# 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.

# Customize your own bibliography style

Since this was my first time to write a paper, I originally used the code

\begin{thebibliography}{99}
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\end{thebibliography}

# 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, its complement in Z. Let be the Grassmannian bundle of d-dimensional subspaces of the tangent bundle TA over A.

Consider the morphism defined by

where is the tangent space of Z at z. Denote the closure of in . Then the morphism

induced by the first projection (locally ) is called the Nash blowing-up of . Note that the construction of is independent of the embedding up to unique -isomorphism.

The first basic result is

# The cotangent complex and virtual fundamental class

Given a morphism of sheaves of rings , we can construct the truncated contangent complex . This complex is a two-term complex , which is equipped with a natural map inducing an isomorphism .

Let A[B] be the free A-algebra with variables , then we have a tautological surjection with the kernel . The complex is set as

# Riemann-Roch for weighted projective stack

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

Let be the inertia stack of the stack . We follow the appendix A of [2].