Lemma 92.12.1. Notation and assumptions as in Cohomology on Sites, Example 21.39.1. Assume $\mathcal{C}$ has a cosimplicial object as in Cohomology on Sites, Lemma 21.39.7. Let $\mathcal{F}$ be a flat $\underline{B}$-module such that $H_0(\mathcal{C}, \mathcal{F}) = 0$. Then $H_ l(\mathcal{C}, \text{Sym}_{\underline{B}}^ k(\mathcal{F})) = 0$ for $l < k$.

## 92.12 A spectral sequence of Quillen

In this section we discuss a spectral sequence relating derived tensor product to the cotangent complex.

**Proof.**
We drop the subscript ${}_{\underline{B}}$ from tensor products, wedge powers, and symmetric powers. We will prove the lemma by induction on $k$. The cases $k = 0, 1$ follow from the assumptions. If $k > 1$ consider the exact complex

with differentials as in the Koszul complex. If we think of this as a resolution of $\text{Sym}^ k\mathcal{F}$, then this gives a first quadrant spectral sequence

By Cohomology on Sites, Lemma 21.39.10 we have

It follows (from the construction of derived tensor products) that the induction hypothesis combined with the vanishing of $H_0(\mathcal{C}, \wedge ^{q + 1}(\mathcal{F})) = 0$ will prove what we want. This is true because $\wedge ^{q + 1}(\mathcal{F})$ is a quotient of $\mathcal{F}^{\otimes q + 1}$ and $H_0(\mathcal{C}, \mathcal{F}^{\otimes q + 1})$ is a quotient of $H_0(\mathcal{C}, \mathcal{F})^{\otimes q + 1}$ which is zero. $\square$

Remark 92.12.2. In the situation of Lemma 92.12.1 one can show that $H_ k(\mathcal{C}, \text{Sym}^ k(\mathcal{F})) = \wedge ^ k_ B(H_1(\mathcal{C}, \mathcal{F}))$. Namely, it can be deduced from the proof that $H_ k(\mathcal{C}, \text{Sym}^ k(\mathcal{F}))$ is the $S_ k$-coinvariants of

Thus our claim is that this action is given by the usual action of $S_ k$ on the tensor product multiplied by the sign character. To prove this one has to work through the sign conventions in the definition of the total complex associated to a multi-complex. We omit the verification.

Lemma 92.12.3. Let $A$ be a ring. Let $P = A[E]$ be a polynomial ring. Set $I = (e; e \in E) \subset P$. The maps $\text{Tor}_ i^ P(A, I^{n + 1}) \to \text{Tor}_ i^ P(A, I^ n)$ are zero for all $i$ and $n$.

**Proof.**
Denote $x_ e \in P$ the variable corresponding to $e \in E$. A free resolution of $A$ over $P$ is given by the Koszul complex $K_\bullet $ on the $x_ e$. Here $K_ i$ has basis given by wedges $e_1 \wedge \ldots \wedge e_ i$, $e_1, \ldots , e_ i \in E$ and $d(e) = x_ e$. Thus $K_\bullet \otimes _ P I^ n = I^ nK_\bullet $ computes $\text{Tor}_ i^ P(A, I^ n)$. Observe that everything is graded with $\deg (x_ e) = 1$, $\deg (e) = 1$, and $\deg (a) = 0$ for $a \in A$. Suppose $\xi \in I^{n + 1}K_ i$ is a cocycle homogeneous of degree $m$. Note that $m \geq i + 1 + n$. Then $\xi = \text{d}\eta $ for some $\eta \in K_{i + 1}$ as $K_\bullet $ is exact in degrees $ > 0$. (The case $i = 0$ is left to the reader.) Now $\deg (\eta ) = m \geq i + 1 + n$. Hence writing $\eta $ in terms of the basis we see the coordinates are in $I^ n$. Thus $\xi $ maps to zero in the homology of $I^ nK_\bullet $ as desired.
$\square$

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

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

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

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

The approach to the theorem is to note that

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

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

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

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

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

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

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

see More on Algebra, Example 15.62.2. Thus it suffices to prove the maps

are zero for all $i$. This is Lemma 92.12.3. $\square$

Remark 92.12.5. In the situation of Theorem 92.12.4 let $I = \mathop{\mathrm{Ker}}(A \to B)$. Then $H^{-1}(L_{B/A}) = H_1(\mathcal{C}_{B/A}, \Omega ) = I/I^2$, see Lemma 92.11.2. Hence $H_ k(\mathcal{C}_{B/A}, \text{Sym}^ k(\Omega )) = \wedge ^ k_ B(I/I^2)$ by Remark 92.12.2. Thus the $E_1$-page looks like

with horizontal differential. Thus we obtain edge maps $\text{Tor}_ i^ A(B, B) \to H^{-i}(L_{B/A})$, $i > 0$ and $\wedge ^ i_ B(I/I^2) \to \text{Tor}_ i^ A(B, B)$. Finally, we have $\text{Tor}_1^ A(B, B) = I/I^2$ and there is a five term exact sequence

of low degree terms.

Remark 92.12.6. Let $A \to B$ be a ring map. Let $P_\bullet $ be a resolution of $B$ over $A$ (Remark 92.5.5). Set $J_ n = \mathop{\mathrm{Ker}}(P_ n \to B)$. Note that

Hence $H_2(L_{B/A})$ is canonically equal to

by Remark 92.11.5. To make this more explicit we choose $P_2$, $P_1$, $P_0$ as in Example 92.5.9. We claim that

Namely, the basis elements $x_ t \wedge x_{t'}$ of the first summand corresponds to the element $x_ t \otimes x_{t'} - x_{t'} \otimes x_ t$ of $J_1 \otimes _{P_1} J_1$. For $f \in J_0$ the element $x_ t \otimes f$ of the second summand corresponds to the element $x_ t \otimes s_0(f) - s_0(f) \otimes x_ t$ of $J_1 \otimes _{P_1} J_1$. Finally, the map $\text{Tor}_2^{P_0}(B, B) \to \text{Tor}_2^{P_1}(B, B)$ is given by $s_0$. The map $d_0 - d_1 : \text{Tor}_2^{P_1}(B, B) \to \text{Tor}_2^{P_0}(B, B)$ is zero on the last summand, maps $x_ t \otimes f$ to $f \otimes f_ t - f_ t \otimes f$, and maps $x_ t \wedge x_{t'}$ to $f_ t \otimes f_{t'} - f_{t'} \otimes f_ t$. All in all we conclude that there is an exact sequence

In this way we obtain a direct proof of a consequence of Quillen's spectral sequence discussed in Remark 92.12.5.

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