Flatderived

Flatderived is a term found in discussions of algebraic geometry and homological algebra to describe a derived analogue or generalization of flatness. It is not a widely standardized notion, and its precise definition can vary between authors. In broad terms, flatderived refers to conditions on complexes of modules or on morphisms of schemes that generalize the classical idea of flatness to the setting of derived categories and derived functors.

One common informal thread is to require that a complex of modules be, up to quasi-isomorphism, built from flat modules or to demand that the derived tensor product with any module behaves in a flat or exact manner. In this spirit, a complex F over a ring R might be described as flatderived if it has Tor-amplitude contained in a nonnegative range, or if it is quasi-isomorphic to a flat complex and the derived tensor product with arbitrary modules preserves exactness. Another perspective uses the notion of flatness in the derived category, aiming for a condition that ensures desirable base-change and deformation properties in derived geometry.

Applications of flatderived ideas appear in deformation theory, derived algebraic geometry, and cohomological base-change results, where controlling how tensoring interacts with derived structures is important. In practice, the term highlights the desire to treat flatness in a setting where higher homological information matters, rather than focusing only on ordinary modules or sheaves.

See also: flat morphism, derived category, Tor, derived functors, base change, derived algebraic geometry.