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 ?