Revisions to Hilbert series and resolution of a surface singularity

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edited Nov 15, 2011 at 16:52

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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$$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!

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!

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!