schemadefinities

Schemadefinities is a term used in some expository contexts to describe a framework for cataloguing finiteness properties that can be assigned to schemes and morphisms in algebraic geometry. A schemadefinity is a predicate P on objects of a chosen category of schemes (and possibly morphisms) that returns true or false when an object satisfies a particular finiteness condition. Examples of schemadefinities include being Noetherian, being of finite type over a base, being finite, being of finite presentation, and being proper.

In this framework, one studies how these finiteness properties behave under standard operations: base change, composition of morphisms, and passage to open or affine covers. A key feature is locality: many schemadefinities can be checked locally on the source or target. The utility of the concept lies in providing a unified language to compare different finiteness notions and to formulate general theorems about stability and descent. For instance, the class of Noetherian schemes is stable under base change and composition, and finite morphisms exhibit well-known finiteness properties that propagate through constructions such as fiber products.

The term is not standard across the literature and may appear primarily in expository or speculative contexts to organize ideas about finiteness phenomena in algebraic geometry and related fields. See also: Noetherian scheme; finite type; finite presentation; finite morphism; proper morphism; base change; descent. References: This article outlines a conceptual framework rather than a canonical, widely adopted terminology.