90.7 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

90.7.0.1
$$\label{cotangent-equation-triangle} L_{B/A} \otimes _ B^\mathbf {L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes _ B^\mathbf {L} C[1]$$

in $D(C)$. This will be proved in Proposition 90.7.4. For an alternative approach see Remark 90.7.5.

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

1. $P$ is a polynomial algebra over $A$,

2. $P \to B$ is an $A$-algebra homomorphism,

3. $Q$ is a polynomial algebra over $P$, and

4. $Q \to C$ 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 90.4. 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

90.7.0.2
$$\label{cotangent-equation-three-maps} \begin{matrix} g_1 : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}), \mathcal{O}_1) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{B/A}), \mathcal{O}) \\ g_2 : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}), \mathcal{O}_2) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/A}), \mathcal{O}) \\ g_3 : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B/A}), \mathcal{O}_3) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{C/B}), \mathcal{O}) \end{matrix}$$

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 90.4.3 and the following lemmas.

Lemma 90.7.1. With notation as in (90.7.0.2) 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 90.7.2. With notation as in (90.7.0.2) 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 21.38.7. 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 90.4.6.

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 90.7.3. With notation as in (90.7.0.2) 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 21.39.2. 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 90.4.4 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 21.39.2. 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 21.38.5 because the fibre categories all have final objects.

The case $i = 3$. In this case we will apply Cohomology on Sites, Lemma 21.39.3 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 90.7.2 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 21.39.3 implies that $Lg_{3, !}(g_3^{-1}\mathcal{F}) = \mathcal{F}$ and the proof is done. $\square$

Proposition 90.7.4. 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 90.7.1 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 90.7.3 and 90.4.3 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 21.38.6 (and flatness of $\Omega _1$ as a sheaf of modules over $\underline{B}$) and the second by Lemma 90.4.3. $\square$

Remark 90.7.5. We sketch an alternative, perhaps simpler, proof of the existence of the fundamental triangle. Let $A \to B \to C$ be ring maps and assume that $B \to C$ is injective. Let $P_\bullet \to B$ be the standard resolution of $B$ over $A$ and let $Q_\bullet \to C$ be the standard resolution of $C$ over $B$. Picture

$\xymatrix{ P_\bullet : & A[A[A[B]]] \ar[d] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[B]] \ar[d] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[B] \ar[d] \ar@<0ex>[l] \ar[r] & B \\ Q_\bullet : & A[A[A[C]]] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[C]] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[C] \ar@<0ex>[l] \ar[r] & C }$

Observe that since $B \to C$ is injective, the ring $Q_ n$ is a polynomial algebra over $P_ n$ for all $n$. Hence we obtain a cosimplicial object in $\mathcal{C}_{C/B/A}$ (beware reversal arrows). Now set $\overline{Q}_\bullet = Q_\bullet \otimes _{P_\bullet } B$. The key to the proof of Proposition 90.7.4 is to show that $\overline{Q}_\bullet$ is a resolution of $C$ over $B$. This follows from Cohomology on Sites, Lemma 21.38.12 applied to $\mathcal{C} = \Delta$, $\mathcal{O} = P_\bullet$, $\mathcal{O}' = B$, and $\mathcal{F} = Q_\bullet$ (this uses that $Q_ n$ is flat over $P_ n$; see Cohomology on Sites, Remark 21.38.11 to relate simplicial modules to sheaves). The key fact implies that the distinguished triangle of Proposition 90.7.4 is the distinguished triangle associated to the short exact sequence of simplicial $C$-modules

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

which is deduced from the short exact sequences $0 \to \Omega _{P_ n/A} \otimes _{P_ n} Q_ n \to \Omega _{Q_ n/A} \to \Omega _{Q_ n/P_ n} \to 0$ of Algebra, Lemma 10.138.9. Namely, by Remark 90.5.5 and the key fact the complex on the right hand side represents $L_{C/B}$ in $D(C)$.

If $B \to C$ is not injective, then we can use the above to get a fundamental triangle for $A \to B \to B \times C$. Since $L_{B \times C/B} \to L_{B/B} \oplus L_{C/B}$ and $L_{B \times C/A} \to L_{B/A} \oplus L_{C/A}$ are quasi-isomorphism in $D(B \times C)$ (Lemma 90.6.4) this induces the desired distinguished triangle in $D(C)$ by tensoring with the flat ring map $B \times C \to C$.

Remark 90.7.6. Let $A \to B \to C$ be ring maps with $B \to C$ injective. Recall the notation $P_\bullet$, $Q_\bullet$, $\overline{Q}_\bullet$ of Remark 90.7.5. Let $R_\bullet$ be the standard resolution of $C$ over $B$. In this remark we explain how to get the canonical identification of $\Omega _{\overline{Q}_\bullet /B} \otimes _{\overline{Q}_\bullet } C$ with $L_{C/B} = \Omega _{R_\bullet /B} \otimes _{R_\bullet } C$. Let $S_\bullet \to B$ be the standard resolution of $B$ over $B$. Note that the functoriality map $S_\bullet \to R_\bullet$ identifies $R_ n$ as a polynomial algebra over $S_ n$ because $B \to C$ is injective. For example in degree $0$ we have the map $B[B] \to B[C]$, in degree $1$ the map $B[B[B]] \to B[B[C]]$, and so on. Thus $\overline{R}_\bullet = R_\bullet \otimes _{S_\bullet } B$ is a simplicial polynomial algebra over $B$ as well and it follows (as in Remark 90.7.5) from Cohomology on Sites, Lemma 21.38.12 that $\overline{R}_\bullet \to C$ is a resolution. Since we have a commutative diagram

$\xymatrix{ Q_\bullet \ar[r] & R_\bullet \\ P_\bullet \ar[u] \ar[r] & S_\bullet \ar[u] \ar[r] & B }$

we obtain a canonical map $\overline{Q}_\bullet = Q_\bullet \otimes _{P_\bullet } B \to \overline{R}_\bullet$. Thus the maps

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

are quasi-isomorphisms (Remark 90.5.5) and composing one with the inverse of the other gives the desired identification.

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