Vol. 5, No. 2, 2020
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On line bundles in derived algebraic geometry

Toni Annala
Vol. 5 (2020), No. 2, 317–325
DOI: 10.2140/akt.2020.5.317
Abstract

We give the first example of a derived scheme X and a line bundle ℒ on the truncation tX so that ℒ does not extend to the original derived scheme X. In other words the pullback map Pic(X) → Pic(tX), and hence also the pullback map K0(X) → K0(tX), is not surjective. The derived schemes we construct have the further property that while their truncations are projective hypersurfaces, they fail to have any nontrivial line bundles, and therefore they are not quasiprojective.

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Keywords
derived algebraic geometry, deformation theory, Picard group
Mathematical Subject Classification 2010
Primary: 14F05
Milestones
Received: 26 April 2019
Revised: 15 October 2019
Accepted: 5 November 2019
Published: 20 June 2020
Authors
Toni Annala
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada