Search results for “commutative algebra”

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.

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.

Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.

An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.