I’d like to discuss today a category-theoretic characterization of Zariski open immersions of rings, which I learned from Toen-Vezzosi’s article.

Theorem 1 If ${f: A \rightarrow B}$ is a finitely presented morphism of commutative rings, then ${\mathrm{Spec} B \rightarrow \mathrm{Spec} A}$ is an open immersion if and only if the restriction functor ${D^-(B) \rightarrow D^-(A)}$ between derived categories is fully faithful.

Toen and Vezzosi use this to define a Zariski open immersion in the derived context, but I’d like to work out carefully what this means in the classical sense. If one has an open immersion ${f: A \rightarrow B}$ (for instance, a localization ${A \rightarrow A_f}$), then the pull-back on derived categories is fully faithful: in other words, the composite of push-forward and pull-back is the identity. (more…)

The next step (in our discussion, started yesterday, of the cotangent complex) will be to define a model structure on the category of algebras over a fixed ring. Model structures allow one to define derived functors in a non-abelian setting. The key idea is that, when you want to derive an additive functor $F$ on an object $X$ in some abelian category, you replace $X$ by a projective resolution and evaluate the functor $F$ on this resolution. (And then, take its homology; in the setting of derived categories, though, one usually just takes $F$ of the projective resolution and leaves it at that.) Because $F$ on projective resolutions is much better behaved than $F$ simply on modules, the derived functor is a nice replacement.

The intuition is that a projective resolution is a cofibrant approximation to the initial object, in the language of model categories (which is often seen as a non-abelian version of classical homological algebra). This is actually precisely true if one imposes the usual model structure on bounded-below chain complexes for modules over a ring, for instance.

In constructing the cotangent complex, we are trying to derive the (highly non-abelian) functor of abelianization, which as we saw was closely related to the construction of differentials. This functor was defined on rings under a fixed ring $A$ and over a fixed ring $B$, which is not anywhere near an abelian category. So we will need the language of model categories, and today we shall construct a model structure on a certain class of categories.

In deriving an additive functor, one ultimately applies it not on the initial abelian category, but the larger category of chain complexes. Here the analogy extends again: by the Dold-Kan correspondence (which I recently talked about), this is equivalent to the category of simplicial objects in that category. The appropriate approach now seems to be to define a model structure not on $A$-algebras over $B$, but on the category of simplicial $A$-algebras over $B$. (more…)

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…)