## 92.18 The cotangent complex

In this section we discuss the cotangent complex of a map of sheaves of rings on a site. In later sections we specialize this to obtain the cotangent complex of a morphism of ringed topoi, a morphism of ringed spaces, a morphism of schemes, a morphism of algebraic space, etc.

Let $\mathcal{C}$ be a site and let $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ denote the associated topos. Let $\mathcal{A}$ denote a sheaf of rings on $\mathcal{C}$. Let $\mathcal{A}\textit{-Alg}$ be the category of $\mathcal{A}$-algebras. Consider the pair of adjoint functors $(U, V)$ where $V : \mathcal{A}\textit{-Alg} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is the forgetful functor and $U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathcal{A}\textit{-Alg}$ assigns to a sheaf of sets $\mathcal{E}$ the polynomial algebra $\mathcal{A}[\mathcal{E}]$ on $\mathcal{E}$ over $\mathcal{A}$. Let $X_\bullet$ be the simplicial object of $\text{Fun}(\mathcal{A}\textit{-Alg}, \mathcal{A}\textit{-Alg})$ constructed in Simplicial, Section 14.34.

Now assume that $\mathcal{A} \to \mathcal{B}$ is a homomorphism of sheaves of rings. Then $\mathcal{B}$ is an object of the category $\mathcal{A}\textit{-Alg}$. Denote $\mathcal{P}_\bullet = X_\bullet (\mathcal{B})$ the resulting simplicial $\mathcal{A}$-algebra. Recall that $\mathcal{P}_0 = \mathcal{A}[\mathcal{B}]$, $\mathcal{P}_1 = \mathcal{A}[\mathcal{A}[\mathcal{B}]]$, and so on. Recall also that there is an augmentation

$\epsilon : \mathcal{P}_\bullet \longrightarrow \mathcal{B}$

where we view $\mathcal{B}$ as a constant simplicial $\mathcal{A}$-algebra.

Definition 92.18.1. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. The standard resolution of $\mathcal{B}$ over $\mathcal{A}$ is the augmentation $\epsilon : \mathcal{P}_\bullet \to \mathcal{B}$ with terms

$\mathcal{P}_0 = \mathcal{A}[\mathcal{B}],\quad \mathcal{P}_1 = \mathcal{A}[\mathcal{A}[\mathcal{B}]],\quad \ldots$

and maps as constructed above.

With this definition in hand the cotangent complex of a map of sheaves of rings is defined as follows. We will use the module of differentials as defined in Modules on Sites, Section 18.33.

Definition 92.18.2. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. The cotangent complex $L_{\mathcal{B}/\mathcal{A}}$ is the complex of $\mathcal{B}$-modules associated to the simplicial module

$\Omega _{\mathcal{P}_\bullet /\mathcal{A}} \otimes _{\mathcal{P}_\bullet , \epsilon } \mathcal{B}$

where $\epsilon : \mathcal{P}_\bullet \to \mathcal{B}$ is the standard resolution of $\mathcal{B}$ over $\mathcal{A}$. We usually think of $L_{\mathcal{B}/\mathcal{A}}$ as an object of $D(\mathcal{B})$.

These constructions satisfy a functoriality similar to that discussed in Section 92.6. Namely, given a commutative diagram

92.18.2.1
\begin{equation} \label{cotangent-equation-commutative-square-sheaves} \vcenter { \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] } } \end{equation}

of sheaves of rings on $\mathcal{C}$ there is a canonical $\mathcal{B}$-linear map of complexes

$L_{\mathcal{B}/\mathcal{A}} \longrightarrow L_{\mathcal{B}'/\mathcal{A}'}$

constructed as follows. If $\mathcal{P}_\bullet \to \mathcal{B}$ is the standard resolution of $\mathcal{B}$ over $\mathcal{A}$ and $\mathcal{P}'_\bullet \to \mathcal{B}'$ is the standard resolution of $\mathcal{B}'$ over $\mathcal{A}'$, then there is a canonical map $\mathcal{P}_\bullet \to \mathcal{P}'_\bullet$ of simplicial $\mathcal{A}$-algebras compatible with the augmentations $\mathcal{P}_\bullet \to \mathcal{B}$ and $\mathcal{P}'_\bullet \to \mathcal{B}'$. The maps

$\mathcal{P}_0 = \mathcal{A}[\mathcal{B}] \longrightarrow \mathcal{A}'[\mathcal{B}'] = \mathcal{P}'_0, \quad \mathcal{P}_1 = \mathcal{A}[\mathcal{A}[\mathcal{B}]] \longrightarrow \mathcal{A}'[\mathcal{A}'[\mathcal{B}']] = \mathcal{P}'_1$

and so on are given by the given maps $\mathcal{A} \to \mathcal{A}'$ and $\mathcal{B} \to \mathcal{B}'$. The desired map $L_{\mathcal{B}/\mathcal{A}} \to L_{\mathcal{B}'/\mathcal{A}'}$ then comes from the associated maps on sheaves of differentials.

Lemma 92.18.3. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $f^{-1}L_{\mathcal{B}/\mathcal{A}} = L_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$.

Proof. The diagram

$\xymatrix{ \mathcal{A}\textit{-Alg} \ar[d]_{f^{-1}} \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar@<1ex>[l] \ar[d]^{f^{-1}} \\ f^{-1}\mathcal{A}\textit{-Alg} \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar@<1ex>[l] }$

commutes. $\square$

Lemma 92.18.4. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $H^ i(L_{\mathcal{B}/\mathcal{A}})$ is the sheaf associated to the presheaf $U \mapsto H^ i(L_{\mathcal{B}(U)/\mathcal{A}(U)})$.

Proof. Let $\mathcal{C}'$ be the site we get by endowing $\mathcal{C}$ with the chaotic topology (presheaves are sheaves). There is a morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ where $f_*$ is the inclusion of sheaves into presheaves and $f^{-1}$ is sheafification. By Lemma 92.18.3 it suffices to prove the result for $\mathcal{C}'$, i.e., in case $\mathcal{C}$ has the chaotic topology.

If $\mathcal{C}$ carries the chaotic topology, then $L_{\mathcal{B}/\mathcal{A}}(U)$ is equal to $L_{\mathcal{B}(U)/\mathcal{A}(U)}$ because

$\xymatrix{ \mathcal{A}\textit{-Alg} \ar[d]_{\text{sections over }U} \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar@<1ex>[l] \ar[d]^{\text{sections over }U} \\ \mathcal{A}(U)\textit{-Alg} \ar[r] & \textit{Sets} \ar@<1ex>[l] }$

commutes. $\square$

Remark 92.18.5. It is clear from the proof of Lemma 92.18.4 that for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there is a canonical map $L_{\mathcal{B}(U)/\mathcal{A}(U)} \to L_{\mathcal{B}/\mathcal{A}}(U)$ of complexes of $\mathcal{B}(U)$-modules. Moreover, these maps are compatible with restriction maps and the complex $L_{\mathcal{B}/\mathcal{A}}$ is the sheafification of the rule $U \mapsto L_{\mathcal{B}(U)/\mathcal{A}(U)}$.

Lemma 92.18.6. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. Then $H^0(L_{\mathcal{B}/\mathcal{A}}) = \Omega _{\mathcal{B}/\mathcal{A}}$.

Lemma 92.18.7. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ and $\mathcal{A} \to \mathcal{B}'$ be homomorphisms of sheaves of rings on $\mathcal{C}$. Then

$L_{\mathcal{B} \times \mathcal{B}'/\mathcal{A}} \longrightarrow L_{\mathcal{B}/\mathcal{A}} \oplus L_{\mathcal{B}'/\mathcal{A}}$

is an isomorphism in $D(\mathcal{B} \times \mathcal{B}')$.

Proof. By Lemma 92.18.4 it suffices to prove this for ring maps. In the case of rings this is Lemma 92.6.4. $\square$

The fundamental triangle for the cotangent complex of sheaves of rings is an easy consequence of the result for homomorphisms of rings.

Lemma 92.18.8. Let $\mathcal{D}$ be a site. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be homomorphisms of sheaves of rings on $\mathcal{D}$. There is a canonical distinguished triangle

$L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{C} \to L_{\mathcal{C}/\mathcal{A}} \to L_{\mathcal{C}/\mathcal{B}} \to L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{C}$

in $D(\mathcal{C})$.

Proof. We will use the method described in Remarks 92.7.5 and 92.7.6 to construct the triangle; we will freely use the results mentioned there. As in those remarks we first construct the triangle in case $\mathcal{B} \to \mathcal{C}$ is an injective map of sheaves of rings. In this case we set

1. $\mathcal{P}_\bullet$ is the standard resolution of $\mathcal{B}$ over $\mathcal{A}$,

2. $\mathcal{Q}_\bullet$ is the standard resolution of $\mathcal{C}$ over $\mathcal{A}$,

3. $\mathcal{R}_\bullet$ is the standard resolution of $\mathcal{C}$ over $\mathcal{B}$,

4. $\mathcal{S}_\bullet$ is the standard resolution of $\mathcal{B}$ over $\mathcal{B}$,

5. $\overline{\mathcal{Q}}_\bullet = \mathcal{Q}_\bullet \otimes _{\mathcal{P}_\bullet } \mathcal{B}$, and

6. $\overline{\mathcal{R}}_\bullet = \mathcal{R}_\bullet \otimes _{\mathcal{S}_\bullet } \mathcal{B}$.

The distinguished triangle is the distinguished triangle associated to the short exact sequence of simplicial $\mathcal{C}$-modules

$0 \to \Omega _{\mathcal{P}_\bullet /\mathcal{A}} \otimes _{\mathcal{P}_\bullet } \mathcal{C} \to \Omega _{\mathcal{Q}_\bullet /\mathcal{A}} \otimes _{\mathcal{Q}_\bullet } \mathcal{C} \to \Omega _{\overline{\mathcal{Q}}_\bullet /\mathcal{B}} \otimes _{\overline{\mathcal{Q}}_\bullet } \mathcal{C} \to 0$

The first two terms are equal to the first two terms of the triangle of the statement of the lemma. The identification of the last term with $L_{\mathcal{C}/\mathcal{B}}$ uses the quasi-isomorphisms of complexes

$L_{\mathcal{C}/\mathcal{B}} = \Omega _{\mathcal{R}_\bullet /\mathcal{B}} \otimes _{\mathcal{R}_\bullet } \mathcal{C} \longrightarrow \Omega _{\overline{\mathcal{R}}_\bullet /\mathcal{B}} \otimes _{\overline{\mathcal{R}}_\bullet } \mathcal{C} \longleftarrow \Omega _{\overline{\mathcal{Q}}_\bullet /\mathcal{B}} \otimes _{\overline{\mathcal{Q}}_\bullet } \mathcal{C}$

All the constructions used above can first be done on the level of presheaves and then sheafified. Hence to prove sequences are exact, or that map are quasi-isomorphisms it suffices to prove the corresponding statement for the ring maps $\mathcal{A}(U) \to \mathcal{B}(U) \to \mathcal{C}(U)$ which are known. This finishes the proof in the case that $\mathcal{B} \to \mathcal{C}$ is injective.

In general, we reduce to the case where $\mathcal{B} \to \mathcal{C}$ is injective by replacing $\mathcal{C}$ by $\mathcal{B} \times \mathcal{C}$ if necessary. This is possible by the argument given in Remark 92.7.5 by Lemma 92.18.7. $\square$

Lemma 92.18.9. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. If $p$ is a point of $\mathcal{C}$, then $(L_{\mathcal{B}/\mathcal{A}})_ p = L_{\mathcal{B}_ p/\mathcal{A}_ p}$.

Proof. This is a special case of Lemma 92.18.3. $\square$

For the construction of the naive cotangent complex and its properties we refer to Modules on Sites, Section 18.35.

Lemma 92.18.10. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $\mathcal{C}$. There is a canonical map $L_{\mathcal{B}/\mathcal{A}} \to \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}}$ which identifies the naive cotangent complex with the truncation $\tau _{\geq -1}L_{\mathcal{B}/\mathcal{A}}$.

Proof. Let $\mathcal{P}_\bullet$ be the standard resolution of $\mathcal{B}$ over $\mathcal{A}$. Let $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{A}[\mathcal{B}] \to \mathcal{B})$. Recall that $\mathcal{P}_0 = \mathcal{A}[\mathcal{B}]$. The map of the lemma is given by the commutative diagram

$\xymatrix{ L_{\mathcal{B}/\mathcal{A}} \ar[d] & \ldots \ar[r] & \Omega _{\mathcal{P}_2/\mathcal{A}} \otimes _{\mathcal{P}_2} \mathcal{B} \ar[r] \ar[d] & \Omega _{\mathcal{P}_1/\mathcal{A}} \otimes _{\mathcal{P}_1} \mathcal{B} \ar[r] \ar[d] & \Omega _{\mathcal{P}_0/\mathcal{A}} \otimes _{\mathcal{P}_0} \mathcal{B} \ar[d] \\ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} & \ldots \ar[r] & 0 \ar[r] & \mathcal{I}/\mathcal{I}^2 \ar[r] & \Omega _{\mathcal{P}_0/\mathcal{A}} \otimes _{\mathcal{P}_0} \mathcal{B} }$

We construct the downward arrow with target $\mathcal{I}/\mathcal{I}^2$ by sending a local section $\text{d}f \otimes b$ to the class of $(d_0(f) - d_1(f))b$ in $\mathcal{I}/\mathcal{I}^2$. Here $d_ i : \mathcal{P}_1 \to \mathcal{P}_0$, $i = 0, 1$ are the two face maps of the simplicial structure. This makes sense as $d_0 - d_1$ maps $\mathcal{P}_1$ into $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{P}_0 \to \mathcal{B})$. We omit the verification that this rule is well defined. Our map is compatible with the differential $\Omega _{\mathcal{P}_1/\mathcal{A}} \otimes _{\mathcal{P}_1} \mathcal{B} \to \Omega _{\mathcal{P}_0/\mathcal{A}} \otimes _{\mathcal{P}_0} \mathcal{B}$ as this differential maps a local section $\text{d}f \otimes b$ to $\text{d}(d_0(f) - d_1(f)) \otimes b$. Moreover, the differential $\Omega _{\mathcal{P}_2/\mathcal{A}} \otimes _{\mathcal{P}_2} \mathcal{B} \to \Omega _{\mathcal{P}_1/\mathcal{A}} \otimes _{\mathcal{P}_1} \mathcal{B}$ maps a local section $\text{d}f \otimes b$ to $\text{d}(d_0(f) - d_1(f) + d_2(f)) \otimes b$ which are annihilated by our downward arrow. Hence a map of complexes.

To see that our map induces an isomorphism on the cohomology sheaves $H^0$ and $H^{-1}$ we argue as follows. Let $\mathcal{C}'$ be the site with the same underlying category as $\mathcal{C}$ but endowed with the chaotic topology. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ be the morphism of topoi whose pullback functor is sheafification. Let $\mathcal{A}' \to \mathcal{B}'$ be the given map, but thought of as a map of sheaves of rings on $\mathcal{C}'$. The construction above gives a map $L_{\mathcal{B}'/\mathcal{A}'} \to \mathop{N\! L}\nolimits _{\mathcal{B}'/\mathcal{A}'}$ on $\mathcal{C}'$ whose value over any object $U$ of $\mathcal{C}'$ is just the map

$L_{\mathcal{B}(U)/\mathcal{A}(U)} \to \mathop{N\! L}\nolimits _{\mathcal{B}(U)/\mathcal{A}(U)}$

of Remark 92.11.4 which induces an isomorphism on $H^0$ and $H^{-1}$. Since $f^{-1}L_{\mathcal{B}'/\mathcal{A}'} = L_{\mathcal{B}/\mathcal{A}}$ (Lemma 92.18.3) and $f^{-1}\mathop{N\! L}\nolimits _{\mathcal{B}'/\mathcal{A}'} = \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}}$ (Modules on Sites, Lemma 18.35.3) the lemma is proved. $\square$

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