derived smooth geometry
Traditionally this has been conceived in terms of model category presentations. This we discuss in the section
The construction of moduli spaces depends strongly on local properties like smoothness and transversality of intersections when trying to represent the functor of assigning families of objects to the varying base of family. When passing to the classes of equivalent objects, one faces the problem of having nontrivial automorphisms.
At the infinitesimal level automorphisms correspond to the derivations. Taking derivations is represented by the module of relative Kähler differentials which sufficies in good cases. Its correct derived replacement is the cotangent complex of Grothendieck-Illusie. One can typically split the information about a map of higher rings into its “discrete part” and infinitesimal obstruction theory governed by the cotangent complex.
To talk about the nonabelian derived functors, Quillen introduced a model structure on the category of simplicial commutative rings.
Given a morphism in CRing, which makes an -algebra, the fiber of the tangent category of abelian group objects in the slice category of -algebras over is equivalent both to the category of -modules and the trivial (= square zero) extensions of by -modules.
All said is true for simplicial commutative rings as well.
The relative cotangent complex functor is the left derived functor
Its value on is the relative cotangent complex
that sends a commutative -algebra to the polynomial algebra on its underlying set.
Forming degreewise the module of Kähler differentials on this yields the simplicial object , which is a -module.
The cotangent complex of is equivalent to
Notice that the universal property of the Kähler differentials is that for a ring and an -module, we have
Accordingly, it follows that the Andre-Quillen cohomology of with values in , which is the cohomology of the cosimplicial object
is equivalently the cohomology of the object
In particular we have the the degree-0 cohomology of this complex is the module of ordinary derivations
This is in section 8.8 of
of the tangent (∞,1)-category projection .
In particular, when , then the cotangent complex assigns … .
For more background see deformation theory.
Apart from simplicial rings we can consider -rings. A map of connective -rings is an equivalence, if it induces an isomorphism at the level of plus a condition on the relative cotangent complex. Similarly, one can express the descent properties of higher stacks via the usual gluing at the bottom level plus the obstruction theory for relative cotangent complex. Study of an appropriate version of the Postnikov tower is a systematic way to do this.
See also deformation theory and references therein.
Alexander Grothendieck, Catégories cofibrées additives et complexe cotangent relatif, Lec. Notes in Math. 79
Luc Illusie, Complexe cotangent et déformations I, Lec. Notes Math. 239, Springer 1971, xv+355 pp.; II, LNM 283, Springer 1972. vii+304 xv+355 pp.
Barbara Fantechi, M. Manetti, Obstruction calculus for functors of Artin rings I, J. Algebra 202 (1998), no. 2, 541–576, MR1617687 (99f:14004).
A short exposition (from the point of view of formal schemes) is in
The abstract cotangent complex and Quillen cohomology of enriched categories (arXiv:1612.02608)