2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/117837We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.15 pages; minor errors were corrected, some new results and references were added, v3: minor revisionsAlgebraic Geometry14F17; 14F25text