The cotangent complex

• Kähler differentials
• Review of derived functors
• Definition of the cotangent complex
• Advertisement for applications

Kähler differentials#

Let $f \colon A \to B$ be a commutative ring map. Then there is the diagonal $$ B \otimes_A B \to B $$ and let $I$ be its kernel.

Definition 1. We define $\Omega_ {B/A} = I/I^2$, with its “natural” $B$-module structure.

From this we have two exact sequences. If $$ X \xrightarrow{f} Y \xrightarrow{g} Z $$ are maps, we get $$ f^\ast \Omega_ {Y/Z} \to \Omega_ {X/Z} \to \Omega_ {X/Y} \to 0. $$ If $f \colon X \hookrightarrow Y$ is a closed immersion, then we will have $\Omega_ {X/Y} = 0$. Then $$ I/I^2 \to f^\ast \Omega_ {Y/Z} \to \Omega_ {X/Z} \to 0. $$

If $A \to B$ is smooth, then $\Omega_ {B/A}$ is locally free. Now what we want to do is to make a good definition of $\Omega_ {B/A}$ that works when $A \to B$ is not smooth as well.

Review of derived functors#

The idea of derived functors is that, when we have $C$ a collection some nice objects in $D$, we can Kan extend a functor $C \to E$ to $D \to E$. For instance, projective modules embed in to the derived category.

Let $\mathcal{C}$ be an $\infty$-category, and let $\mathcal{P}(\mathcal{C})$ be the $\infty$-category of $\infty$-presheaves. We defined $\mathcal{P} _ \Sigma(\mathcal{C})$ as the subcategory of sheaves preserving finite products. Then HTT, Proposition 5.5.8.15 told us that $$ j^\ast \colon \mathsf{Fun} _ \Sigma(\mathcal{P} _ \Sigma(\mathcal{C}), \mathcal{D}) \to \mathsf{Fun}(\mathcal{C}, \mathcal{D}) $$ is an equivalence, so that we can extend.

To make this abstract category $\mathcal{P} _ \Sigma(\mathcal{C})$ explicit, we made the following definitions.

Definition 2. An object $P \in \mathcal{C}$ is compact if $\Hom(P, -)$ preserves filtered colimits and projective if it preserves geometric realizations.

Then for a functor $F \colon \mathcal{C} \to \mathcal{D}$, the derived functor $LF \colon \mathcal{P} _ \Sigma(\mathcal{C}) \to \mathcal{D}$ is an equivalence if and only if $F$ is fully faithful and the essential image of $F$ is a set of compact projective generators.

Example 3. For $\mathcal{C}$ finite sets, we have $\mathcal{P} _ \Sigma(\mathcal{C})$ the category of spaces.

Example 4. For $\mathcal{C}$ the category of finitely generated free $A$-modules, we get the category of simplicial $A$-modules. For $A = \mathbb{Z}$, we will get a $D^{\le 0}(A)$.

Example 5. For $\mathcal{C}$ the category of polynomial $A$-algebras, we get simplicial commutative $A$-algebras.

Definition of the cotangent complex#

So here is our definition. Note that we have a functor $$ F = \Omega_ {-/A} \colon \mathsf{Poly} _A \to \mathsf{Ab}. $$ Now $\mathsf{Ab} \hookrightarrow D(\mathsf{Ab})$, and this derived category has filtered colimits and geometric realizations. Therefore we can make the Kan extension $$ LF = \mathbb{L} _ {-/A} \colon \mathsf{sCAlg} _A \to D(\mathsf{Ab}). $$ This is what we call the cotangent complex.

Can we make this $L_ {B/A}$ explicit? We can arrange $$ \dotsb \to A[A[B]] \rightrightarrows A[B] $$ to a simplicial object. This is an explicit resolution of $A$, and then what we can do this is to map this through $\Omega_ {-/A}$, and use Dold–Kan.

Proposition 6. The cotangent complex $\mathbb{L} _ {B/A}$ turns out to have a natural structure of a $B$-module, i.e., is in $D^{\le 0}(B)$.

Proposition 7 (Stacks, 91.7). Let $f \colon A \to B$ and $g \colon B \to C$ be (ordinary) commutative ring maps. Then there exists a fiber sequence $$ \mathbb{L} _ {B/A} \otimes_B^\mathbb{L} C \to \mathbb{L} _ {C/A} \to \mathbb{L} _ {C/B}. $$

Proposition 8. Let $A \to B$ be a (ordinary) ring map. Then $$ H^0(\mathbb{L} _ {B/A}) = \Omega_ {B/A}^1. $$ If $A \twoheadrightarrow B$ is moreover surjective, then $$ H^{-1}(\mathbb{L} _ {B/A}) = I/I^2. $$ If $I$ is moreover generated by a regular sequence, then $$ H^i(\mathbb{L} _ {B/A}) = 0 $$ for all $i \neq -1$. If $A \to B$ is étale, then $$ \mathbb{L} _ {B/A} = 0. $$

Proposition 9. If $A \to B$ is smooth of relative dimension $d$, then $$ \mathbb{L} _ {B/A} \cong \Omega_ {B/A}[0]. $$

Advertisement for applications#

Apparently, the cotangent complex gives a nice homological characterization of locally complete intersections, which are morphisms $$ X \hookrightarrow Y \to Z $$ where $Y \to Z$ is smooth and $X \to Y$ is cut out by a regular sequence. Let’s restrict to Noetherian schemes.

Theorem 10 (Quillen). A morphism $f \colon A \to B$ is a locally complete intersection if and only if

1. $\mathbb{L} _ {B/A}$ is perfect, and
2. $\mathbb{L} _ {B/A}$ has “Tor amplitude” in $[-1, 0]$.

Perfect means that it is quasi-isomorphic to a finite complex consisting of finite projective modules. The second condition just means $H^i(\mathbb{L} _ {B/A} \otimes_B^\mathbb{L} M) = 0$ for $i \neq 0, -1$ for all modules $M$ over $B$.

Theorem 11 (Avramov). If $\mathbb{L} _ {B/A}$ has bounded $H^\bullet$, then $A \to B$ is locally complete intersection.

Another application is to deformation theory. Suppose we have $X \to S$ and a square-zero thickening $S \hookrightarrow S^\prime$. The question is, can we classify lifts of $X \to S$ to $X^\prime \to S^\prime$?

Theorem 12 (Illusie?). There is an obstruction class $o \in \Ext^2(\mathbb{L} _ {X/S}, \mathscr{O} _X)$ such that $o = 0$ if and only if a lift exists. If $o = 0$, then liftings are classified by $\Ext^1(\mathbb{L} _ {X/S}, \mathscr{O} _X)$. Finally, automorphisms of some lift $X^\prime$ corresponds to $\Hom(\mathbb{L} _ {X/S}, \mathscr{O} _X)$.

Using this, we can actually construct the Witt vector lift of a perfect scheme. (This is in a MathOverflow post by Bhatt.)

Proposition 13. For $\mathbb{F} _p \to A$ be a perfect algebra, there exists a lift $W(A)$ of $A$ to a flat $\mathbb{Z} _p$-algebra, and this lift is unique up to unique isomorphism.