2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97261In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $σ$-ample divisor, where $σ$ is an automorphism of a projective scheme X. Many open questions regarding $σ$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $σ$-ample. As a consequence, we show right and left $σ$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $σ$ yield a $σ$-ample divisor.16 pages, LaTeX2e, to appear in J. of the AMS, minor errors corrected (esp. in 1.4 and 3.1), proofs simplifiedAlgebraic GeometryRings and Algebras14A22, 14F17, 14J50, 16P90, 16S38, 16W50text