4 clarified Noeterian hypotheses

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 let's specialize to thinking of we can just look at primary decompositions of $0$. So suppose let $M$ is be a finitely generated module over a Noetherian ring $A$, and $0=N_1\cap\cdots\cap N_n$ is 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 (theorems, if you take the right definitions)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)

3 clarification

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. WLOG let's specialize to thinking of primary decompositions of $0$.

So suppose $M$ is a finitely generated module over a Noetherian ring $A$, and $0=N_1\cap\cdots\cap N_n$ is 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), 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 to make sense of the above (theorems, if you take the right definitions):

• 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)

2 fixed reference / remotivated

Primary

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 for of idealsin a ring has a pretty standard geometric interpretation: the primary components of an ideal $I \triangleleft A$ represent 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).

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 sub-$A$-modulevery good visualization. WLOG let's specialize to thinking of primary decompositions of $N\subseteq 0$.
So suppose $M$ ? How should I picture this in algebraic geometry land? Lets restrict to is a finitely generated modules module over a Noetherian ring , and WLOG we can look at $N=0$.
So say A$, and$0=N_1\cap\cdots\cap N_n$is 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 particular that the$N_i$are primary), and (2) the relationship of the modules$N_i$to their co-associated primes, say {$P_i$}$ = Ass(M/N_i) \subseteq Spec(A)$? Some useful background to make sense of the above (theorems, if you take the right definitions): • The primes$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, Theorem Corollary 1.3.11)