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):
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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)
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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)
Best Answer
I will assume that everything in sight is Noetherian and finitely generated. Then the primary decomposition amounts to a description of the geometric support of the module $M$ if you view it as a sheaf on $\text{Spec}(A)$. The scheme $\text{Spec}(A/\text{Ann}(M))$ is a subscheme of $\text{Spec}(A)$ that, by definition, supports $M$. If $N_i$ is minimal, then $\text{Spec}(A/P_i)$ is an irreducible component of $\text{Spec}(A/\text{Ann}(M))$. The module $M/N_i$ is also a quotient of $M \otimes (A/P_i^n)$ for $n$ large enough (and in a reduced decomposition I think they are equal), so it is part of $M$ on that irreducible component of its support. If $N_i$ is not minimal, then $\text{Spec}(A/P_i)$ is an irreducible scheme inside of a component of $\text{Spec}(A/\text{Ann}(M))$.