Doctor of Philosophy
Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois cohomology is a Koszul algebra. In particular, we show that for all known cases where the maximal pro-$p$-quotient of the absolute Galois group is finitely generated, Galois cohomology is universally Koszul. Assuming the Elementary Type conjecture, this gives us infinitely many refinements of the Bloch-Kato Conjecture. We moreover obtain several unconditional results. Lastly, we show that all forms of enhanced Koszulity are preserved under certain natural operations, which generalizes results that were only known to hold in the commutative case.
Palaisti, Marina, "Enhanced Koszulity in Galois cohomology" (2019). Electronic Thesis and Dissertation Repository. 6038.
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