Let $R$ be a Noetherian positively graded $k$-algebra of dimension $1$, where $k$ is an algebraically closed field. If length $l(R_n)=H (R, n) > 1$ for some $n$, show $R$ is not a domain.
I assume $R$ is a domain, then there exists a homogeneous element $x$ of degree $d$ such that we have the exact sequence $0\longrightarrow R[-d]\stackrel{x}\longrightarrow R\longrightarrow R/(x)\longrightarrow0$. Therefore $H(R,n)+H(R/(x),n+d)=H(R,n+d)$. Since $\dim R/(x)=0$, $H(R,n)=l(R/(x))_n=0$ for $n\gg0$. I don't know if this is helpful and I am also not sure how to use the algebraically closed condition of the field.