Search results for “commutative algebra”

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.

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.

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.

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.

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