### 111.5.3 Intersection theory

Papers discussing intersection theory on algebraic stacks.

Vistoli:

*Intersection theory on algebraic stacks and on their moduli spaces*[vistoli_intersection]This paper develops the foundations for intersection theory with rational coefficients for Deligne-Mumford stacks. If $\mathcal{X}$ is a separated Deligne-Mumford stack, the chow group $\mathop{\mathrm{CH}}\nolimits _*(\mathcal{X})$ with rational coefficients is defined as the free abelian group of integral closed substacks of dimension $k$ up to rational equivalence. There is a flat pullback, a proper push-forward and a generalized Gysin homomorphism for regular local embeddings. If $\phi : \mathcal{X} \to Y$ is a moduli space (ie. a proper morphism with is bijective on geometric points), there is an induced push-forward $\mathop{\mathrm{CH}}\nolimits _*(\mathcal{X}) \to \mathop{\mathrm{CH}}\nolimits _ k(Y)$ which is an isomorphism.

Edidin, Graham:

*Equivariant Intersection Theory*[edidin-graham]The purpose of this article is to develop intersection theory with integral coefficients for a quotient stack $[X/G]$ of an action of an algebraic group $G$ on an algebraic space $X$ or, in other words, to develop a $G$-equivariant intersection theory of $X$. Equivariant chow groups defined using only invariant cycles does not produce a theory with nice properties. Instead, generalizing Totaro's definition in the case of $BG$ and motivated by the fact that if $V \to X$ is a vector bundle then $\mathop{\mathrm{CH}}\nolimits _ i(X) \cong \mathop{\mathrm{CH}}\nolimits _ i(V)$ naturally, the authors define $\mathop{\mathrm{CH}}\nolimits _ i^ G(X)$ as follows: Let $\dim (X) = n$ and $\dim (G) = g$. For each $i$, choose a $l$-dimensional $G$-representation $V$ where $G$ acts freely on an open subset $U \subset V$ whose complement as codimension $d > n - i$. So $X_ G = [X \times U / G]$ is an algebraic space (it can even be chosen to be a scheme). Then they define $\mathop{\mathrm{CH}}\nolimits _ i^ G(X) = \mathop{\mathrm{CH}}\nolimits _{i + l - g}(X_ G)$. For the quotient stack, one defines $\mathop{\mathrm{CH}}\nolimits _ i( [X/G]) = \mathop{\mathrm{CH}}\nolimits _{i + g}^ G(X) = \mathop{\mathrm{CH}}\nolimits _{i + l}(X_ G)$. In particular, $\mathop{\mathrm{CH}}\nolimits _ i([X/G]) = 0$ for $i > \dim [X/G] = n - g$ but can be non-zero for $i \ll 0$. For example $\mathop{\mathrm{CH}}\nolimits _ i(B \mathbf{G}_ m) = \mathbf{Z}$ for $i \le 0$. They establish that these equivariant Chow groups enjoy the same functorial properties as ordinary Chow groups. Furthermore, they establish that if $[X / G] \cong [Y / H]$ that $\mathop{\mathrm{CH}}\nolimits _ i([X/G]) = \mathop{\mathrm{CH}}\nolimits _ i([Y/H])$ so that the definition is independent on how the stack is presented as a quotient stack.

Kresch:

*Cycle Groups for Artin Stacks*[kresch_cycle]Kresch defines Chow groups for arbitrary Artin stacks agreeing with Edidin and Graham's definition in [edidin-graham] in the case of quotient stack. For algebraic stacks with affine stabilizer groups, the theory satisfies the usual properties.

Behrend and Fantechi:

*The intrinsic normal cone*[behrend-fantechi]Generalizing a construction due to Li and Tian, Behrend and Fantechi construct a virtual fundamental class for a Deligne-Mumford stack.

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