92.7 The fundamental triangle
In this section we consider a sequence of ring maps $A \to B \to C$. It is our goal to show that this triangle gives rise to a distinguished triangle
92.7.0.1
\begin{equation} \label{cotangent-equation-triangle} L_{B/A} \otimes _ B^\mathbf {L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes _ B^\mathbf {L} C[1] \end{equation}
in $D(C)$. This will be proved in Proposition 92.7.4. For an alternative approach see Remark 92.7.5.
Consider the category $\mathcal{C}_{C/B/A}$ wich is the opposite of the category whose objects are $(P \to B, Q \to C)$ where
$P$ is a polynomial algebra over $A$,
$P \to B$ is an $A$-algebra homomorphism,
$Q$ is a polynomial algebra over $P$, and
$Q \to C$ is a $P$-algebra-homomorphism.
We take the opposite as we want to think of $(P \to B, Q \to C)$ as corresponding to the commutative diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(C) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(Q) \ar[d] \\ \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(P) \ar[dl] \\ \mathop{\mathrm{Spec}}(A) } \]
Let $\mathcal{C}_{B/A}$, $\mathcal{C}_{C/A}$, $\mathcal{C}_{C/B}$ be the categories considered in Section 92.4. There are functors
\[ \begin{matrix} u_1 : \mathcal{C}_{C/B/A} \to \mathcal{C}_{B/A},
& (P \to B, Q \to C) \mapsto (P \to B)
\\ u_2 : \mathcal{C}_{C/B/A} \to \mathcal{C}_{C/A},
& (P \to B, Q \to C) \mapsto (Q \to C)
\\ u_3 : \mathcal{C}_{C/B/A} \to \mathcal{C}_{C/B},
& (P \to B, Q \to C) \mapsto (Q \otimes _ P B \to C)
\end{matrix} \]
These functors induce corresponding morphisms of topoi $g_ i$. Let us denote $\mathcal{O}_ i = g_ i^{-1}\mathcal{O}$ so that we get morphisms of ringed topoi
92.7.0.2
\begin{equation} \label{cotangent-equation-three-maps} \begin{matrix} g_1 : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}), \mathcal{O}_1) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{B/A}), \mathcal{O})
\\ g_2 : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}), \mathcal{O}_2) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/A}), \mathcal{O})
\\ g_3 : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}), \mathcal{O}_3) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B}), \mathcal{O})
\end{matrix} \end{equation}
Let us denote $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}) \to \mathop{\mathit{Sh}}\nolimits (*)$, $\pi _1 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{B/A}) \to \mathop{\mathit{Sh}}\nolimits (*)$, $\pi _2 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/A}) \to \mathop{\mathit{Sh}}\nolimits (*)$, and $\pi _3 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B}) \to \mathop{\mathit{Sh}}\nolimits (*)$, so that $\pi = \pi _ i \circ g_ i$. We will obtain our distinguished triangle from the identification of the cotangent complex in Lemma 92.4.3 and the following lemmas.
Lemma 92.7.1. With notation as in (92.7.0.2) set
\[ \begin{matrix} \Omega _1 = \Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B} \text{ on }\mathcal{C}_{B/A}
\\ \Omega _2 = \Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{C} \text{ on }\mathcal{C}_{C/A}
\\ \Omega _3 = \Omega _{\mathcal{O}/B} \otimes _\mathcal {O} \underline{C} \text{ on }\mathcal{C}_{C/B}
\end{matrix} \]
Then we have a canonical short exact sequence of sheaves of $\underline{C}$-modules
\[ 0 \to g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C} \to g_2^{-1}\Omega _2 \to g_3^{-1}\Omega _3 \to 0 \]
on $\mathcal{C}_{C/B/A}$.
Proof.
Recall that $g_ i^{-1}$ is gotten by simply precomposing with $u_ i$. Given an object $U = (P \to B, Q \to C)$ we have a split short exact sequence
\[ 0 \to \Omega _{P/A} \otimes Q \to \Omega _{Q/A} \to \Omega _{Q/P} \to 0 \]
for example by Algebra, Lemma 10.138.9. Tensoring with $C$ over $Q$ we obtain a short exact sequence
\[ 0 \to \Omega _{P/A} \otimes C \to \Omega _{Q/A} \otimes C \to \Omega _{Q/P} \otimes C \to 0 \]
We have $\Omega _{P/A} \otimes C = \Omega _{P/A} \otimes B \otimes C$ whence this is the value of $g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C}$ on $U$. The module $\Omega _{Q/A} \otimes C$ is the value of $g_2^{-1}\Omega _2$ on $U$. We have $\Omega _{Q/P} \otimes C = \Omega _{Q \otimes _ P B/B} \otimes C$ by Algebra, Lemma 10.131.12 hence this is the value of $g_3^{-1}\Omega _3$ on $U$. Thus the short exact sequence of the lemma comes from assigning to $U$ the last displayed short exact sequence.
$\square$
Lemma 92.7.2. With notation as in (92.7.0.2) suppose that $C$ is a polynomial algebra over $B$. Then $L\pi _!(g_3^{-1}\mathcal{F}) = L\pi _{3, !}\mathcal{F} = \pi _{3, !}\mathcal{F}$ for any abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{C/B}$
Proof.
Write $C = B[E]$ for some set $E$. Choose a resolution $P_\bullet \to B$ of $B$ over $A$. For every $n$ consider the object $U_ n = (P_ n \to B, P_ n[E] \to C)$ of $\mathcal{C}_{C/B/A}$. Then $U_\bullet $ is a cosimplicial object of $\mathcal{C}_{C/B/A}$. Note that $u_3(U_\bullet )$ is the constant cosimplicial object of $\mathcal{C}_{C/B}$ with value $(C \to C)$. We will prove that the object $U_\bullet $ of $\mathcal{C}_{C/B/A}$ satisfies the hypotheses of Cohomology on Sites, Lemma 21.39.7. This implies the lemma as it shows that $L\pi _!(g_3^{-1}\mathcal{F})$ is computed by the constant simplicial abelian group $\mathcal{F}(C \to C)$ which is the value of $L\pi _{3, !}\mathcal{F} = \pi _{3, !}\mathcal{F}$ by Lemma 92.4.6.
Let $U = (\beta : P \to B, \gamma : Q \to C)$ be an object of $\mathcal{C}_{C/B/A}$. We may write $P = A[S]$ and $Q = A[S \amalg T]$ by the definition of our category $\mathcal{C}_{C/B/A}$. We have to show that
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_{C/B/A}}(U_\bullet , U) \]
is homotopy equivalent to a singleton simplicial set $*$. Observe that this simplicial set is the product
\[ \prod \nolimits _{s \in S} F_ s \times \prod \nolimits _{t \in T} F'_ t \]
where $F_ s$ is the corresponding simplicial set for $U_ s = (A[\{ s\} ] \to B, A[\{ s\} ] \to C)$ and $F'_ t$ is the corresponding simplicial set for $U_ t = (A \to B, A[\{ t\} ] \to C)$. Namely, the object $U$ is the product $\prod U_ s \times \prod U_ t$ in $\mathcal{C}_{C/B/A}$. It suffices each $F_ s$ and $F'_ t$ is homotopy equivalent to $*$, see Simplicial, Lemma 14.26.10. The case of $F_ s$ follows as $P_\bullet \to B$ is a trivial Kan fibration (as a resolution) and $F_ s$ is the fibre of this map over $\beta (s)$. (Use Simplicial, Lemmas 14.30.3 and 14.30.8). The case of $F'_ t$ is more interesting. Here we are saying that the fibre of
\[ P_\bullet [E] \longrightarrow C = B[E] \]
over $\gamma (t) \in C$ is homotopy equivalent to a point. In fact we will show this map is a trivial Kan fibration. Namely, $P_\bullet \to B$ is a trivial can fibration. For any ring $R$ we have
\[ R[E] = \mathop{\mathrm{colim}}\nolimits _{\Sigma \subset \text{Map}(E, \mathbf{Z}_{\geq 0})\text{ finite}} \prod \nolimits _{I \in \Sigma } R \]
(filtered colimit). Thus the displayed map of simplicial sets is a filtered colimit of trivial Kan fibrations, whence a trivial Kan fibration by Simplicial, Lemma 14.30.7.
$\square$
Lemma 92.7.3. With notation as in (92.7.0.2) we have $Lg_{i, !} \circ g_ i^{-1} = \text{id}$ for $i = 1, 2, 3$ and hence also $L\pi _! \circ g_ i^{-1} = L\pi _{i, !}$ for $i = 1, 2, 3$.
Proof.
Proof for $i = 1$. We claim the functor $\mathcal{C}_{C/B/A}$ is a fibred category over $\mathcal{C}_{B/A}$ Namely, suppose given $(P \to B, Q \to C)$ and a morphism $(P' \to B) \to (P \to B)$ of $\mathcal{C}_{B/A}$. Recall that this means we have an $A$-algebra homomorphism $P \to P'$ compatible with maps to $B$. Then we set $Q' = Q \otimes _ P P'$ with induced map to $C$ and the morphism
\[ (P' \to B, Q' \to C) \longrightarrow (P \to B, Q \to C) \]
in $\mathcal{C}_{C/B/A}$ (note reversal arrows again) is strongly cartesian in $\mathcal{C}_{C/B/A}$ over $\mathcal{C}_{B/A}$. Moreover, observe that the fibre category of $u_1$ over $P \to B$ is the category $\mathcal{C}_{C/P}$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{B/A}$. Since we have a fibred category we may apply Cohomology on Sites, Lemma 21.40.2. Thus $L_ ng_{1, !}g_1^{-1}\mathcal{F}$ is the (pre)sheaf which assigns to $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_{B/A})$ the $n$th homology of $g_1^{-1}\mathcal{F}$ restricted to the fibre category over $U$. Since these restrictions are constant the desired result follows from Lemma 92.4.4 via our identifications of fibre categories above.
The case $i = 2$. We claim $\mathcal{C}_{C/B/A}$ is a fibred category over $\mathcal{C}_{C/A}$ is a fibred category. Namely, suppose given $(P \to B, Q \to C)$ and a morphism $(Q' \to C) \to (Q \to C)$ of $\mathcal{C}_{C/A}$. Recall that this means we have a $B$-algebra homomorphism $Q \to Q'$ compatible with maps to $C$. Then
\[ (P \to B, Q' \to C) \longrightarrow (P \to B, Q \to C) \]
is strongly cartesian in $\mathcal{C}_{C/B/A}$ over $\mathcal{C}_{C/A}$. Note that the fibre category of $u_2$ over $Q \to C$ has an final (beware reversal arrows) object, namely, $(A \to B, Q \to C)$. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{C/A}$. Since we have a fibred category we may apply Cohomology on Sites, Lemma 21.40.2. Thus $L_ ng_{2, !}g_2^{-1}\mathcal{F}$ is the (pre)sheaf which assigns to $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_{C/A})$ the $n$th homology of $g_1^{-1}\mathcal{F}$ restricted to the fibre category over $U$. Since these restrictions are constant the desired result follows from Cohomology on Sites, Lemma 21.39.5 because the fibre categories all have final objects.
The case $i = 3$. In this case we will apply Cohomology on Sites, Lemma 21.40.3 to $u = u_3 : \mathcal{C}_{C/B/A} \to \mathcal{C}_{C/B}$ and $\mathcal{F}' = g_3^{-1}\mathcal{F}$ for some abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{C/B}$. Suppose $U = (\overline{Q} \to C)$ is an object of $\mathcal{C}_{C/B}$. Then $\mathcal{I}_ U = \mathcal{C}_{\overline{Q}/B/A}$ (again beware of reversal of arrows). The sheaf $\mathcal{F}'_ U$ is given by the rule $(P \to B, Q \to \overline{Q}) \mapsto \mathcal{F}(Q \otimes _ P B \to C)$. In other words, this sheaf is the pullback of a sheaf on $\mathcal{C}_{\overline{Q}/C}$ via the morphism $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\overline{Q}/B/A}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\overline{Q}/B})$. Thus Lemma 92.7.2 shows that $H_ n(\mathcal{I}_ U, \mathcal{F}'_ U) = 0$ for $n > 0$ and equal to $\mathcal{F}(\overline{Q} \to C)$ for $n = 0$. The aforementioned Cohomology on Sites, Lemma 21.40.3 implies that $Lg_{3, !}(g_3^{-1}\mathcal{F}) = \mathcal{F}$ and the proof is done.
$\square$
Proposition 92.7.4. Let $A \to B \to C$ be ring maps. There is a canonical distinguished triangle
\[ L_{B/A} \otimes _ B^\mathbf {L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes _ B^\mathbf {L} C[1] \]
in $D(C)$.
Proof.
Consider the short exact sequence of sheaves of Lemma 92.7.1 and apply the derived functor $L\pi _!$ to obtain a distinguished triangle
\[ L\pi _!(g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C}) \to L\pi _!(g_2^{-1}\Omega _2) \to L\pi _!(g_3^{-1}\Omega _3) \to L\pi _!(g_1^{-1}\Omega _1 \otimes _{\underline{B}} \underline{C})[1] \]
in $D(C)$. Using Lemmas 92.7.3 and 92.4.3 we see that the second and third terms agree with $L_{C/A}$ and $L_{C/B}$ and the first one equals
\[ L\pi _{1, !}(\Omega _1 \otimes _{\underline{B}} \underline{C}) = L\pi _{1, !}(\Omega _1) \otimes _ B^\mathbf {L} C = L_{B/A} \otimes _ B^\mathbf {L} C \]
The first equality by Cohomology on Sites, Lemma 21.39.6 (and flatness of $\Omega _1$ as a sheaf of modules over $\underline{B}$) and the second by Lemma 92.4.3.
$\square$
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