flatsimplicitly

Flatsimplicitly is a term that has appeared in informal discussions within algebraic geometry and related fields. It refers to a heuristic or criterion by which flatness of a morphism or family is inferred indirectly from local or fiberwise data, rather than verified by the classical flatness criterion (for example, by checking that tensoring with arbitrary modules preserves exact sequences). The term is not standard in textbooks or references, and its meaning can vary with context.

In practice, a morphism f: X -> S might be described as flatsimplicitly if, after an appropriate base change and under certain hypotheses, the fibers X_s are flat over their residue fields and the variation of invariants (such as the Hilbert polynomial of fibers, dimension, or cohomology) is controlled in a way that supports flatness of X over S. This sense emphasizes that flatness is being deduced from fiberwise data or from known descent and base-change phenomena rather than from a direct verification of the flatness condition.

Usage is largely informal and appears in lecture notes, expository articles, and some research discussions. It is often used to summarize that a family is flat by applying standard theorems (such as miracle flatness, generic flatness, or descent) that connect fiberwise properties to global flatness.

Examples are typically drawn from families of schemes over a base where fibers are equidimensional or have constant Hilbert polynomial. In such cases, one may say the family is flatsimplicitly flat to indicate that the conclusion follows from the behavior of the fibers rather than from an explicit flatness check.

See also: flat morphism, fiber, base change, miracle flatness, descent theory, Hilbert polynomial.