Hilbert-Samuel multiplicity of standard graded $k$-algebra which is an integral domain and $k$ is algebraically closed
By John Thompson
Let $R=\oplus_{i\ge 0} R_i $ be a graded domain such that $R_0=k$ is an algebraically closed field, $R$ is finitely generated $k$-algebra and $R=k[R_1]$. Let $d=\dim R>0$. Let $\mathfrak m=\oplus_{i>0}R_i$ be the unique homogeneous maximal ideal of $R$, so that $\dim_k R_i=\mu(\mathfrak m^i),\forall i\ge 1$. Let $e(R)=(d-1)!\lim_{n\to \infty}\dfrac {\mu(\mathfrak m^n)}{n^{d-1}}$ denote the Hilbert-Samuel multiplicity of $R$.
My question is: Is it true that $e(R)\ge \dim_k R_1 - \dim R +1 $ ?
I know this is true if $R$ is Cohen-Macaulay by Abhyankar's inequality, but even if $R$ is not Cohen-Macaulay, I think I have seen this inequality being used in articles/papers dealing with projective varieties, hence my question .
$\endgroup$1 Answer
$\begingroup$Yes, this is true. See, for example, Theorem 4.3 here. The proof is also in Abhyankar's original paper, but this is harder to access.
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