The Stacks project

Theorem 92.12.4 (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 92.4. 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 92.4.5) as $A \to B$ is surjective (Algebra, Lemma 10.131.4). By Lemma 92.12.1 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 92.5.7, the second equality by Cohomology on Sites, Lemma 21.39.6, 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 92.5.6. 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 15.62.2. 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 92.12.3. $\square$

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

Comments (1)

Comment #7429 by Hossein Faridian on

How does one obtain the dual to Quillen's spectral sequence involving Hom and Ext? The statement is in "On the Co(homology) of Commutative Rings (Quillen)" with no proof.

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