### 111.5.3 Intersection theory

Papers discussing intersection theory on algebraic stacks.

• Vistoli: Intersection theory on algebraic stacks and on their moduli spaces

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

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 defines Chow groups for arbitrary Artin stacks agreeing with Edidin and Graham's definition in 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

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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