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 ?



Support of a module over a polynomial algebra
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: 


1. $\forall f: f\cdot E = 0 \Rightarrow f(a)=0$
2. $\exists g: g \cdot D=0 \wedge g(a)\neq 0$
3. $\exists h: h \cdot F=0 \wedge h(a)\neq 0$


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. 


