The Going-Up and Going-Down Theorems, also known as the Cohen-Seidenberg Theorems, describe how chains of prime ideals behave under an integral ring extension, establishing conditions under which a chain of prime ideals in the smaller ring can be extended to a corresponding chain in the larger ring, either by upward inclusion or by downward inclusion. Proved by Irvin S. Cohen and Abraham Seidenberg, they are foundational results of commutative algebra used throughout the study of integral extensions and algebraic geometry's dimension theory.
Facts
StatementFor an extension A contained in B of commutative rings, the going-up and going-down theorems give sufficient conditions for a chain of prime ideals in B, each member of which lies over members of a longer chain of prime ideals in A, to be extended to the length of the chain in A. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Going up and going down (Wikipedia)
Sources
1. Going up and going down (Wikipedia)
Going up and going down
give sufficient conditions for a chain of prime ideals in B, each member of which lies over members of a longer chain of prime ideals in A
In Branch: Commutative Algebra, Lead sentence
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of
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