# A cotangent complex

Let me quickly explain how to define the cotangent complex of a ring map R —> S using the ideas of the previous 3 posts. Again we will work in the affine case.

Consider again the category C which is opposite to the category of surjections of R-algebras φ : P —> S where P is a polynomial R-algebra. Endow C with the chaotic topology (where all presheaves are sheaves). There is a surjection of sheaves of rings O —> B where O associates the value P to the object (P, φ) and where B is the constant sheaf with value S. (Please excuse the weird notation.) We consider as before the morphism of sites

π : (C, B) ——> (point, S)

and we will use the existence of the derived funtor Lπ!. Then I claim that

LS/R = Lπ!O/RO B)

In fact, it doesn’t matter if this is actually true or not, because arguing similarly to the previous post we see that inf autos, defos, and obstructions are computed by taking ext groups out of the object defined on the right hand side of the equation, so we can take it as our cotangent complex. Ha!

Still I am pretty sure the two sides are (canonically) equal (see update below). For example

H0(Lπ!O/RO B)) = π!O/RO B) = ΩS/R

by a direct computation of the colimit of the modules ΩP/R ⊗ S over the category of pairs (P, φ). Maybe there is a reference?

[Edit 3 hours later: Both Bhargav Bhatt and Jack Hall have pointed out that this is very similar to what happens in Quillen’s notes “homology of commutative rings”, and that there is further work by Gaitsgory and Jonathan Wise. As usual, I am looking for something very simple that I can add to the Stacks project without first developing a huge amount of theory. The approach above seems short and sweet, but I am sure there’s all kinds of problems with it — it might even be BALONEY!]

[Edit next day: OK, so now I’ve had time to glance at Quillen, Gaitsgory, and Wise. As Bhargav pointed out in his email, I have now discovered that the arrows in Quillen and Wise go in the opposite direction. For example, what Quillen says is that you take the category C of all R-algebra maps X —> S for varying R-algebras X. You endow C with a topology by declaring coverings to be surjective maps {X’ —> X} in C. Given an S-module M you get a sheaf DerR(-, M) which assigns to X —> S the module DerR(X, M). Then you define Dq(S/R, M) to be the q-th cohomology of this sheaf. Finally, you show that there is a complex LS/R so that ExtqS(LS/R, M) = Dq(S/R, M). Thus it seems that our thing above is at least technically different. The Gaitsgory thing seems to work *very very* roughly (I would be more than happy to be corrected on this) by having spaces be derived themselves, then representing it by a simplicial (or whatnot) smooth thing, and then taking the usual Omega. This is exactly what I am trying to avoid doing.]

[Update OK, I think the agreement holds. Sketch proof. Let P* —> S be a simplicial resolution of S by polynomial R-algebras (as in Quillen, Illusie, and everywhere). To show that the = sign in the post is true it suffices to prove that the left derived functors of π! of an abelian sheaf B on our category C are computed by the complex F(P*). It is OK for H0 by direct argument (this is one place where you really need all the algebras in C to be polynomial algebras). It is clear that the functors F |—> Hn(F(P*)) form a delta functor. Finally, you show you get zero for higher n when you apply it to the projective B-modules on C defined by the formula

(P, φ) |—-> free S-module on MorC((Q, ψ), (P, φ))

where ψ : Q –> S is a fixed object of C. Applying this to P* you get

free S-module on the simplicial set Mor(Q, P*)

which is contractible to a point by our choice of P*.]