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Dec 26, 2012 at 13:52 comment added Donu Arapura I'm not sure about the origin of the notion either, but certainly "pro-algebraic hulls" can be at least traced back to Hochschild & Mostow, Pro-affine algebraic groups, Amer. J. Math 1969.
Dec 26, 2012 at 0:51 comment added Qiaochu Yuan The (maximal) spectrum is of an $I$-indexed product of copies of a field $k$ contains, and ought to just be equal to, the Stone-Cech compactification $\beta I$ of $I$ (sketch: look at what happens to idempotents); identifying $\beta I$ with the set of ultrafilters on $I$, the quotient by the maximal ideal corresponding to an ultrafilter is the corresponding ultrapower of $k$.
Dec 26, 2012 at 0:35 history answered S. Carnahan CC BY-SA 3.0