We study the cohomology of an elliptic differential complex arising from the infinitesimal moduli of heterotic string theory in the supergravity approximation. We compute these cohomology groups at the standard embedding, and show that they decompose into a direct sum of cohomologies. While this is often assumed in the literature, it had not been explicitly demonstrated. Given a stable gauge bundle over a complex threefold with trivial canonical bundle and no holomorphic vector felds, we also show that the Euler characteristic of this differential complex is zero. This points towards a perfect obstruction theory for the heterotic moduli problem, at least for the most physically relevant compactifcations.