Hilbert's Theorem 90 and algebraic spaces

Details Hilbert's Theorem 90 and algebraic spaces Hilbert's Theorem 90 and algebraic spaces

2001-10-22

arxiv.org/abs/math/0110243v1

J. Pure Appl. Algebra 173 (2002), 339-345

Mathematics

Algebraic Geometry

6 pages, to appear in J. Pure Appl. Algebra

Scientific paper

In 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.

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