Learning about the Krull dimension of modules for the first time. The standard definition for an $A$-module $M$ seems to be $\dim(M)=\dim(A/ann(M))$. This definition seems to make sense for finitely generated modules, since in that case $Supp(M) = V(ann(M))$. To me it seemed that the information we are after is the topological dimension of $Supp(M)$.
Yet this breaksdown in the non-f.g. case. In particular, let $M=\oplus_{n \geq 1} \mathbb{Z}/3^n\mathbb{Z}$. Then $ann(M)=0$ so we should have that $\dim(M)=1$, yet $Supp(M)$ consists of a single point the ideal $(3)$. So its dimension as a topological space is $0$.
So what explains this discrepancy, are we not actually after the dimension of $Supp(M)$? Do we even care about Krull dimension in the non-finitely generated case?
As you noticed, there are two different Krull dimensions for arbitrary modules which agree for finitely generated ones, but in general they can differ. (With respect to the question if we care about this in general, the answer is positive: the local cohomology modules are usually not finitely generated, but we care about their Krull dimension.)