  We study a large deviation principle for a system of stochastic
reaction--diffusion equations (SRDEs) with a separation of fast and slow
components and small noise in the slow component. The derivation of the large
deviation principle is based on the weak convergence method in infinite
dimensions, which results in studying averaging for controlled SRDEs. By
appropriate choice of the parameters, the fast process and the associated
control that arises from the weak convergence method decouple from each other.
We show that in this decoupling case one can use the weak convergence method to
characterize the limiting process via a "viable pair" that captures the
limiting controlled dynamics and the effective invariant measure
simultaneously. The characterization of the limit of the controlled slow-fast
processes in terms of viable pair enables us to obtain a variational
representation of the large deviation action functional. Due to the
infinite--dimensional nature of our set--up, the proof of tightness as well as
the analysis of the limit process and in particular the proof of the large
deviations lower bound is considerably more delicate here than in the
finite--dimensional situation. Smoothness properties of optimal controls in
infinite dimensions (a necessary step for the large deviations lower bound)
need to be established. We emphasize that many issues that are present in the
infinite dimensional case, are completely absent in finite dimensions.
