Theorem 89.11.4 (Quillen spectral sequence). Let $A \to B$ be a surjective ring map. Consider the sheaf $\Omega = \Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B}$ of $\underline{B}$-modules on $\mathcal{C}_{B/A}$, see Section 89.4. Then there is a spectral sequence with $E_1$-page

$E_1^{p, q} = H_{- p - q}(\mathcal{C}_{B/A}, \text{Sym}^ p_{\underline{B}}(\Omega )) \Rightarrow \text{Tor}^ A_{- p - q}(B, B)$

with $d_ r$ of bidegree $(r, -r + 1)$. Moreover, $H_ i(\mathcal{C}_{B/A}, \text{Sym}^ k_{\underline{B}}(\Omega )) = 0$ for $i < k$.

Proof. Let $I \subset A$ be the kernel of $A \to B$. Let $\mathcal{J} \subset \mathcal{O}$ be the kernel of $\mathcal{O} \to \underline{B}$. Then $I\mathcal{O} \subset \mathcal{J}$. Set $\mathcal{K} = \mathcal{J}/I\mathcal{O}$ and $\overline{\mathcal{O}} = \mathcal{O}/I\mathcal{O}$.

For every object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we can choose an isomorphism $P = A[E]$ such that the map $P \to B$ maps each $e \in E$ to zero. Then $J = \mathcal{J}(U) \subset P = \mathcal{O}(U)$ is equal to $J = IP + (e; e \in E)$. Moreover $\overline{\mathcal{O}}(U) = B[E]$ and $K = \mathcal{K}(U) = (e; e \in E)$ is the ideal generated by the variables in the polynomial ring $B[E]$. In particular it is clear that

$K/K^2 \xrightarrow {\text{d}} \Omega _{P/A} \otimes _ P B$

is a bijection. In other words, $\Omega = \mathcal{K}/\mathcal{K}^2$ and $\text{Sym}_ B^ k(\Omega ) = \mathcal{K}^ k/\mathcal{K}^{k + 1}$. Note that $\pi _!(\Omega ) = \Omega _{B/A} = 0$ (Lemma 89.4.5) as $A \to B$ is surjective (Algebra, Lemma 10.130.5). By Lemma 89.11.1 we conclude that

$H_ i(\mathcal{C}_{B/A}, \mathcal{K}^ k/\mathcal{K}^{k + 1}) = H_ i(\mathcal{C}_{B/A}, \text{Sym}^ k_{\underline{B}}(\Omega )) = 0$

for $i < k$. This proves the final statement of the theorem.

The approach to the theorem is to note that

$B \otimes _ A^\mathbf {L} B = L\pi _!(\mathcal{O}) \otimes _ A^\mathbf {L} B = L\pi _!(\mathcal{O} \otimes _{\underline{A}}^\mathbf {L} \underline{B}) = L\pi _!(\overline{\mathcal{O}})$

The first equality by Lemma 89.5.7, the second equality by Cohomology on Sites, Lemma 21.38.6, and the third equality as $\mathcal{O}$ is flat over $\underline{A}$. The sheaf $\overline{\mathcal{O}}$ has a filtration

$\ldots \subset \mathcal{K}^3 \subset \mathcal{K}^2 \subset \mathcal{K} \subset \overline{\mathcal{O}}$

This induces a filtration $F$ on a complex $C$ representing $L\pi _!(\overline{\mathcal{O}})$ with $F^ pC$ representing $L\pi _!(\mathcal{K}^ p)$ (construction of $C$ and $F$ omitted). Consider the spectral sequence of Homology, Section 12.24 associated to $(C, F)$. It has $E_1$-page

$E_1^{p, q} = H_{- p - q}(\mathcal{C}_{B/A}, \mathcal{K}^ p/\mathcal{K}^{p + 1}) \quad \Rightarrow \quad H_{- p - q}(\mathcal{C}_{B/A}, \overline{\mathcal{O}}) = \text{Tor}_{- p - q}^ A(B, B)$

and differentials $E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$. To show convergence we will show that for every $k$ there exists a $c$ such that $H_ i(\mathcal{C}_{B/A}, \mathcal{K}^ n) = 0$ for $i < k$ and $n > c$1.

Given $k \geq 0$ set $c = k^2$. We claim that

$H_ i(\mathcal{C}_{B/A}, \mathcal{K}^{n + c}) \to H_ i(\mathcal{C}_{B/A}, \mathcal{K}^ n)$

is zero for $i < k$ and all $n \geq 0$. Note that $\mathcal{K}^ n/\mathcal{K}^{n + c}$ has a finite filtration whose successive quotients $\mathcal{K}^ m/\mathcal{K}^{m + 1}$, $n \leq m < n + c$ have $H_ i(\mathcal{C}_{B/A}, \mathcal{K}^ m/\mathcal{K}^{m + 1}) = 0$ for $i < n$ (see above). Hence the claim implies $H_ i(\mathcal{C}_{B/A}, \mathcal{K}^{n + c}) = 0$ for $i < k$ and all $n \geq k$ which is what we need to show.

Proof of the claim. Recall that for any $\mathcal{O}$-module $\mathcal{F}$ the map $\mathcal{F} \to \mathcal{F} \otimes _\mathcal {O}^\mathbf {L} B$ induces an isomorphism on applying $L\pi _!$, see Lemma 89.5.6. Consider the map

$\mathcal{K}^{n + k} \otimes _\mathcal {O}^\mathbf {L} B \longrightarrow \mathcal{K}^ n \otimes _\mathcal {O}^\mathbf {L} B$

We claim that this map induces the zero map on cohomology sheaves in degrees $0, -1, \ldots , - k + 1$. If this second claim holds, then the $k$-fold composition

$\mathcal{K}^{n + c} \otimes _\mathcal {O}^\mathbf {L} B \longrightarrow \mathcal{K}^ n \otimes _\mathcal {O}^\mathbf {L} B$

factors through $\tau _{\leq -k}\mathcal{K}^ n \otimes _\mathcal {O}^\mathbf {L} B$ hence induces zero on $H_ i(\mathcal{C}_{B/A}, -) = L_ i\pi _!( - )$ for $i < k$, see Derived Categories, Lemma 13.12.5. By the remark above this means the same thing is true for $H_ i(\mathcal{C}_{B/A}, \mathcal{K}^{n + c}) \to H_ i(\mathcal{C}_{B/A}, \mathcal{K}^ n)$ which proves the (first) claim.

Proof of the second claim. The statement is local, hence we may work over an object $U = (P \to B)$ as above. We have to show the maps

$\text{Tor}_ i^ P(B, K^{n + k}) \to \text{Tor}_ i^ P(B, K^ n)$

are zero for $i < k$. There is a spectral sequence

$\text{Tor}_ a^ P(P/IP, \text{Tor}_ b^{P/IP}(B, K^ n)) \Rightarrow \text{Tor}_{a + b}^ P(B, K^ n),$

see More on Algebra, Example 15.60.2. Thus it suffices to prove the maps

$\text{Tor}_ i^{P/IP}(B, K^{n + 1}) \to \text{Tor}_ i^{P/IP}(B, K^ n)$

are zero for all $i$. This is Lemma 89.11.3. $\square$

 A posteriori the “correct” vanishing $H_ i(\mathcal{C}_{B/A}, \mathcal{K}^ n) = 0$ for $i < n$ can be concluded.

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