I’d now like to begin a series of posts on the cotangent complex, following Daniel Quillen’s paper “Homology of Commutative Rings.” (There are also two very nice articles from a 2004 summer school on “Homotopy and algebra” on the subject, those by Goerss-Schemmerhorn and Iyengar, that discuss the topic.) While the cotangent complex can be defined quite cleanly once one has the appropriate categorical setting, it will be useful to spend a brief period formulating that.
Let be a commutative ring. Ultimately, we are going to think of the cotangent complex of an -algebra as a “linearization” or “abelianization.” Viewed more precisely, the cotangent complex will be the derived functor of abelianization (this is the general means of defining “homology” in a model category). It will turn out that abelianization will correspond to taking the module of Kähler differentials, so that the cotangent complex will also be a derived functor of those.
The problem is the category of -algebras does not exactly admit a nice abelianization functor. Recall:
Definition 1 If is a category with finite products, then an abelian monoid object in is an object together with a multiplication morphism and a unit (where is the terminal object). These are required to satisfy the usual commutativity and associativity constraints. For instance,
should be the identity.
The terminal object in the category is the zero ring, and it cannot map on any nonzero ring. So there are no nontrivial abelian group objects in this category! (more…)