I'm reading Andrew Kresch's paper, Cycle groups in Artin Stacks.

The author defined Chow groups of Artin stacks by very technical way, instead of ordinary ways which he called 'naive chow group', quotient of free group generated by integral substacks by rational equivalence.

I don't understand why he used technical definition instead of ordinary one. What is the main failure arises when using 'naive chow group's?


Already for a finite group $\Gamma$, the only integral closed substacks of $B\Gamma$ are the empty stack and all of $B\Gamma$. So your naive Chow group would be cyclic (typically zero) in each degree. On the other hand, the "correct" Chow group of divisor classes should be the Pontrjagin dual group $\text{Hom}_{\text{Group}}(\Gamma,k^\times)$ (here $k$ is the ground field, which I guess I am assuming is characteristic $0$ and algebraically closed). Typically this is not cyclic, e.g., for $\Gamma = (\mathbb{Z}/r\mathbb{Z})^n$, the Pontrjagin dual group is $(\mu_r)^n$, and this is not cyclic for $n>1$.

The correct Chow groups of $B\Gamma$, i.e., the $\Gamma$-equivariant Chow groups of $\text{Spec}(k)$, were defined by Totaro and Edidin-Graham. Their theory works for every Deligne-Mumford stack of the form $[X/\Gamma]$ for $X$ a quasi-projective $k$-scheme with a $k$-action of $\Gamma$. The paper of Kresch extends these Chow groups to all Deligne-Mumford stacks that have a stratification whose associated strata are each of the form $[X_i/\Gamma_i]$.

This was essential at that time, since it was not then known that the stacks of stable maps could be written as $[X/\Gamma]$, yet it was known that the stacks have stratifications with strata of the form $[X_i/\Gamma_i].$ The Behrend-Fantechi construction of a virtual fundamental class for (closed) Gromov-Witten invariants in algebraic geometry depends on the existence of Chow groups for stacks satisfying certain axioms (in particular, Vistoli's rational equivalence). Kresch proved that the stacks of stable maps have such a Chow theory.


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