If a module is a limit of two inverse systems, then the two systems are isomorphic.

Question

The original problem comes from corollary (10.10.6), chapter 10, Volumn I, EGA.

I state it in the language of modules here for convenience.

Claim. If an $R$-module $F$ is a limit of two inverse(or projective) systems $\{F_n\}_{n \geq 1}$ and $\{G_n\}_{n \geq 1}$, where the morphisms $\phi_{m,n} : F_n \longrightarrow F_m, \psi_{m,n} : G_n \longrightarrow G_m, m \leq n$ satisfy $\phi_{m,n}(F_n) = F_m$ and $\psi_{m,n}(G_n) = G_m$, then the two inverse systems are isomorphic.

I believe that this claim is wrong. Are there any examples?

Edit: EGA I corollary(1, 10.10.6) is correct due to (0,7.2.9).

Answer 1

The claim is false already for vector spaces.

For example, one inverse system can have increasing even dimensions and the other can have increasing odd dimensions, and the inverse limits will both have dimension $2^{\aleph_0}$.

There may still be map between inverse systems that induces an isomorphism on the inverse limit though.

EDIT:

For example, define two inverse systems by projection onto the leading factors ($m\leq n$)

$$\phi_{m,n}: k^{2n} \rightarrow k^{2m} $$

$$\psi_{m,n}: k^{2n+1} \rightarrow k^{2m+1}$$

The inverse limits are both isomorphic to $\prod_\omega k$.