I recently learnt about profinite group cohomology to do class field theory and I am looking for a proof of the profinite version of the inflation-restriction sequence which hopefully just uses the version for usual group cohomology (in particular $0\to Br(L/K)\to Br(K)\to Br(L)\to 0$ for $L/K$ a Galois and finite field extension) for profinite groups. Is there a proof not using the Hochschild–Serre spectral sequence. I found a proof in Cassel Frohlich p126 which uses the inflation-restriction sequence for usual group cohomology, which seems be to incorrect by Kevin Buzzard https://www.ma.ic.ac.uk/~buzzard/errata.pdf.
My idea is that maybe the inflation-restriction sequence for usual group cohomology can be naturally generalizes since the usual group cohomology and profinite group cohomology are both defined using derived functors, one of general modules and the other for discrete modules. In this case, I need a more abstract version of the inflation-restriction sequence, for example for general derived functors, but I didn't find anything about this generalization.
Alternatively, I guess I can also develop the theory of profinite group cohomology from scratch. This takes a lot of work though.
Edit: is this correct? to generally prove the inflation-restriction sequence might be hard without spectral sequences, but I guess we don't need it to show the special case applying to Brauer groups.
$$0\to Br(L/K)\to Br(K)\to Br(L)\to 0.$$
The Brauer group can be thought of direct limit of those of the finite subextensions with inflation being the transition maps. By Hilbert 90 and the usual inflation-restriction sequence, this direct limit is just direct sum. Since the maps in the above sequence are induced by component-wise inflation and restriction, the kernels and images in this sequence are direct sum of those in the inflation-restriction sequence for finite subextensions. Then the exactness naturally generalizes to the above seqeunce.