Abstract: A basic and old Folklore Problem in number theory asks: If $O_K$ is the ring of integers of a number field $K$, is every rank $d > 1$ projective $O_K$-module isomorphic to the ring of integers in a field extension $L/K$? Every rank $d$ projective module over a Dedekind domain like $O_K$ is isomorphic to a direct sum of a rank $d-1$ free module and a well defined ideal class $I$. The class $I$ is sometimes called the Steinitz class of the module, and hence the folklore problem is often restated: Which Steinitz classes are realized by degree $d$ field extensions? Although every Steinitz class is expected to be realizable, a proof of this statement seems far beyond the reach of current methods. This is probably why I made no progress on it. In joint work with Anand Deopurkar, however, we solve the (complex) geometric version of the Folklore Problem. I will speak about these things and more.
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