“In mathematics the art of proposing a question must be held of higher value than solving it.”
George Cantor
This week I finally finished my final project for my PMATH 499 reading course in arithmetic geometry. My project is called “Galois cohomology and weak Mordell–Weil theorem”, which is available here. Galois cohomology is the application of group cohomology to Galois groups. In particular, we know that for a perfect field \(k\) with an algebraic closure \(K\), the Galois group \(\mathrm{Gal}(K/k)\) is isomorphic to an inverse limit of topological Galois groups \(\mathrm{Gal}(L/k)\) ranging over the finite Galois extensions \(L/k\) with the natural projections. This fact makes \(G\) a profinite group, and the Krull topology, the topology of \(G\) endowed by the inverse limit process by viewing each finite Galois group as a topological group with the discrete topology, is a topology where a basis for the neighborhood at the identity is the collection of normal subgroups having finite index in \(G\). This is a motivation to use homological algebra in such situation.