Support of a module over polynomials: intuitive meaning I find myself working with modules over complex polynomials in more variables, say $C[u_1,\dots,u_l]$. One can introduce the concept of support of the module $H$ as
\begin{equation}
supp H = \cap_{f:fH=0} V_f
\end{equation}
with $V_f\subset \mathbb C^l$ the set of points $u\in \mathbb C^l$ such that $f(u)=0$.
Now: what's the intuitive idea behind the support? Does it have anyhting to do with the support of a function, conceptually speaking?
For example, when the module is free the support is the whole space. Now say $H=C[u_1,u_2]/(u_1-u_2)$, then the support is the line $u_1=u_2$; I would be tempted to say, in a complete uninformal way, that if we restrict the domain of the polynomials from $C^2$ to $supp H$, then the module is free. Can this be formalized, in any sense? Is then the support a subset of the domain making the module free, in some sense?
Follow-up question. Suppose that I pick $H=C[u]/(u-2)\otimes C[u]/(u-3)$. Then the support is the empty set. What would this mean?
 A: My way of thinking about support of modules is in the context of schemes (but you don't need the full generality to get something out of it).
Basically, if $R$ is a commutative ring, then you think of $Spec(R)$ as a geometric space whose ring of regular functions is $R$. Then any $R$-module $M$ gives a sheaf of modules $\widetilde{M}$ over $Spec(R)$. 
In a simpler language it means that when you localize $R$, you can also localize $M$ ($S^{-1}M$ is a $S^{-1}R$-module), so you can think of $M$ as local data that glue together.
In particular, if $\mathfrak{p}$ is a prime ideal (ie a point of $Spec(R)$), then you can localize at this point : $M_\mathfrak{p}$ is a $R_\mathfrak{p}$-module. In the case $R = \mathbb{C}[x_1,\dots,x_n]$ this has a strong geometric intuition : if $\mathfrak{p}$ is a maximal ideal, then it canonically corresponds to a point $a=(a_1,\dots,a_n)\in \mathbb{C}^n$ and $R_\mathfrak{p}$ is the ring of germs of regular functions at $a$.
Now the support of $M$ is the set of $\mathfrak{p}$ such that $M_\mathfrak{p}\neq 0$. So it's the set of points at which $M$ is non-zero. If you think of elements of $M$ as functions, you can interpret $M(\mathfrak{p}) = M_\mathfrak{p}/\mathfrak{p}M_\mathfrak{p}$ as the vector space (over $k(\mathfrak{p}) = R_\mathfrak{p}/\mathfrak{p}R_\mathfrak{p}$, which in your case is just $\mathbb{C}$) of values at $\mathfrak{p}$, and the natural function $M\to M_\mathfrak{p}\to M(\mathfrak{p})$ as the evaluation map. Then the support would be the set of points over which there is at least one function in $M$ that doesn't vanish.
Your definition amounts to considering only maximal ideals of $R = \mathbb{C}[x_1,\dots,x_n]$ instead of prime ideals, which is a reasonable first approximation of the finer notion I used here (considering prime ideals allows you to localize not only at points of $\mathbb{C}^n$ but also at irreducible subvarieties).
Note that the examples you put in your question are all of the form $R/I$ for an ideal $I$. In this case the support is just $V(I)$, the variety defined by $I$, and $R/I$ is the ring of regular functions on $V(I)$. But this is really a special case because in this case your module has a natural algebra structure.
