Holmstrom Derived stack

nLab

Toen remark in Seville: The inclusion from stacks to derived stacks is a full embedding which admits a right adjoint, but does NOT commute with holim. That’s why we need derived stacks to get “correct” limits.

arXiv:1101.3300 Constructing derived moduli stacks from arXiv Front: math.AG by J. P. Pridham We introduce frameworks for constructing global derived moduli stacks associated to a broad range of problems, bridging the gap between the concrete and abstract conceptions of derived moduli. Our three approaches are via differential graded Lie algebras, via cosimplicial groups, and via quasi-comonoids, each more general than the last. Explicit examples of derived moduli problems addressed here are finite schemes, polarised projective schemes, torsors, coherent sheaves, and finite group schemes.

arXiv:1102.1150 Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes from arXiv Front: math.KT by Timo Schürg, Bertrand Toën, Gabriele Vezzosi We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack naturally endows the stack with a functorial perfect obstruction theory in the sense of Behrend-Fantechi. This result is then applied to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, for X=S a smooth algebraic K3-surface, $g\geq 0$, and $\beta \neq 0$ a curve class, we construct a derived stack whose truncation is the usual stack of pointed stable maps from curves of genus g to S hitting the class $\beta$, and such that the inclusion of the truncation into the full derived stack induces on the stack a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory - not relying on any result on semiregularity maps - but also a new global geometric interpretation. For moduli of complexes, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X) to the derived stack of simple perfect complexes on X with vanishing negative Ext’s, and show how this map induces a morphism of the correponding obstruction theories, in the case X is a Calabi-Yau threefold. We relate this result to the conjectural comparison between Gromov-Witten snd Donaldson-Thomas invariants. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is, pointwise, Illusie’s trace map for perfect complexes.

nLab page on Derived stack