The Stacks project

41.10 Topological properties of flat morphisms

We “recall” below some openness properties that flat morphisms enjoy.

Theorem 41.10.1. Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The set of points in $X$ where $\mathcal{F}$ is flat over $Y$ is an open set. In particular the set of points where $f$ is flat is open in $X$.

Proof. See More on Morphisms, Theorem 37.15.1. $\square$

Theorem 41.10.2. Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is flat and locally of finite type. Then $f$ is (universally) open.

Proof. See Morphisms, Lemma 29.25.10. $\square$

Theorem 41.10.3. A faithfully flat quasi-compact morphism is a quotient map for the Zariski topology.

Proof. See Morphisms, Lemma 29.25.12. $\square$

An important reason to study flat morphisms is that they provide the adequate framework for capturing the notion of a family of schemes parametrized by the points of another scheme. Naively one may think that any morphism $f : X \to S$ should be thought of as a family parametrized by the points of $S$. However, without a flatness restriction on $f$, really bizarre things can happen in this so-called family. For instance, we aren't guaranteed that relative dimension (dimension of the fibres) is constant in a family. Other numerical invariants, such as the Hilbert polynomial, too may change from fibre to fibre. Flatness prevents such things from happening and, therefore, provides some “continuity” to the fibres.