Let’s look at some interesting sums. We start with a simple one. Suppose is a nth primitive root of unity, then

The proof is simple. It comes from the fact that

and

when is even.

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Let’s look at some interesting sums. We start with a simple one. Suppose is a nth primitive root of unity, then

The proof is simple. It comes from the fact that

and

when is even.

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.

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

\begin{thebibliography}{99}

\bibitem{*****}

\end{thebibliography}

to add references manually.

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

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

In this post, let’s use tools of generating functions and ordinary differential equation to prove that

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].

A partition is a sequence of non-increasing non-negative integers . We can identify partitions with their corresponding Young diagrams.

For example, if , then the corresponding Young diagram is

Consider the fan in consisting of the four cones shown in the below figure with all of their faces. The four rays are , , and .