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
\[ \ldots \to \wedge ^2\mathcal{F} \otimes \text{Sym}^{k - 2}\mathcal{F} \to \mathcal{F} \otimes \text{Sym}^{k - 1}\mathcal{F} \to \text{Sym}^ k\mathcal{F} \to 0 \]
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
\[ E_1^{p, q} = H_ p(\mathcal{C}, \wedge ^{q + 1}\mathcal{F} \otimes \text{Sym}^{k - q - 1}\mathcal{F}) \Rightarrow H_{p + q}(\mathcal{C}, \text{Sym}^ k(\mathcal{F})) \]
By Cohomology on Sites, Lemma 21.39.10 we have
\[ L\pi _!(\wedge ^{q + 1}\mathcal{F} \otimes \text{Sym}^{k - q - 1}\mathcal{F}) = L\pi _!(\wedge ^{q + 1}\mathcal{F}) \otimes _ B^\mathbf {L} L\pi _!(\text{Sym}^{k - q - 1}\mathcal{F})) \]
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.
\[ H^{-k}(L\pi _!(\mathcal{F}) \otimes _ B^\mathbf {L} L\pi _!(\mathcal{F}) \otimes _ B^\mathbf {L} \ldots \otimes _ B^\mathbf {L} L\pi _!(\mathcal{F})) = H_1(\mathcal{C}, \mathcal{F})^{\otimes k} \]
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 $i < k$.
\[ E_1^{p, q} = H_{- p - q}(\mathcal{C}_{B/A}, \text{Sym}^ p_{\underline{B}}(\Omega )) \Rightarrow \text{Tor}^ A_{- p - q}(B, B) \]
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. $\square$
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.
\[ \begin{matrix} B
\\ 0
\\ 0
& I/I^2
\\ 0
& H^{-2}(L_{B/A})
\\ 0
& H^{-3}(L_{B/A})
& \wedge ^2(I/I^2)
\\ 0
& H^{-4}(L_{B/A})
& H_3(\mathcal{C}_{B/A}, \text{Sym}^2(\Omega ))
\\ 0
& H^{-5}(L_{B/A})
& H_4(\mathcal{C}_{B/A}, \text{Sym}^2(\Omega ))
& \wedge ^3(I/I^2)
\end{matrix} \]
\[ \text{Tor}_3^ A(B, B) \to H^{-3}(L_{B/A}) \to \wedge ^2_ B(I/I^2) \to \text{Tor}_2^ A(B, B) \to H^{-2}(L_{B/A}) \to 0 \]
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.
\[ \text{Tor}_2^{P_ n}(B, B) = \text{Tor}_1^{P_ n}(J_ n, B) = \mathop{\mathrm{Ker}}(J_ n \otimes _{P_ n} J_ n \to J_ n^2). \]
\[ \mathop{\mathrm{Coker}}(\text{Tor}_2^{P_1}(B, B) \to \text{Tor}_2^{P_0}(B, B)) \]
\[ \text{Tor}_2^{P_1}(B, B) = \wedge ^2(\bigoplus \nolimits _{t \in T} B)\ \oplus \ \bigoplus \nolimits _{t \in T} J_0\ \oplus \ \text{Tor}_2^{P_0}(B, B) \]
\[ \wedge ^2_ B(J_0/J_0^2) \to \text{Tor}_2^{P_0}(B, B) \to H^{-2}(L_{B/A}) \to 0 \]
[1] 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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