I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group.

Let $A$ be a noetherian graded $K$-algebra over a ring $K$. We write $C$ for the category of noetherian $A$-modules. Let $K_0 (C)$ be the Grothendieck group of $C$. The translation functor on noetherian $A$-modules gives an action of $\mathbb{Z}[x, x^{-1}]$ on $K_0 (C)$.

If $\lambda$ is an additive function on the set of isomorphism classes of noetherian modules (edit: noetherian $K$-modules, not noetherian $A$-modules) taking values in $\mathbb{Z}$, then $\lambda$ arranges into an additive function $\lambda_* = \sum_{i \in \mathbb{Z}} \lambda (-)$ on the isomorphism classes of $C$, sending a graded module $\oplus_{i \in \mathbb{Z}} M_i$ to the formal sum $\sum_{i \in \mathbb{Z}} \lambda( M_i)$. $\lambda_*: \text{iso}(C) \rightarrow \mathbb{Z}[[x]]$ must then factor through $K_0 (C)$.

Usually we use $\lambda_*$ in the following theorem:

**Theorem:** (Hilbert, Serre) Let $A$ be a noetherian graded $K$-algebra, and let $M$ be a noetherian module. Then there is a $m \in \mathbb{Z}$ such that $\lambda(M)(n) = f(n) \prod_{i = 1}^n (1 - n^{d_i})$ for $n > m$, where $d_i$ occur as the degrees of generators of $A$ over $K$.

If generators of degree $1$ can be chosen, then this leads to a notion of dimension, where the dimension is the degree of the pole at $1$ minus $1$.

But I wonder whether the grothendieck group can be used here instead. Is this possible? I'm hoping someone can point me towards a more theoretical approach along these lines.

For instance, one might try to characterize the function $K_0 (C) \rightarrow \mathbb{N}_{\geq 0}$ which sends a class of modules to its dimension. The standard way of getting a function (but which does not characterize it) is by going through $\mathbb{Z}[[t]]$ and look at the degree of the pole at $1$, but maybe there are properties which define this uniquely. Another way of looking at it is to ask, "what sort of equivalence relation do we put on $K_0(C)$ to get $\mathbb{N}_{\geq 0}$, with the quotient map giving the dimension?"