The cotangent complex revisited

• Review of the cotangent complex
• Relative adjunction
• The tangent bundle
• The relative cotangent complex
• Properties of the cotangent complex
• The algebraic cotangent complex

Review of the cotangent complex#

We’ve already talked about the cotangent complex. Let me remind you how we constructed it. Let $A \in \mathsf{CAlg}^\heartsuit$ be a classical commutative ring. We had this functor $$ \Omega_{-/A} \colon \mathsf{CAlg}_ {A/}^\heartsuit \to \mathsf{Mod}_ A^\heartsuit $$ but this only had right-exactness and so that one may postulate that functor should really land in $\mathsf{Mod}_ {A,\ge 0}$ instead of $\mathsf{Mod}_ A^\heartsuit$.

The idea was that

• the $A$-modules $\Omega_ {B/A}$ seems to be the correct object when $B$ is smooth over $A$, in particular when $B \cong A[x_1, \dotsc, x_n]$,
• if we enlarge $\mathsf{CAlg}_ {A/}^\heartsuit$ to $\mathsf{CAlg}_ {A}^\Delta$, this is a category freely generated by $\mathsf{Poly}_ A$ under sifted colimits.

Then using these nonabelian derived category business, we were able to enhance the functor $\Omega_{-/A}$ to the functor $$ \mathbb{L}_ {-/A} \colon \mathsf{CAlg}_ {A/}^\Delta \to \mathsf{Mod}_ {A, \ge 0}$$ that preserves all sifted colimits. Then we argued that $\mathbb{L}_ {B/A}$ somehow has a structure of a $B$-module.

The observation is that from the perspective of higher algebra, the definition is $\Omega_{-/A}$ might be wrong to start with. Recall that one defines it as the module that represents derivations: for any $B$-module $M$, we have $$ \operatorname{Der}_ A(B, M) = \Hom_B(\Omega_{B/A}, M). $$ In a sense, this can never see higher information because we are only asking $\Omega_{B/A}$ to be corepresenting a certain functor in $\mathsf{Mod}_ B^\heartsuit$. If we truly want access to the cotangent complex, we should use test modules $M$ that live in $\mathsf{Mod}_ B$ (or maybe just $\mathsf{Mod}_ {B, \ge 0}$) rather than $\mathsf{Mod}_ B^\heartsuit$.

On the other hand, if one allows $M$ to be general (derived) modules, then the corepresenting object is just the cotangent complex $\mathbb{L}_ {B/A}$. In that sense, $\mathbb{L}_ {B/A}$ is a very natural object in the setting of higher algebra.

Relative adjunction#

So the question is, how do we understand derivations in the setting of higher algebra? For discrete rings, we have the following interpretation. Start with a map of rings $i \colon A \to B$ in $\mathsf{CAlg}^\heartsuit$ and also $M \in \mathsf{Mod}_ B^\heartsuit$. We can form an $A$-algebra $B \oplus M$ that is a split square-zero thickening of $B$ via $$ B \xrightarrow{s} B \oplus M \xrightarrow{p} B. $$ Now an $A$-derivation $B \to M$ is simply an $A$-algebra homomorphism $$ d \colon B \to B \oplus M $$ that is a section to $p$. So if we understand these split square-zero thickenings, we can understand derivations. In particular, we can take $$ \operatorname{Der}_ A(B, M) = \Hom_{\mathsf{CAlg}_ {A//B}}(B, B \oplus M) $$ to be a definition of a derivation.

Let’s try to do this in a family, so that we can define $\Omega_{B/A}$ functorially. Consider the $1$-category $\mathsf{CAlg}^{\heartsuit+}$ of pairs $(A, M)$ where $A$ is a ring and $M$ is an $A$-module. Morphisms are defined by $$ \Hom_{\mathsf{CAlg}^{\heartsuit+}}((A, M), (B, N)) = \lbrace (f \colon A \to B, g \colon M \to N) \rbrace $$ where $f$ is a ring homomorphism and $g$ is $A$-linear. There is a functor $$ G \colon \mathsf{CAlg}^{\heartsuit+} \to \mathsf{Fun}(\Delta^1, \mathsf{CAlg}^\heartsuit); \quad (A, M) \mapsto (A \oplus M \to A). $$ This lives over $\mathsf{CAlg}^\heartsuit$ by evaluation over $\lbrace 1 \rbrace \subset \Delta^1$. $$ \begin{CD} \mathsf{CAlg}^{\heartsuit+} @>{G}>> \mathsf{Fun}(\Delta^1, \mathsf{CAlg}^\heartsuit) \br @VVV @V{\mathrm{ev}_1}VV \br \mathsf{CAlg}^\heartsuit @= \mathsf{CAlg}^\heartsuit \end{CD} $$

Remark 1. Note that both vertical functors are Cartesian. Let’s describe the Cartesian lifts of $A \to B$ on the base. On the left vertical functor $\mathsf{CAlg}^{\heartsuit+} \to \mathsf{CAlg}^\heartsuit$, the lifts look like $(A, N) \mapsto (B, N)$. On the right vertical functor $\mathrm{ev}_ 1 \colon \mathsf{Fun}(\Delta^1, \mathsf{CAlg}^\heartsuit) \to \mathsf{CAlg}^\heartsuit$, the lifts look like $(C \times_B A \to A) \to (C \to B)$. So the top horizontal map preserves Cartesian edges.

Proposition 2 (HA 7.3.2.6). Consider a commutative diagram $$ \begin{CD} \mathcal{C} @>{G}>> \mathcal{D} \br @V{p}VV @V{q}VV \br \mathcal{E} @= \mathcal{E} \end{CD} $$ of ∞-categories. Assume the following:

• $p$ and $q$ are locally Cartesian,
• $G$ carries $p$-locally Cartesian edges to $q$-locally Cartesian edges,
• for each $E \in \mathcal{E}$, the functor on fibers $G_E \colon \mathcal{C}_ E \to \mathcal{D}_ E$ has a left adjoint.

Then $G$ has a left adjoint $F$ which lives over $\mathcal{E}$, so that $F_E$ is the left adjoint of $G_E$ for each $E \in \mathcal{E}$.

So now, we just look at the left adjoint $F$ of $G$, and it can be computed as $$ F \colon \mathsf{Fun}(\Delta^1, \mathsf{CAlg}^\heartsuit) \to \mathsf{CAlg}^{\heartsuit+}; \quad (A \to B) \mapsto (B, B \otimes_A \Omega_{A/\mathbb{Z}}). $$ Then we can define the absolute sheaf of differentials as the composition $$ \Omega_{-/\mathbb{Z}} \colon \mathsf{CAlg}^\heartsuit \xrightarrow{\Delta} \mathsf{Fun}(\Delta^1, \mathsf{CAlg}^\heartsuit) \xrightarrow{F} \mathsf{CAlg}^{\heartsuit+}. $$ Of course, we also want a theory of relative cotangent complex, but let’s deal with this later.

The tangent bundle#

The above was to illustrate that once we build an analogue of the category $\mathsf{CAlg}^{\heartsuit+}$, then we are in very good shape and can proceed pretty formally by looking at a relative left adjoint. So can we define a category $$ \mathsf{CAlg}^+ \to \mathsf{CAlg} $$ where the fiber over $A$ is $\mathsf{Mod}_ A$? Given an $A$-module $M$, we can try to define an $\mathbb{E}_ \infty$ structure on $A \oplus M$ by $$ (A \oplus M) \otimes (A \oplus M) \to A \oplus M $$ in the obvious way, but this formula is not enough to give an $\mathbb{E}_ \infty$ structure. Also we need to worry about arranging them into a category and so forth.

At this point, Lurie pulls a Lurie and does something completely insane. His idea is that at the end, we want to have $\mathsf{CAlg}_ A^+ \to \mathsf{CAlg}_ {/A}$, but the left hand side is stable, so it necessarily factors as $$ \mathsf{CAlg}_ A^+ \to \mathsf{Sp}(\mathsf{CAlg}_ {/A}) \to \mathsf{CAlg}_ A, $$ where $\mathsf{Sp}$ means stabilization and the second functor is just evaluation at $S^0$.

Theorem 3 (HA 7.3.4.14). For $A \in \mathsf{CAlg}$, there is an equivalence of categories $$ \mathsf{Sp}(\mathsf{CAlg}_ {/A}) \simeq \mathsf{Mod}_ A. $$

Example 4. If we have a discrete ring $A$ and a discrete $A$-module $M$, this would correspond to a functor $\mathcal{S}_ \ast^\mathrm{fin} \to \mathsf{CAlg}_ {/A}$ satisfying $$ S^n \mapsto A \oplus M[n]. $$

But we somehow need to do this for the category of $A$-algebra for each $A$ and glue them together. So we need a way of making sense of stabilization in families. There is even a relative stabilization construction for general inner fibrations, but we can do something simpler.

Definition 5 (HA 7.3.1.10). Let $\mathcal{C}$ be a presentable ∞-category. The tangent bundle to $\mathcal{C}$ is the full subcategory $$ T_\mathcal{C} \subseteq \mathsf{Fun}(\mathcal{S}_ \ast^\mathrm{fin}, \mathcal{C}) $$ consisting of excisive functors (recall this means sending pushout squares to pullback squares). Define the functor $$ G \colon T_\mathcal{C} \to \mathsf{Fun}(\Delta^1, \mathcal{C}) $$ that is obtained from evaluating at $S^0 \to \ast$.

Of course, we will be applying this to $\mathcal{C} = \mathsf{CAlg}$. But somehow this formalism works for an arbitrary presentable $\mathcal{C}$. Here are some facts.

• The fiber of the functor $\mathrm{ev}_ 1 \circ G \colon T_\mathcal{C} \to \mathcal{C}$ over $A \in \mathcal{C}$ is identified with $\mathsf{Sp}(\mathcal{C}_ {/A})$.
• The functor $\mathrm{ev}_ 1 \circ G$ is both a Cartesian fibration and a coCartesian fibration.
• The functor $G$ sends Cartesian edges for $\mathrm{ev}_ 1 \circ G$ to Cartesian edges for $\mathrm{ev}_ 1$.
• On each fiber over $A \in \mathcal{C}$, the functor $\Omega^\infty \colon \mathsf{Sp}(\mathcal{C}_ {/A}) \to \mathcal{C}_ {/A}$ admits a left adjoint.

This is enough to tell us that $G$ has a left adjoint $$ F \colon \mathsf{Fun}(\Delta^1, \mathcal{C}) \to T_\mathcal{C} $$ lying over $\mathcal{C}$. Now we can define the absolute cotangent complex as $$ L_{(-)} \colon \mathcal{C} \xrightarrow{\Delta} \mathsf{Fun}(\Delta^1, \mathcal{C}) \xrightarrow{F} T_\mathcal{C}. $$

Example 6. What happens if $\mathcal{C}$ is just $\mathcal{S}$, the simplest presentable ∞-category? I think $\operatorname{Sp}(\mathcal{S}_ {/A})$ is just $\operatorname{Fun}(A, \mathsf{Sp})$, and then $L_A$ is the constant diagram $A \to \ast \to \lbrace \mathbb{S} \rbrace$. I’m not super sure, but I think for $f \colon A \to B$, we then have $f_! L_A$ the functor that sends $b \in B$ to the suspension spectrum of the homotopy fiber $A_b$, and so $L_{B/A}$ seem to be the functor sending $b \in B$ to $\Sigma^\infty$ of some kind of pointed unreduced suspension of $A_b$.

The relative cotangent complex#

Lurie’s definition of $L_{B/A}$ is pretty simple. For a coCartesian fibration and an edge $f$ in the base, denote by $f_!$ the induced functor between the fibers.

Definition 7 (HA 7.3.3.1). Let $\mathcal{C}$ be a presentable ∞-category. We define a functor $$ L_{-/-} \colon \mathsf{Fun}(\Delta^1, \mathcal{C}) \to T_\mathcal{C} $$ such that its composition with $T_\mathcal{C} \to \mathcal{C}$ is $\mathrm{ev}_ 1$, defined so that $$ f_! L_A \to L_B \to L_{B/A} $$ is a fiber sequence in $\mathsf{Sp}(\mathcal{C}_ {/B})$ for all $f \colon A \to B$ in $\mathcal{C}$.

When $\mathcal{C} = \mathsf{CAlg}$, this $f_!$ is just tensoring and therefore the fundamental exact sequence is designed to work. What is not so trivial is that this definition of $T_{B/A}$ agrees with the absolute cotangent complex of $\mathcal{C}_ {A/}$.

Proposition 8 (HA 7.3.3.8). Let $\mathcal{C}$ be a presentable ∞-category and let $A \in \mathcal{C}$ be an object. Then $$ T_{\mathcal{C}_ {A/}} \simeq T_\mathcal{C} \times_\mathcal{C} \mathcal{C}_ {A/}. $$

Theorem 9 (HA 7.3.3.15). The composition $$ \mathcal{C}_ {A/} \xrightarrow{L} T_{\mathcal{C}_ {A/}} \to T_\mathcal{C} $$ can be identified with $B \mapsto L_{B/A}$.

The reason isn’t too complicated, but writing down introduces some technicalities. Let me instead try to illustrate the basic idea using the example of $\mathsf{CAlg}$ and being sloppy about square-zero thickenings. The image of $B$ under the above functor the object corepresenting the functor $$ M \mapsto \Hom_{\mathsf{CAlg}_ {A//B}}(B, B \oplus M). $$ An element in the right is just an element $\phi \in \Hom_{\mathsf{CAlg}_ {/B}}(B, B \oplus M)$ together with a path between $$ \phi \vert_A \in \Hom_{\mathsf{CAlg}_ {/A}}(A, A \oplus M) $$ and zero. So this is the fiber of $$ \Hom_{\mathsf{Mod}_ B}(L_B, M) \to \Hom_{\mathsf{Mod}_ A}(L_A, M) = \Hom_{\mathsf{Mod}_ B}(B \otimes_A L_A, M) $$ and therefore is corepresented by the cofiber of $B \otimes_A L_A \to L_B$.

Once we have the absolute fundamental sequence, the relative fundamental sequence immediately follows.

Proposition 10 (HA 7.3.3.6). Let $\mathcal{C}$ be a presentable ∞-category, and let $A \xrightarrow{f} B \xrightarrow{g} C$ be in $\mathcal{C}$. Then $$ g_! L_{B/A} \to L_{C/A} \to L_{C/B} $$ is a fiber sequence in $\mathsf{Sp}(\mathcal{C}_ {/C})$.

Properties of the cotangent complex#

Let’s now prove some facts about the cotangent complex. They will all follow pretty formally from the cotangent formalism.

Proposition 11. The absolute cotangent complex functor $L \colon \mathcal{C} \to T_\mathcal{C}$ preserves all small colimits.

On the other hand, because $T_\mathcal{C} \to \mathcal{C}$ is a coCartesian fibrations, colimits in $T_\mathcal{C}$ are computable.

Proposition 12 (HTT 4.3.1.11). Let $p \colon \mathcal{E} \to \mathcal{C}$ be a functor that is both a coCartesian fibration and a Cartesian fibration. Then the colimit of a diagram $f \colon K \to \mathcal{E}$ can be computed via the following algorithm:

1. find a colimit $K^\vartriangleright \to \mathcal{C}$ of $p \circ f$ with cone point $C \in \mathcal{C}$,
2. find a natural transformation from $f$ to $g \colon K \to \mathcal{E}_ C$ where each edge of $\mathcal{E}$ corresponding to vertices of $K$ are coCartesian,
3. compute the colimit of $g$.

Applying this fact to the functor $L$ immediately gives the following.

Theorem 13. Let $A = \varinjlim A_i$ for any diagram in $\mathcal{C}$. Denote by $f_i \colon A_i \to A$ the canonical map. Then $$ L_A = \varinjlim (f_i)_ ! L_{A_i} $$ where the colimit is computed in $\mathsf{Sp}(\mathcal{C}_ {/A})$.

Corollary 14. Consider a diagram $K \times \Delta^1 \to \mathcal{C}$, written as a system of $A_i \to B_i$, and consider the induced map $$ A = \varinjlim A_i \to \varinjlim B_i = B. $$ Denote by $f_i \colon B_i \to B$ the natural map. Then $$ L_{B/A} = \varinjlim (f_i)_ ! L_{B_i/A_i}. $$

This really justifies the fact that in the earlier definition, we defined $L_{B/A}$ to commute with sifted colimits in $B$; the reason is that it should commute with arbitrary colimits.

Corollary 15 (HA 7.3.3.7). Consider a pushout diagram $$ \begin{CD} A @>>> B \br @VVV @VV{f}V \br A^\prime @>>> B^\prime \end{CD} $$ in a presentable category $\mathcal{C}$. Then $$ f_! L_{B/A} \simeq L_{B^\prime/A^\prime}. $$

The algebraic cotangent complex#

I should probably tell you that I’m actually being a bit sneaky. This cotangent complex $L_{B/A}$ does satisfy all the nice properties, but it doesn’t quite agree with the cotangent complex that was constructed before. The reason is that the definition of the cotangent complex that is usually given in algebraic geometry (e.g., the Stacks project) works with the category $\mathsf{CAlg}^\Delta$ instead of $\mathsf{CAlg}$.

Example 16. I think $L_{\mathbb{Z}[x] / \mathbb{Z}}$ has higher homotopy groups. On the other hand, what is true is that $L_{\mathbb{Z}\lbrace x \rbrace / \mathbb{Z}}$ is rank $1$ over $\mathbb{Z}\lbrace x \rbrace$ where this ring is the free $\mathbb{E}_ \infty$-ring generated by a single element over $\mathbb{Z}$, namely $$ \mathbb{Z}\lbrace x \rbrace = \bigoplus_{n \ge 0} \mathbb{Z} / \Sigma_n $$ where we need to take a homotopy coinvariants. This is related to how $\mathbb{Z}[t]$ is not smooth over $\mathbb{Z}$ in spectral algebraic geometry.

To get the other definition of the cotangent complex, we need to work with $$ \mathsf{SCRMod}^\mathrm{cn} = \mathcal{P}_ \Sigma(\lbrace (A, M) \rbrace) $$ where $\lbrace (A, M) \rbrace$ is the category consisting of $A \cong \mathbb{Z}[x_1, \dotsc, x_n]$ and $M$ a finite free module over $A$. Then using nonabelian derived functors, we can define $$ \mathsf{SCRMod}^\mathrm{cn} \to \mathsf{Fun}(\Delta^1, \mathsf{CAlg}^\Delta) $$ by deriving the construction on the polynomial pairs $(A, M)$.

Once we have this definition, we can do the thing we did above and obtain $$ L_{B/A}^\mathrm{alg} \in \mathsf{Mod}^\mathrm{cn}(B). $$ This will now be the definition of the cotangent complex that appears in algebraic geometry.