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 , the Chow group of X is the same as the G-equivariant Chow group of Y.
![{\displaystyle X=[Y/G]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b32a8e64d11248e98a57c53de5cb5397ef8c2a92)
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).
Vistoli (1989) 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 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 Edidin-Graham and Totaro. Kresch (1999) later extended the theory to a stack admitting a stratification by quotient stacks.
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