I am studying a paper which uses the following lemma. The context is irrelevant, as the lemma is only used as a technical trick and has no pointer to a reference or hint in the proof but its link to theory of heights in $p$-torsion abelian groups (which I ignore).
There exists a sequence $B_\bullet = (B_\alpha)$ of $p$-torsion abelian groups, one for each ordinal number $\alpha\in \text{Ord}$, satisfying the following properties:
- each $B_\alpha$ contains an element $x_\alpha$ such that $p x_\alpha = 0$;
- when $\alpha < \beta$ every function of sets $f_{\alpha\beta} \colon B_\alpha \to B_\beta$ such that $f(px)=p f(x)$ sends $x_\alpha$ to zero.
Also, I would like to know if the following refinement of the result is true.
There exists a sequence $B_\bullet = (B_\alpha)$ of $p$-torsion abelian groups, one for each ordinal number $\alpha\in \text{Ord}$, satisfying the following properties:
- each $B_\alpha$ contains an element $x_\alpha$ such that $p x_\alpha = 0$;
- when $\alpha < \beta$ every function of sets $f_{\alpha\beta} \colon B_\alpha \to B_\beta$ such that $f(px)=p f(x)$ sends $x_\alpha$ to zero.
- There exists B such that each $B_\alpha$ has a map $B_\alpha \to B$ of groups that does not vanish on $x_\alpha$.
I am pretty sure that function of sets is not needed, so if someone has the answer with the hypotesis morphisms of group, please don't be shy.
