Inertial Chow rings of toric stacks

Abstract

For any vector bundle V on a toric Deligne-Mumford stack X the formalism of [EJK2] defines two intertial products V + and V − on the Chow group of the inertia stack. We give an explicit presentation for the integral V + and V − Chow rings, extending earlier work of Boris-Chen-Smith [BCS] and Jiang-Tsen [JT] in the orbifold Chow ring case, which corresponds to V = 0.We also describe an asymptotic product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle V go to infinity.

The chow cohomology of affine toric varieties

“…The Chow ring of X (∆) admits a simple presentation using equivariant Chow groups. Following [CE,Section 2] Thus, by [CE, Propositions 2.1 and 2.2], the Chow ring of X (∆) has the following presentation:…”

The vanishing of the chow cohomology ring for affine toric varieties and additional results

“…With these in hand, the construction of the Chow homology and cohomology groups of a quotient stack are defined as the equivariant Chow homology and cohomology groups. We recall an important computation due to [3] to be used later in the paper concerning computing the equivariant Chow rings of certain open subschemes of A n .…”

“…We end this section with a computation due to [3] computing the G-equivariant Chow rings of certain open subschemes of A n where G is diagonalizable. The usefulness of this result will be seen in later sections dealing with computing Chow rings of canonical toric stacks.…”

“…Indeed, if n = |∆(1)|, then as discussed in Chapter 2, the toric variety X(Σ ∆ ) ∼ = A n − Z(∆) where Z(∆) is a finite union of G β -invariant, linear subspaces. Thus, by [3], we can directly compute the Chow ring of any canonical toric stack. We begin by re-defining the ideals discussed in Chapter 2 relative to the current situation.…”