Support of a module over a polynomial algebra In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$
$$D \to E \to F,$$
we have that
Supp $E \subset$ Supp $D \cup$ Supp $F$,
where Supp$E$ is defined to be $\cap V_f$ where $f$ runs over all polynomials such that $fE=0$.
This must be quite elementary, but I can't manage to prove it.
I tried to show that if some $f$ annihilates $E$, then it anihilates either $F$ or $D$ (which, if I'm not mistaken, implies what we want), but I didn't manage this either. Is it too strong ? Is the fact that the ground ring is $\mathbb{C}[u_1,...,u_l]$ that important ?
 A: If I understand the definitions properly, the following should work: 
Let $D \xrightarrow{\alpha} E \xrightarrow{\beta} F$ be exact and suppose $\text{Supp} E \not\subseteq \text{Supp}D \cup \text{Supp}F$. Hence there is $a \in \text{Supp} E$ such that $a \notin \text{Supp}D$ and $a \notin \text{Supp} F$. This means: 


*

*$\forall f: f\cdot E = 0 \Rightarrow f(a)=0$

*$\exists g: g \cdot D=0 \wedge g(a)\neq 0$

*$\exists h: h \cdot F=0 \wedge h(a)\neq 0$


For $x\in E$ we have $\beta(hx)=h\beta(x)=0$, i.e. $hx\in \ker \beta=\text{im}\;\alpha$. Thus  there is $y\in D$ such that $hx=\alpha(y)$. So $ghx=g\alpha(y)=\alpha(gy)=\alpha(0)=0$ and consequently $ghE=0$. Now, by 1., $0=(gh)(a)=g(a)h(a)$, contradicting 2. and 3. q.e.d.  
A: As Ralph mentioned, if I understand the definitions properly, there is another way to see this by using a short exact sequence. I believe that $V_f = V(f)$ where $R = \mathbb{C}[u_1, \dots, u_l]$ and $V(f) = \{ p \in \hbox{spec(R)} \mid f \in p \}$ since we are over an algebraically closed field. If so we have $\hbox{Supp(-)} = \cap V_f = V(I)$ for some ideal $I$.

Lemma. Let $R$ be a commutative Noetherian ring. Let $N,M,L$ be $R$-modules such that $$0 \to N \to M \to L \to 0$$ is exact. Then we have $\hbox{Supp} (M) = \hbox{Supp} (L) \cup \hbox{Supp} (N)$.
  Here $\hbox{Supp} (M) = \{ p \in \hbox{Spec}(R) \mid M_p \neq 0 \}$. 

We give a name to the maps in your exact sequence, i.e,.
$$ D \stackrel{d}{\to} E \stackrel{e}{\to} F. $$
Then $$ 0 \to \hbox{im}(d) \to E \to \hbox{coker}(d) \to 0$$ is exact where $\hbox{im}(-)$ denotes the image of a map.
Hence $\hbox{Supp}(E) = \hbox{Supp}(\hbox{im}(d)) \cup \hbox{Supp}(\hbox{coker} (d))$ by the lemma. Since the sequence $ D \stackrel{d}{\to} E \stackrel{e}{\to} F$ is exact, we have $\hbox{im} (d) = \hbox{ker} (e)$, and this implies that $\hbox{coker}(d) = E / \hbox{im}(d) = E / \hbox{ker}(e)$. 
Applying the lemma again to the exact sequences $D \to \hbox{im}(d) \to 0$ and $0 \to \hbox{coker}(d) \to F$ we have $\hbox{Supp}(\hbox{im}(d)) \subseteq \hbox{Supp}(D)$ and $\hbox{Supp}(\hbox{coker}(d)) \subseteq \hbox{Supp}(F)$, respectively. This shows the containment in the question.
A: This is more or less a classical result but I believe  some comments are warranted.  Atiyah and Bott  are   first and foremost   geometers/topologists and they   opted for a more  geometric description of the concept of support better adapted to the purposes of that (wonderful) paper.
One can define  the support of a module over an arbitrary ring $R$. It is a subset of  $\mathrm{spec} (R) =$ the   set of all prime ideals   of $R$.     When $R$ is the ring of polynomials over $\mathbb{C}$ (more generally an  algebraically closed field) a remarkable accident  happens, and it goes by the name  Hilbert Nullstellensatz. This  theorem establishes a correspondence between varieties  of $\mathbb{C}^n$ and ideals of $\mathbb{C}[z_1,\dotsc,z_n]$.  Each variety decomposes into irreducible components  and the irreducible components   correspond  to prime ideals.
As for the  localization theorem, I taught this subject a while ago in a graduate course. I only looked at the special case of Hamiltonian $S^1$-actions which already has  many nontrivial consequences,  and all the ideas needed in the general case  are needed in this special as well.
You might want to look over  section 3.5 of   these course notes   where  I go  through the proof in great detail and  at a slower speed  than  Atiyah-Bott.      In the  case $n=1$ the question you ask  has a more familiar  interpretation and the spectrum  can be identified with the spectrum of a matrix, whence the  the word  spectrum.  As I point in the notes,   the key technical result of the  localization theorem was known to A. Borel, almost two decades prior to Atiyah-Bott's work.
