Improvement on the proposition 4.2 of the toric stack paper [1]

When I tried to use Proposition 4.2 of the paper [1] few months ago, I realized that it has to satisfy the condition 4.1. I came up with a way to avoid this awkward condition using the idea from [2]. I emailed Prof. Borisov and he agreed that there are two ways, including the one I mentioned, to fix this. Let me explain my approach.

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Nash blowing-up

Let Z be a d-dimensional subscheme which can be embedded into a smooth variety A of dimension n. Let S(Z) be the singular points of Z, Z_0 its complement in Z. Let \text{Gr}_d(TA) be the Grassmannian bundle of d-dimensional subspaces of the tangent bundle TA over A.

Consider the morphism defined by

    \[\begin{array}{ll} \eta: & Z_0 \to \text{Gr}_d(TA) \\ & z \to (z, T_{Z,z}) \end{array}\]

where T_{Z,z} is the tangent space of Z at z. Denote Z^* the closure of \eta(Z_0) in \text{Gr}_d(TA). Then the morphism

    \[p: Z^* \to Z\]

induced by the first projection (locally \text{Gr}_d(TA) \cong Z \times \text{Gr}(d,n)) is called the Nash blowing-up of X. Note that the construction of (Z^*,p) is independent of the embedding up to unique Z-isomorphism.

The first basic result is

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The cotangent complex and virtual fundamental class

Given a morphism of sheaves of rings A \to B, we can construct the truncated contangent complex \mathbb{L}_{B/A}. This complex is a two-term complex \mathbb{L}_{B/A}=[L_1 \to L_0], which is equipped with a natural map \mathbb{L}_{B/A} \to \Omega_{B/A} inducing an isomorphism H_0(\mathbb{L}_{B/A})=\Omega_{B/A}.

Let A[B] be the free A-algebra with variables x_b, then we have a tautological surjection A[B] \to B with the kernel I. The complex \mathbb{L}_{B/A} is set as

    \[\mathbb{L}_{B/A}:=[I/I^2 \to \Omega_{A[B]/A} \otimes_{A[B]} B]\]

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Riemann-Roch for weighted projective stack

Grothendieck-Riemann-Roch on Deligne-Mumford stacks are proved by Toen [1] using cohomology theories.

Let I\mathcal{X} be the inertia stack of the stack \mathcal{X}. We follow the appendix A of [2].

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