Flat module

Summary

A flat module is a module $M$ such that tensoring with $M$ keeps injective linear maps injective. Equivalently, the functor $- \otimes_R M$ preserves exact sequences. Free and projective modules are flat, and so are localizations such as $\Q$ over $\Z$. Flatness is the algebraic form of "varying continuously in a family". Flat ring maps are the basic tool of descent and base change in algebraic geometry.