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Commutative Algebra

The theory of commutative rings and their ideals, and the algebraic ground on which algebraic geometry is built.

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A non-Noetherian local ring with finitely generated maximal ideal

2026NotesBogdan Zavyalov

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.

The torsion component of the Picard scheme

2026OtherBogdan Zavyalov

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.