Hilbert series and resolution of a surface singularity

Question

I have a question about the following theorem in Stanley's paper "Invariants of Finite Groups and Their Applications to Combinatorics".

Suppose that the Cohen-Macaulay $N$-graded $k$-algebra $B$ is generated by elements $\gamma_1, \dots, \gamma_{m+p}$ all of the same degree $e$, and that the Hilbert series $F(B,\lambda)=(1+p\lambda^e)(1-\lambda^e)^{-m}$, where $p, m$ are some constants. Then in the minimal free resolution of $B$ (with respect to $\lambda_1, \dots, \lambda_{m+p}$) $0\rightarrow M_h\rightarrow M_{h-1}\rightarrow\dots\rightarrow M_0\rightarrow B\rightarrow0$, we have $h=p$ and $M_i$ has a basis consisting of $i\binom{p+1}{i+1}$ elments of degree $e(i+1)$.

He refers to Wahl's paper "Equations defining rational singularities" for the proof, and in this paper there is a similar theorem says:

Let $R=P/I$ be a rational surface singularity of embedding dimension $e$, with $P$ a regular local $k$-algebra of dimension $e$. Then there is a minimal resolution $0\rightarrow P^{b_{e-2}}\rightarrow\dots\rightarrow P^{b_1}\rightarrow P\rightarrow P/I\rightarrow0$ so that $b_i=i\binom{e-1}{i+1}$.

My question is: How does $F(B,\lambda)=(1+p\lambda^e)(1-\lambda^e)^{-m}$ grantee that $B$ can be realized to be the coordinate ring of the cone over a rational surface singularity? Or is there some reference I can read? It seems Eisenbud has some work, can anyone tell me some related results?

Thanks a lot!

Answer 1

Assume, for now, that $e=1$. Then there exists an $(m+p)$-dimensional polynomial ring $S$ over $k$ with generators of degree $1$ and an $S$-ideal $I$ with $\mathrm{codim}(I) = p$ such that $B = S/I$. Let $0 \rightarrow M_h \rightarrow \cdots \rightarrow M_0 \rightarrow B \rightarrow 0$ be a minimal graded free resolution of $B$ as an $S$-module. Since $B$ is Cohen-Macaulay, it follows from the Auslander-Buchsbaum formula that $h=p$. The Castelnuovo-Mumford regularity $\mathrm{reg}(B)$ of $B$ is the maximum of $j-i$ such that $R(-j)$ is a free summand of $M_i$. (Eisenbud and Goto (J. Algebra, 1984) showed that this definition agrees with the definition using cohomology.) Since $B$ is Cohen-Macaulay, there exist linear forms, $\mathbf{l} = l_1, \ldots, l_m$ in $S$ that form a regular sequence on $B$. (We need that $k$ is an infinite field for this, but base change does not affect the hypotheses or conclusions, so we may replace $k$ by an algebraic closure, for instance.) Hence $\mathrm{reg}(B/\mathbf{l}B) = \mathrm{reg}(B)$ and such that $F(B/\mathbf{l}B, \lambda) = (1+p\lambda)$.

Since $B/\mathbf{l}B$ is Artinian, its regularity is the highest degree in which it has a nonzero component, which in this case is one. Hence $\mathrm{reg}(B) = 1$, which means that $I$ is generated by quadrics and has a linear resolution, i.e, the maps in the minimal free resolution are matrices of linear forms. Then all the minimal homogeneous generators of $M_i$ have degree $(i+1)$. Moreover, $\mathrm{rank}(M_i) = i\binom{p+1}{i+1}$. I know this from a paper of Herzog and K\"uhl (Commun. Algebra, 1984), which, in turn, cites Wahl's paper. I'd guess that Wahl first shows that rational surface singularities have a linear resolution and then shows that the ranks are as above.

Finally, let me point out that the assumption that $e=1$ is not really a restriction; the general case reduces to this, by changing the degrees of the homogeneous generators. The assertions about regularity would appear, in some way or the other, in Eisenbud's commutative algebra book or syzygies book.