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We give a rigorous mathematical proof for the validity of the toric sheafcohomology algorithm conjectured in the recent paper by R. Blumenhagen, B.Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only theoriginal algorithm but also a speed-up version of it. Our proof is independentfrom (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T.Rahn (arXiv:1006.2392), and has several advantages such as being shorter andcleaner and can also settle the additional conjecture on "Serre duality forBetti numbers" which was raised but unresolved in arXiv:1006.2392.

Cohomology of toric line bundles via simplicial Alexander duality