The Stacks project

Example 10.82.6. Non-split and non-flat universally exact sequences.

1. In spite of Lemma 10.82.4, it is possible to have a short exact sequence of $R$-modules

   $$0 \to M_1 \to M_2 \to M_3 \to 0$$

   that is universally exact but non-split. For instance, take $R = \mathbf{Z}$, let $M_1 = \bigoplus _{n=1}^{\infty } \mathbf{Z}$, let $M_{2} = \prod _{n = 1}^{\infty } \mathbf{Z}$, and let $M_{3}$ be the cokernel of the inclusion $M_1 \to M_2$. Then $M_1, M_2, M_3$ are all flat since they are torsion-free (More on Algebra, Lemma 15.22.11), so by Lemma 10.82.5,

   $$0 \to M_1 \to M_2 \to M_3 \to 0$$

   is universally exact. However there can be no section $s: M_3 \to M_2$. In fact, if $x$ is the image of $(2, 2^2, 2^3, \ldots ) \in M_2$ in $M_3$, then any module map $s: M_3 \to M_2$ must kill $x$. This is because $x \in 2^ n M_3$ for any $n \geq 1$, hence $s(x)$ is divisible by $2^ n$ for all $n \geq 1$ and so must be $0$.

2. In spite of Lemma 10.82.5, it is possible to have a short exact sequence of $R$-modules

   $$0 \to M_1 \to M_2 \to M_3 \to 0$$

   that is universally exact but with $M_1, M_2, M_3$ all non-flat. In fact if $M$ is any non-flat module, just take the split exact sequence

   $$0 \to M \to M \oplus M \to M \to 0.$$

   For instance over $R = \mathbf{Z}$, take $M$ to be any torsion module.

3. Taking the direct sum of an exact sequence as in (1) with one as in (2), we get a short exact sequence of $R$-modules

   $$0 \to M_1 \to M_2 \to M_3 \to 0$$

   that is universally exact, non-split, and such that $M_1, M_2, M_3$ are all non-flat.