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.

1. Generalities

Let ${A}$ be a commutative ring. Ultimately, we are going to think of the cotangent complex of an ${A}$-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 ${\mathbf{Alg}^{A}}$ of $A$-algebras does not exactly admit a nice abelianization functor. Recall:

Definition 1 If ${\mathcal{C}}$ is a category with finite products, then an abelian monoid object in ${\mathcal{C}}$ is an object ${X}$ together with a multiplication morphism ${\mu: X \times X \rightarrow X}$ and a unit ${e: \ast \rightarrow X}$ (where ${\ast}$ is the terminal object). These are required to satisfy the usual commutativity and associativity constraints. For instance,

$\displaystyle \mu \circ ( e \times 1): X \rightarrow \ast \times X \rightarrow X$

should be the identity.

The terminal object in the category ${\mathbf{Alg}^{A}}$ is the zero ring, and it cannot map on any nonzero ring. So there are no nontrivial abelian group objects in this category! (more…)