Hilbert's Theorem 90 and algebraic spaces

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62466In modern form, Hilbert's Theorem 90 tells us that R^1f_*(G_m)=0, where f is the canonical map between the etale site and the Zariski site of a scheme X. I construct examples showing that the corresponding statement for algebraic spaces does not hold. The first example is a nonseparated smooth 1-dimensional bug-eyed cover in Kollar's sense. The second example is a nonnormal proper algebraic space obtained by identifying points on suitable nonprojective smooth proper schemes.6 pages, to appear in J. Pure Appl. AlgebraAlgebraic Geometry14A20, 14C22, 14F20Hilbert's Theorem 90 and algebraic spacestext