Presenting higher stacks as simplicial schemes

Details Presenting higher stacks as simplicial schemes Presenting higher stacks as simplicial schemes

2009-05-25

arxiv.org/abs/0905.4044v2

Mathematics

Algebraic Geometry

56 pages; v2 Proposition 8.6 added, Lemma 8.36 expanded, minor presentational changes

Scientific paper

We show that an n-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n-stacks, Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This formulation adapts to most HAG contexts, so in particular works for derived n-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich's dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.

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