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User fbakhshi deleted the following question:

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\geq 1$ for all $i$, then prove that $$ R[It]/\mathfrak m R[It]\simeq K[f_1t,\ldots, f_q t]\simeq K[f_1,\ldots,f_q] \text{ (as $K$-algebras).} $$ $R[It]$ denotes the Rees algebra of $I$ and $\mathfrak m=(x_1,\ldots,x_n)$.

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1 Answers1

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$R[It]/\mathfrak mR[It]$ is $\oplus_{n\geq 0}{I^n/\mathfrak mI^n}.$ Let $\phi$ be any homogeneous polynomial of degree $l$. Consider $I_l$ to be the $k$-vector space generated by all $\phi(f_1,\ldots,f_q).$ Then $k[f_1,\ldots,f_q]=\oplus_{l\geq 0}{I_l}.$ Now $\dim_{k}{I_l}=\dim_{k}{I^l/\mathfrak mI^l}.$ Hence $k[f_1,\ldots,f_q]\simeq {R[It]/\mathfrak mR[It]}.$

user85835
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  • So, if the graded pieces of two graded $k$-algebras are isomorphic as $k$-vector spaces, that is, these rings have the same Hilbert series, then the rings are isomorphic, too. What do you think, is this right? –  Jul 11 '13 at 10:40