Basic properties of complete intersection ideals

Question

The title is quite self-explanatory. I'm studying, for an exam, the paper "The solution to Waring's problem for monomials" (this is the link https://arxiv.org/abs/1110.0745 ).

At the bottom of page 2, Lemma 2.2, we consider $J=(y_1,y_2^{a_2},\ldots, y_n^{a_n}) \subset T=k[y_1,\ldots,y_n]$ ideal with $2\leq a_2\leq\ldots\leq a_n$.

The authors note that $J$ is homogeneous (and I agree) and complete intersection, for which they assert $$A=T/J$$ is an artinian Gorenstein ring, with $\dim A_\tau \neq 0$ for $\tau=a_2+\ldots+a_n-(n-1)$.

My question are:

1) Why we can simply note $J$ is complete intersection?

2) Where can I find a proof of the statement "$J$ homogeneous and complete intersection $\implies$ A artinian and Gorenstein"?

3) I know what a Gorenstein ring is, but I can't see why $\dim A_\tau \neq 0$ for this $\textit{specific}$ $\tau$.

I know there's a lot of stuff in this question, but I hope someone can help me understaning, or suggesting me some references. Thanks in advance.

Answer 1

I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.

1. The ideal $J$ is a complete intersection ideal iff $ht(J) = \mu(J)$. Here $\mu(J)$ stands for the minimal number of generators of the ideal $J$. It is clear that $ht(J) = n$ and $\mu(J) \leq n$, since $J$ can evidently be generated by $n$ elements. But we know that $ht(J) \leq \mu(J)$. If we combine the previous observations together we have $n = ht(J) \leq \mu(J) \leq n$. Thus $ht(J) = \mu(J) = n$.

2. Consider the ideal $\mathfrak{m} = (y_1, \dots , y_n)$. Then $T_{\mathfrak{m}}$ is a regular local ring and hence Gorenstein. Since $y_1, y_2^{a_2}, \dots y_n^{a_n}$ form a regular sequence, we have that $T_{\mathfrak{m}}/J$ is gorenstein. Now note that since $y_i$ are nilpotent in $A$, every element not in image of $\mathfrak{m}$ is already invertible in $A$, thus we get that $A = T_{\mathfrak{m}}/J$. Thus $A$ is Gorenstien. Here I have used Lemma 45.21.3. and Lemma 45.21.6. from the following link https://stacks.math.columbia.edu/tag/0DW6.

3. I claim that for $a =y_2^{a_2-1}. y_3^{a_3-1} \dots y_n^{a_n-1}$, we have that the image $\overline{a}$ is a non-zero element in $A$. If not then $a \in J$. That will give a polynomial equation in $y_i$ but they are transcendental.