes_{\mathbb{Z}} \mathbb{Q}\end{align*}

When \(X\) is a variety, \({\operatorname{CH}}_{\dim X - 1} X \cong \operatorname{Div} \operatorname{Cl} (X)\), the divisor class group. If $X\in {\mathsf{sm}}{\mathsf{Var}}_{/ {k}} $, then \({\operatorname{CH}}_{\dim X - 1}X \cong \operatorname{Pic}(X)\), the Picard group.

Products

Relation to K theory

See algebraic K theory:

Examples