I want to know how I should visualize *modules* in algebraic geometry. The way we visualize *rings*, via their spectra, automatically (or by the beauty of its design) depicts primary decomposition of *ideals*: the primary components of an ideal $I \triangleleft A$ cut out "primary subschemes" (irreducible and embedded components) whose union is $Z(I)=Spec(A/I)$. (See, for example, Eisenbud and Harris, *The Geometry of Schemes*, II.3.3, pp. 66-70). This aspect of scheme theory is essential to what makes it "geometric."

By this standard, I think however we visualize *modules* should allow us to depict primary decomposition of *submodules*; otherwise I would say it's not a very good visualization.

If we're happy taking quotients, WLOG we can just look at primary decompositions of $0$. So let $M$ be a finitely generated module over a Noetherian ring $A$, and $0=N_1\cap\cdots\cap N_n$ be a primary decomposition of $0$ in $M$, with primes $P_i$ co-associated to the primary modules $N_i$, i.e. associated to the coprimary modules $M/N_i$.

How can one visualize the modules $M,N_1,\ldots,N_n$ in relation to $Spec(A)$ in a way that meaningfully depicts:

(1)the primary decomposition of $0$ in $M$ (in particular that the $N_i$ are primary in $M$), and(2)the relationship of the modules $N_i$ to their co-associated primes, say

{ $P_i$ } $ = Ass(M/N_i) \subseteq Spec(A)$?

Some useful background results to make sense of the above (all rings and modules are Noetherian):

The primes $P_i$ co-associated to $N_i$ are precisely the associated primes of $M$ (see R. Ash,

*Comutative Algebra*, Theorem 1.3.9)A module $Q$ is coprimary iff it has exactly one associated prime $P$, and then $P=\sqrt{ann Q}$. (see R. Ash,

*Comutative Algebra*, Corollary 1.3.11)