Power grids use a constantly changing mix of fossil fuels and clean renewables like solar and wind. Because of this, the {\em carbon footprint} fluctuates drastically based on time and geographic location. Time-varying carbon intensity creates opportunities to reduce carbon emissions by shifting workloads to cleaner periods, but deadlines, task dependencies, and limited edge capacity constrain this flexibility. We study carbon-aware scheduling of jobs with release times, deadlines, and arbitrary precedence constraints under piecewise-constant carbon costs. Jobs execute preemptively on a single edge server or are outsourced at job-dependent costs to an unlimited-capacity cloud, modeled as providing instantaneous execution. The objective is to minimize the sum of edge carbon costs and cloud outsourcing costs while satisfying all timing and precedence constraints. For edge-only execution, we give an exact polynomial-time algorithm. Allowing cloud outsourcing makes the problem NP-hard, but we obtain a fully polynomial-time approximation scheme through a reduction to preemptive scheduling with rejection penalties (weighted tardiness). In contrast, even two edge servers without job migration admit no polynomially bounded approximation when carbon-cost variation is unrestricted, unless \(P=NP\). We complement these theoretical results with experiments on controlled synthetic workloads and production-derived Alibaba traces. On synthetic instances, our incremental max-flow implementation is $4.96\times$ faster on average than a direct min-cost flow, while experiements on production-derived workloads evaluate its performance under realistic job structures.

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