Faithful Flatness

Abstract

This is primarily a technical chapter introducing another algebraic tool. We will use it at once to complete the proof of the smoothness theorem (11.6) and then draw on it throughout the rest of the book. To begin, we call a ring homomorphism A → B flat if, whenever M → N is an injection of A-modules, then M ⊗ _A B → N ⊗ _A B is also an injection. For example, any localization A →S⁻¹ A is flat. Indeed, an element m⊗a/s in M⊗S⁻¹ A= S⁻¹ M is zero iff tam = 0 for some t in S; if M injects into N and tam is zero in N, it is zero in M. What we really want, however, is a condition stronger than flatness and not satisfied by localizations.