Derived moduli of complexes and derived Grassmannians

In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some commutative ring $R$ and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we construct derived versions of Grassmannians and flag varieties.Comment: 54 pages, Section 2.4 significantly extended, minor corrections to the rest of the pape

Galois actions on homotopy groups

We study the Galois actions on the l-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the l-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever l is not equal to the residue characteristic p of K. For quasi-projective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When l=p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic homotopy types. Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin-Mazur homotopy groups.Comment: 72 pages; v2 corrections to Section 3; v3 references updated; v4 final versio

An introduction to derived (algebraic) geometry

These are notes from an introductory lecture course on derived geometry, given by the second author, mostly aimed at an audience with backgrounds in geometry and homological algebra. The focus is on derived algebraic geometry, mainly in characteristic $0$, but we also see the tweaks which extend most of the content to analytic and differential settings. The main motivating applications given are in moduli theory, with practically applicable representability theorems.Comment: 93pp; v2 minor changes; v3 minor additions, mostly reference

Shifted Poisson and symplectic structures on derived N-stacks

We show that on a derived Artin N-stack, there is a canonical equivalence between the spaces of n-shifted symplectic structures and non-degenerate n-shifted Poisson structures.Comment: 34 pages; v2 details added, several simplifications; v3 further simplifications, Artin details added; v4 several changes (mostly cosmetic, including notation and terminology), Examples 3.31 added (2-shifted structures on BG), final version (to appear in J. Topol.); v5 typos fixed and refs updated; v6 corrected Lemma 3.9 and dependent proof

Constructing derived moduli stacks

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.Comment: 53 pages; v2 final version, to appear in Geometry & Topolog