References for some analogs of the Picard group. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:42:10Z http://mathoverflow.net/feeds/question/64577 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64577/references-for-some-analogs-of-the-picard-group References for some analogs of the Picard group. Alex Gavrilov 2011-05-11T08:28:16Z 2012-02-08T20:32:38Z <p>Let $X$ be a compact complex manifold. By definition, $Pic(X)={\rm H^1}(X,\mathcal{O}^\times)$. We know a lot about this group. What is known about the groups ${\rm H^n}(X,\mathcal{O}^\times)$ for $n\ge 2$? </p> <p>A bit more specialized question. It is well known that for a nonsingular projective complex variety $X$ the natural map $${\rm H^1}(X,\mathcal{O}^\times)\to{\rm H^1}(X,\mathcal{M}^\times)$$ is trivial. What is known about the kernel of the same map for $n=2$ or $n=3$? (Here $\mathcal{M}^\times$ is the sheaf of nonzero meromorphic functions, and the topology is the strong one).</p> http://mathoverflow.net/questions/64577/references-for-some-analogs-of-the-picard-group/64586#64586 Answer by Piotr Achinger for References for some analogs of the Picard group. Piotr Achinger 2011-05-11T09:14:06Z 2011-05-11T09:14:06Z <p>First of all, it probably depends on how you define $H^1(X, \mathcal{O}^\times)$. I don't see any reason why derived functor cohomology should agree here with Cech cohomology. </p> <p>I think that $H^i(X, \mathcal{O}^\times)$ is a functor of order $i+1$ in the sense of Mumford "Abelian Varieties" (2.6, Remark preceding the proof of the theorem of the cube), at least for complex projective varieties. That is, there is a higher analogue of the theorem of the cube for $H^i(X, \mathcal{O}^\times)$. For this, we look at the exponential sequence as in the aforementioned Remark. </p> http://mathoverflow.net/questions/64577/references-for-some-analogs-of-the-picard-group/87932#87932 Answer by Piotr Achinger for References for some analogs of the Picard group. Piotr Achinger 2012-02-08T20:32:38Z 2012-02-08T20:32:38Z <p>Here is a reference: Grothendieck's <a href="http://www.ams.org/mathscinet-getitem?mr=244269" rel="nofollow">three exposés</a> in <em>Dix Exposés sur la Cohomologie des Schémas</em> (and the references therein). One can find there e.g. computation of $H^i_{ét}({\rm Spec}\text{ } \mathcal{O}_K, \mathbb{G}_m)$ for spectra of rings of integers in number fields.</p> <p>MR0244269 (39 #5586a) Grothendieck, Alexander Le groupe de Brauer. I. Algèbres d'Azumaya et interprétations diverses. (French) 1968 Dix Exposés sur la Cohomologie des Schémas pp. 46–66 North-Holland, Amsterdam; Masson, Paris, 14.55</p> <p>MR0244270 (39 #5586b) Grothendieck, Alexander Le groupe de Brauer. II. Théorie cohomologique. (French) 1968 Dix Exposés sur la Cohomologie des Schémas pp. 67–87 North-Holland, Amsterdam; Masson, Paris, 14.55</p> <p>MR0244271 (39 #5586c) Grothendieck, Alexander Le groupe de Brauer. III. Exemples et compléments. (French) 1968 Dix Exposés sur la Cohomologie des Schémas pp. 88–188 North-Holland, Amsterdam; Masson, Paris (Reviewer: J. S. Milne), 14.55</p>