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In algebraic geometry, the '''Chow group of a stack''' is a generalization of the [[Chow group]] of a variety or scheme to [[stack (mathematics)|stack]]s. For a [[quotient stack]] <math>X = [Y/G]</math>, 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 ==
{{harvs|txt|last=Vistoli|first=Angelo|year=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]] <math>X = [Y/G]</math> 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 [[Andrew Kresch]] (1999) extended the theory to a stack admitting a stratification by quotient stacks.

For [[higher Chow group]]s (precursor of [[motivic homology|motivic homologies]]) of algebraic stacks, see Roy Joshua's Intersection Theory on Stacks:I and II. [https://people.math.osu.edu/joshua.1/pub.html]

== 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'',
# (homotopy-invariance) if ''E'' is a rank-''n'' vector bundle on ''X'', then <math>A_{p}(E) = A_{p-n}(X)</math>.
# for each integral substack ''Z'' of dimension < ''p'', <math>A_p(X - Z) = A_p(X)</math>, a corollary of a localization sequence.
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 <math>B G</math>, the stack of principal ''G''-bundles for a smooth linear algebraic group ''G''. By definition, it is the quotient stack <math>[*/G]</math>, where * is viewed as the stack associated to * = Spec ''k''. We approximate it as follows. Given an integer ''p'', choose a representation <math>G \to GL(V)</math> such that there is a ''G''-invariant open subset ''U'' of ''V'' on which ''G'' acts freely and the complement <math>Z = V - U</math> has codimension <math>> - \operatorname{dim} G - p</math>. Let <math>* \times_G V</math> be the quotient of <math>* \times V</math> by the action <math>(x, v) \cdot g = (xg, g^{-1} v)</math>. Note the action is free and so <math>* \times_G V</math> is a vector bundle over <math>BG</math>. By Property 1 applied to this vector bundle,
:<math>A_p(BG) = A_{p+\dim V} (* \times_G V).</math>
Then, since <math>* \times_G U = U/G</math>, by Property 2,
:<math>A_{p + \dim V}(* \times_G V) = A_{p + \dim V}(U/G)</math>
since <math>\dim[Z/G] = \dim Z - \dim G < \dim V + p</math>.

As a concrete example, let <math>G = \mathbb{G}_m</math> and let it act on <math>\mathbb{A}^n</math> by scaling. Then <math>\mathbb{G}_m</math> acts freely on <math>U = \mathbb{A}^n - \{ 0 \}</math>. By the above calculation, for each pair of integers ''n'', ''p'' such that <math>n+p \ge 0</math>,
:<math>A_p(B \mathbb{G}_m) = A_{p+n}(\mathbb{P}^{n-1}).</math>
In particular, for every integer ''p'' ≥ 0, <math>A_p(B \mathbb{G}_m) = 0</math>. In general, <math>A_{n-k}(\mathbb{P}^n) = \mathbb{Q}h^k</math> for the hyperplane class ''h'', <math>h^k</math> ''k''-times self-intersection and <math>h^k =0</math> for negative ''k'' and so
:<math>A_p(B \mathbb{G}_m) = \mathbb{Q} h^{-1-p}</math>
where the right-hand side is independent of models used in the calculation (since different ''h''{{'}}s correspond under the [[projective variety#Projections|projections]] between projective spaces.) For <math>p = -1 = \dim B \mathbb{G}_m</math>, the class <math>h^0 = [\mathbb{P}^{n}]</math>, any ''n'', may be thought of as the fundamental class of <math>B \mathbb{G}_m</math>.

Similarly, we have
:<math>A^*(B \mathbb{G}_m) = \mathbb{Q}[c]</math>
where <math>c = c_1(h)</math> 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 <math>h^k = c \cdot h^{k-1}</math>, we have that <math>A_*(B \mathbb{G}_m)</math> is the free <math>\mathbb{Q}[c]</math>-module generated by <math>h^0</math>.

== Virtual fundamental class ==
The notion originates in the [[Kuranishi theory]] in [[symplectic geometry]].<ref name="Fukaya-Ono">{{cite journal | last1=Fukaya | first1=Kenji | author1-link=Kenji Fukaya|last2=Ono | first2=Kaoru | title=Arnold conjecture and Gromov-Witten invariant | journal=[[Topology (journal)|Topology]] | volume=38 | issue=5 | year=1999 | doi=10.1016/s0040-9383(98)00042-1 | pages=933–1048|mr=1688434| doi-access=free }}</ref><ref>{{Cite journal|last=Pardon|first=John|authorlink=John Pardon|date=2016-04-28|title=An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves|journal=[[Geometry & Topology]]|volume=20|issue=2|pages=779–1034| doi=10.2140/gt.2016.20.779|issn=1364-0380|arxiv=1309.2370|s2cid=119171219}}</ref>

In § 2. of {{harvtxt|Behrend|2009}}, given a DM stack ''X'' and ''C''<sub>''X''</sub> the [[intrinsic normal cone]] to ''X'', K. Behrend defines the '''virtual fundamental class''' of ''X'' as
:<math>[X]^{\text{vir}} = s_0^![C_X]</math>
where ''s''<sub>0</sub> is the zero-section of the cone determined by the [[perfect obstruction theory]] and ''s''<sub>0</sub><sup>!</sup> 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]].<ref>§ 1.2.1. of {{cite arXiv|last1=Cisinski|first1=Denis-Charles|last2=Khan|first2=Adeel A.|date=2017-05-09|title=Brave new motivic homotopy theory II: Homotopy invariant K-theory|eprint=1705.03340|class=math.AT}}</ref>
<!-- use the material below to discuss EG, BG =[pt/G]
== Equivariant Chow ring ==
Let ''X'' be an equidimensional [[smooth scheme]] with the action of a [[linear algebraic group]] ''G''. We note:
# A universal ''G''-bundle ''EG'' on ''X'' admits an approximation by [[algebraic bundle]]s of arbitrary large codimension. (Note <math>EG</math> is an [[ind-scheme]].)
# ''X'' can be covered by invariant quasi-projective open subsets.
(1. is always true and 2. holds by [[Sumihiro's theorem]] as ''X'' is normal.) Then one defines the equivariant Chow group to be
:<math>A^*_G(X) = \bigoplus A^n(EG(n) \times_G X)</math>
where <math>EG(n)</math> is an invariant open subset with codimension at least ''n''. (The definition is an analog of [[equivariant cohomology]].) Like the usual [[Chow ring]], <math>A^*_G(X)</math> is a [[graded ring]] with respect to [[intersection product]]. Moreover, it is a graded algebra over <math>A^*_G(pt)</math> via the map induced by <math>X \times_G EG \to EG/G</math>.

There is the cycle map
:<math>\operatorname{cl}: A^*_G(X) \to H_G(X; \mathbb{Z})</math>
from the equivariant Chow ring to the [[equivariant cohomology ring]]. The map is an isomorphism if ''X'' admits a cell decomposition (e.g., ''X'' is a ''T''-variety by the [[Białynicki-Birula decomposition]].)-->

== See also ==
* [[Perfect obstruction theory]]
* [[Gromov–Witten invariant]]
* [[Cohomology of a stack]]

== Notes ==
{{reflist}}

==References==
* {{citation
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}}
* {{cite journal | last1=Ciocan-Fontanine | first1=Ionuț | last2=Kapranov | first2=Mikhail | title=Virtual fundamental classes via dg–manifolds | journal=[[Geometry & Topology]] | volume=13 | issue=3 | year=2009 |  doi=10.2140/gt.2009.13.1779 | pages=1779–1804 | arxiv=math/0703214|mr=2496057
| s2cid=1211344 }}
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* {{citation
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* {{citation
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* {{citation
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* {{citation
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* {{citation
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}}

== External links ==
*[[arxiv:1804.06048|Virtual classes for the working mathematician]]
*[https://mathoverflow.net/q/160561 The classical number 2875 of lines on the quintic, as a DT invariant]
*[https://mathoverflow.net/q/259128 What is the main failure in using Naive Chow group in Artin Stack]
*[https://mathoverflow.net/q/122045 Local model of virtual fundamental cycle]
*https://ncatlab.org/nlab/show/virtual+fundamental+class
*[https://www.math.ubc.ca/~behrend/talks/seoul14.pdf On the Virtual Fundamental Class] - a slide by Kai Behrend

[[Category:Algebraic geometry]]