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Deformation theory of representable morphisms of algebraic stacks. (English) Zbl 1096.14007
The central statement of deformation theory goes as follows: The obstruction for the existence of first-order deformations lies in the group \(\text{ Ext}^2(L_{X/Y}^\bullet, f^*I)\). In case this obstruction vanishes, the set of isomorphism classes of first-order deformations is a torsor under \(\text{ Ext}^1(L_{X/Y}^\bullet, f^*I)\), and the automorphism group of a given first-order deformation is \(\text{ Ext}^0(L_{X/Y}^\bullet, f^*I)\). This results, and in fact the whole theory of the cotangent complex \(L_{X/Y}^\bullet\), is due to L. Illusie [“Complexe cotangent et déformations I”. Lect. Notes Math. 239 (1971; Zbl 0224.13014)], and worked out in the general context of ringed topoi.
Olsson extends these and related results further to representable morphisms between Artin stacks, using the cotangent complex for Artin stacks as defined by G. Laumon and L. Moret-Bailly [“Champs algébriques”, Ergebn. Math. Grenzgeb. 39 (2000; Zbl 0945.14005)]. The point is that the latter cotangent complex is not defined as the cotangent complex of a morphism of ringed topoi. In fact, it is not a complex at all, rather a projective system of complexes. Thus Illusie’s theory does not immediately apply to this situation.
The idea of this paper is to replace the involved Artin stacks \(\mathcal{S}\) by the simplicial algebraic space \(S^\bullet\) coming from a smooth cover \(S\rightarrow \mathcal{S}\), and then relate Illusie’s cotangent complex of this simplicial algebraic space to the cotangent complex of Laumon and Moret-Bailly.

14D15 Formal methods and deformations in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
Full Text: DOI
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