Search results for “Noetherian rings”

A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.

Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.

A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.

A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.