The Stacks project

110.33 A finite flat module which is not projective

This is a copy of Algebra, Remark 10.78.4. It is not true that a finite $R$-module which is $R$-flat is automatically projective. A counter example is where $R = \mathcal{C}^\infty (\mathbf{R})$ is the ring of infinitely differentiable functions on $\mathbf{R}$, and $M = R_{\mathfrak m} = R/I$ where $\mathfrak m = \{ f \in R \mid f(0) = 0\} $ and $I = \{ f \in R \mid \exists \epsilon , \epsilon > 0 : f(x) = 0\ \forall x, |x| < \epsilon \} $.

The morphism $\mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R)$ is also an example of a flat closed immersion which is not open.

Lemma 110.33.1. Strange flat modules.

1. There exists a ring $R$ and a finite flat $R$-module $M$ which is not projective.

2. There exists a closed immersion which is flat but not open.

Proof. See discussion above. $\square$