I'm trying to solve Liu's exercise 2.1 Chapter 5 which states:
Here is what I'm thinking so far:
a) This is vacuously true since J is a single element set so there is no $i<j$ and this is trivial.
b) 1) Consider $(df)_{ijk\alpha}$. Without loss of generality, assume $i<j$. This gives $(df)_{ijk\alpha} = 0 = f_{jk\alpha} - f_{ik\alpha} + f_{ijk} - f_{jk\hat{\alpha}} \implies f_{jk\alpha} + f_{ij\alpha} = f_{ik\alpha} + f_{ijk\hat{\alpha}}$. Now since $f_{ij\alpha} = g_{j\alpha} - (dg_i)_{j\alpha}$, we get that $f_{jk\alpha} + g_{j\alpha} - (dg_i)_{j\alpha} = f_{ik\alpha} + f_{ijk\hat{\alpha}}$. I'm having trouble dealing with this last $f_{ijk\hat{\alpha}}$ term and I don't know how we can develop it to get everything to cancel out nicely.
- I'm guessing here is where we use the note mentioned in part (a) that $\mathcal{C}^{p}(\mathcal{U,F}) = \mathcal{C}^{p-1} (\mathcal{U_i,F}) \times \prod_{i \neq i'} \mathcal{C}^{p-1} (\mathcal{U_{i'}},F)$ to glue together the $g_i$s we fund before for each $i$ but I haven't had too much insight with that. I don't quite understand this and the final part of b.
c) I'm honestly quite confused by this. Why would we need to use Zorn's lemma here? wouldn't part a) and b) be enough by induction?
I'm mainly looking for either help with these questions or a reference. Thanks