On line bundles in derived algebraic geometry

Abstract

We give the first example of a derived scheme $X$ and a line bundle $\mathcal{ℒ}$ on the truncation $tX$ so that $\mathcal{ℒ}$ does not extend to the original derived scheme $X$. In other words the pullback map $Pic\left(X\right)\to Pic\left(tX\right)$, and hence also the pullback map $K^0\left(X\right)\to K^0\left(tX\right)$, 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.