The Stacks project

90.3 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.

Definition 90.3.1. 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.

It will turn out that we can use the standard resolution to compute left derived functors in certain settings.

Definition 90.3.2. 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 90.3.3. 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 90.3.4. 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 (2)

Comment #1859 by falkland on

in 76.3.1, should the map be instead


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