Definition 92.3.1. Let A \to B be a ring map. The standard resolution of B over A is the augmentation \epsilon : P_\bullet \to B with terms
P_0 = A[B],\quad P_1 = A[A[B]],\quad \ldots
and maps as constructed above.
92.3 The cotangent complex of a ring map
Let A be a ring. Let \textit{Alg}_ A be the category of A-algebras. Consider the pair of adjoint functors (U, V) where V : \textit{Alg}_ A \to \textit{Sets} is the forgetful functor and U : \textit{Sets} \to \textit{Alg}_ A assigns to a set E the polynomial algebra A[E] on E over A. Let X_\bullet be the simplicial object of \text{Fun}(\textit{Alg}_ A, \textit{Alg}_ A) constructed in Simplicial, Section 14.34.
Consider an A-algebra B. Denote P_\bullet = X_\bullet (B) the resulting simplicial A-algebra. Recall that P_0 = A[B], P_1 = A[A[B]], and so on. In particular each term P_ n is a polynomial A-algebra. Recall also that there is an augmentation
\epsilon : P_\bullet \longrightarrow B
where we view B as a constant simplicial A-algebra.
It will turn out that we can use the standard resolution to compute left derived functors in certain settings.
Definition 92.3.2. The cotangent complex L_{B/A} of a ring map A \to B is the complex of B-modules associated to the simplicial B-module
\Omega _{P_\bullet /A} \otimes _{P_\bullet , \epsilon } B
where \epsilon : P_\bullet \to B is the standard resolution of B over A.
In Simplicial, Section 14.23 we associate a chain complex to a simplicial module, but here we work with cochain complexes. Thus the term L_{B/A}^{-n} in degree -n is the B-module \Omega _{P_ n/A} \otimes _{P_ n, \epsilon _ n} B and L_{B/A}^ m = 0 for m > 0.
Remark 92.3.3. Let A \to B be a ring map. Let \mathcal{A} be the category of arrows \psi : C \to B of A-algebras and let \mathcal{S} be the category of maps E \to B where E is a set. There are adjoint functors V : \mathcal{A} \to \mathcal{S} (the forgetful functor) and U : \mathcal{S} \to \mathcal{A} which sends E \to B to A[E] \to B. Let X_\bullet be the simplicial object of \text{Fun}(\mathcal{A}, \mathcal{A}) constructed in Simplicial, Section 14.34. The diagram
\xymatrix{ \mathcal{A} \ar[d] \ar[r] & \mathcal{S} \ar@<1ex>[l] \ar[d] \\ \textit{Alg}_ A \ar[r] & \textit{Sets} \ar@<1ex>[l] }
commutes. It follows that X_\bullet (\text{id}_ B : B \to B) is equal to the standard resolution of B over A.
Lemma 92.3.4. Let A_ i \to B_ i be a system of ring maps over a directed index set I. Then \mathop{\mathrm{colim}}\nolimits L_{B_ i/A_ i} = L_{\mathop{\mathrm{colim}}\nolimits B_ i/\mathop{\mathrm{colim}}\nolimits A_ i}.
Proof. This is true because the forgetful functor V : A\textit{-Alg} \to \textit{Sets} and its adjoint U : \textit{Sets} \to A\textit{-Alg} commute with filtered colimits. Moreover, the functor B/A \mapsto \Omega _{B/A} does as well (Algebra, Lemma 10.131.5). \square
Comments (3)
Comment #1859 by falkland on
Comment #1896 by Johan on
Comment #9543 by S on