Chow group of a stack

In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack formula_1, the Chow group of "X" is the same as the "G"-equivariant Chow group of "Y".
A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers).
Definitions.
 develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack. There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence.
If a stack "X" can be written as the quotient stack formula_1 for some quasi-projective variety "Y" with a linearized action of a linear algebraic group "G", then the Chow group of "X" is defined as the "G"-equivariant Chow group of "Y". This approach is introduced and developed by Dan Edidin and William A. Graham, as well as Burt Totaro. later extended the theory to a stack admitting a stratification by quotient stacks.
For higher Chow groups (precursor of motivic homologies) of algebraic stacks, see Roy Joshua's Intersection Theory on Stacks:I and II. 
Examples.
The calculations depend on definitions. Thus, here, we proceed somehow axiomatically. Specifically, we assume: given an algebraic stack "X" locally of finite type over a base field "k",
These properties are valid if "X" is Deligne–Mumford and are expected to hold for any other reasonable theory.
We take "X" to be the classifying stack formula_5, the stack of principal "G"-bundles for a smooth linear algebraic group "G". By definition, it is the quotient stack formula_6, where * is viewed as the stack associated to * = Spec "k". We approximate it as follows. Given an integer "p", choose a representation formula_7 such that there is a "G"-invariant open subset "U" of "V" on which "G" acts freely and the complement formula_8 has codimension formula_9. Let formula_10 be the quotient of formula_11 by the action formula_12. Note the action is free and so formula_10 is a vector bundle over formula_14. By Property 1 applied to this vector bundle,
Then, since formula_16, by Property 2,
since formula_18.
As a concrete example, let formula_19 and let it act on formula_20 by scaling. Then formula_21 acts freely on formula_22. By the above calculation, for each pair of integers "n", "p" such that formula_23,
In particular, for every integer "p" ≥ 0, formula_25. In general, formula_26 for the hyperplane class "h", formula_27 "k"-times self-intersection and formula_28 for negative "k" and so
where the right-hand side is independent of models used in the calculation (since different "h"s correspond under the projections between projective spaces.) For formula_30, the class formula_31, any "n", may be thought of as the fundamental class of formula_32.
Similarly, we have
where formula_34 is the first Chern class of "h" (and "c" and "h" are identified when Chow groups and Chow rings of projective spaces are identified). Since formula_35, we have that formula_36 is the free formula_37-module generated by formula_38.
Virtual fundamental class.
The notion originates in the Kuranishi theory in symplectic geometry.
In § 2. of , given a DM stack "X" and "C""X" the intrinsic normal cone to "X", K. Behrend defines the virtual fundamental class of "X" as
where "s"0 is the zero-section of the cone determined by the perfect obstruction theory and "s"0! is the refined Gysin homomorphism defined just as in Fulton's "Intersection theory". The same paper shows that the degree of this class, morally the integration over it, is equal to the weighted Euler characteristic of the Behrend function of "X".
More recent (circa 2017) approaches do this type of construction in the context of derived algebraic geometry.