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.

Local algebra. – Liviu Nicolaescu Apr 24 '13 at 14:28