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Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...

Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...

If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...

I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...

Let $A=\bigoplus A_i$ be a finitely generated commutative unital graded algebra over a field $k$. Let $d(i)=\dim A_i$. In general $d(i)$ is not a polynomial in $i$ (even not eventually polynomial). ...