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
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 and maps as constructed above.
\[ \mathcal{P}_0 = \mathcal{A}[\mathcal{B}],\quad \mathcal{P}_1 = \mathcal{A}[\mathcal{A}[\mathcal{B}]],\quad \ldots \]
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 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 $D(\mathcal{B})$.
\[ \Omega _{\mathcal{P}_\bullet /\mathcal{A}} \otimes _{\mathcal{P}_\bullet , \epsilon } \mathcal{B} \]
These constructions satisfy a functoriality similar to that discussed in Section 92.6. Namely, given a commutative diagram
of sheaves of rings on $\mathcal{C}$ there is a canonical $\mathcal{B}$-linear map of complexes
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
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
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
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 $H^0(L_{\mathcal{B}/\mathcal{A}}) = \Omega _{\mathcal{B}/\mathcal{A}}$.
Proof. Follows from Lemmas 92.18.4 and 92.4.5 and Modules on Sites, Lemma 18.33.4. $\square$
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 is an isomorphism in $D(\mathcal{B} \times \mathcal{B}')$.
\[ L_{\mathcal{B} \times \mathcal{B}'/\mathcal{A}} \longrightarrow L_{\mathcal{B}/\mathcal{A}} \oplus L_{\mathcal{B}'/\mathcal{A}} \]
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 in $D(\mathcal{C})$.
\[ L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{C} \to L_{\mathcal{C}/\mathcal{A}} \to L_{\mathcal{C}/\mathcal{B}} \to L_{\mathcal{B}/\mathcal{A}} \otimes _\mathcal {B}^\mathbf {L} \mathcal{C}[1] \]
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
$\mathcal{P}_\bullet $ is the standard resolution of $\mathcal{B}$ over $\mathcal{A}$,
$\mathcal{Q}_\bullet $ is the standard resolution of $\mathcal{C}$ over $\mathcal{A}$,
$\mathcal{R}_\bullet $ is the standard resolution of $\mathcal{C}$ over $\mathcal{B}$,
$\mathcal{S}_\bullet $ is the standard resolution of $\mathcal{B}$ over $\mathcal{B}$,
$\overline{\mathcal{Q}}_\bullet = \mathcal{Q}_\bullet \otimes _{\mathcal{P}_\bullet } \mathcal{B}$, and
$\overline{\mathcal{R}}_\bullet = \mathcal{R}_\bullet \otimes _{\mathcal{S}_\bullet } \mathcal{B}$.
The distinguished triangle is the distinguished triangle associated to the short exact sequence of simplicial $\mathcal{C}$-modules
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
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
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
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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