Definition 92.13.1. Let $A \to B$ be a ring map. Let $M$ be a $(B, B)$-bimodule over $A$. An $A$-biderivation is an $A$-linear map $\lambda : B \to M$ such that $\lambda (xy) = x\lambda (y) + \lambda (x)y$.
92.13 Comparison with Lichtenbaum-Schlessinger
Let $A \to B$ be a ring map. In [Lichtenbaum-Schlessinger] there is a fairly explicit determination of $\tau _{\geq -2}L_{B/A}$ which is often used in calculations of versal deformation spaces of singularities. The construction follows. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to B$ with kernel $I$. Choose generators $f_ t$, $t \in T$ for $I$ which induces a surjection $F = \bigoplus _{t \in T} P \to I$ with $F$ a free $P$-module. Let $Rel \subset F$ be the kernel of $F \to I$, in other words $Rel$ is the set of relations among the $f_ t$. Let $TrivRel \subset Rel$ be the submodule of trivial relations, i.e., the submodule of $Rel$ generated by the elements $(\ldots , f_{t'}, 0, \ldots , 0, -f_ t, 0, \ldots )$. Consider the complex of $B$-modules
92.13.0.1
\begin{equation} \label{cotangent-equation-lichtenbaum-schlessinger} Rel/TrivRel \longrightarrow F \otimes _ P B \longrightarrow \Omega _{P/A} \otimes _ P B \end{equation}
where the last term is placed in degree $0$. The first map is the obvious one and the second map sends the basis element corresponding to $t \in T$ to $\text{d}f_ t \otimes 1$.
For a polynomial algebra the biderivations are easy to describe.
Lemma 92.13.2. Let $P = A[S]$ be a polynomial ring over $A$. Let $M$ be a $(P, P)$-bimodule over $A$. Given $m_ s \in M$ for $s \in S$, there exists a unique $A$-biderivation $\lambda : P \to M$ mapping $s$ to $m_ s$ for $s \in S$.
Proof. We set
\[ \lambda (s_1 \ldots s_ t) = \sum s_1 \ldots s_{i - 1} m_{s_ i} s_{i + 1} \ldots s_ t \]
in $M$. Extending by $A$-linearity we obtain a biderivation. $\square$
Here is the comparison statement. The reader may also read about this in [page 206, Proposition 12, Andre-Homologie] or in the paper [Doncel] which extends the complex (92.13.0.1) by one term and the comparison to $\tau _{\geq -3}$.
Lemma 92.13.3. In the situation above denote $L$ the complex (92.13.0.1). There is a canonical map $L_{B/A} \to L$ in $D(B)$ which induces an isomorphism $\tau _{\geq -2}L_{B/A} \to L$ in $D(B)$.
Proof. Let $P_\bullet \to B$ be a resolution of $B$ over $A$ (Remark 92.5.5). We will identify $L_{B/A}$ with $\Omega _{P_\bullet /A} \otimes B$. To construct the map we make some choices.
Choose an $A$-algebra map $\psi : P_0 \to P$ compatible with the given maps $P_0 \to B$ and $P \to B$.
Write $P_1 = A[S]$ for some set $S$. For $s \in S$ we may write
\[ \psi (d_0(s) - d_1(s)) = \sum p_{s, t} f_ t \]
for some $p_{s, t} \in P$. Think of $F = \bigoplus _{t \in T} P$ as a $(P_1, P_1)$-bimodule via the maps $(\psi \circ d_0, \psi \circ d_1)$. By Lemma 92.13.2 we obtain a unique $A$-biderivation $\lambda : P_1 \to F$ mapping $s$ to the vector with coordinates $p_{s, t}$. By construction the composition
\[ P_1 \longrightarrow F \longrightarrow P \]
sends $f \in P_1$ to $\psi (d_0(f) - d_1(f))$ because the map $f \mapsto \psi (d_0(f) - d_1(f))$ is an $A$-biderivation agreeing with the composition on generators.
For $g \in P_2$ we claim that $\lambda (d_0(g) - d_1(g) + d_2(g))$ is an element of $Rel$. Namely, by the last remark of the previous paragraph the image of $\lambda (d_0(g) - d_1(g) + d_2(g))$ in $P$ is
\[ \psi ((d_0 - d_1)(d_0(g) - d_1(g) + d_2(g))) \]
which is zero by Simplicial, Section 14.23).
The choice of $\psi $ determines a map
\[ \text{d}\psi \otimes 1 : \Omega _{P_0/A} \otimes B \longrightarrow \Omega _{P/A} \otimes B \]
Composing $\lambda $ with the map $F \to F \otimes B$ gives a usual $A$-derivation as the two $P_1$-module structures on $F \otimes B$ agree. Thus $\lambda $ determines a map
\[ \overline{\lambda } : \Omega _{P_1/A} \otimes B \longrightarrow F \otimes B \]
Finally, We obtain a $B$-linear map
\[ q : \Omega _{P_2/A} \otimes B \longrightarrow Rel/TrivRel \]
by mapping $\text{d}g$ to the class of $\lambda (d_0(g) - d_1(g) + d_2(g))$ in the quotient.
The diagram
\[ \xymatrix{ \Omega _{P_3/A} \otimes B \ar[r] \ar[d] & \Omega _{P_2/A} \otimes B \ar[r] \ar[d]_ q & \Omega _{P_1/A} \otimes B \ar[r] \ar[d]_{\overline{\lambda }} & \Omega _{P_0/A} \otimes B \ar[d]_{\text{d}\psi \otimes 1} \\ 0 \ar[r] & Rel/TrivRel \ar[r] & F \otimes B \ar[r] & \Omega _{P/A} \otimes B } \]
commutes (calculation omitted) and we obtain the map of the lemma. By Remark 92.11.4 and Lemma 92.11.3 we see that this map induces isomorphisms $H_1(L_{B/A}) \to H_1(L)$ and $H_0(L_{B/A}) \to H_0(L)$.
It remains to see that our map $L_{B/A} \to L$ induces an isomorphism $H_2(L_{B/A}) \to H_2(L)$. Choose a resolution of $B$ over $A$ with $P_0 = P = A[u_ i]$ and then $P_1$ and $P_2$ as in Example 92.5.9. In Remark 92.12.6 we have constructed an exact sequence
\[ \wedge ^2_ B(J_0/J_0^2) \to \text{Tor}_2^{P_0}(B, B) \to H^{-2}(L_{B/A}) \to 0 \]
where $P_0 = P$ and $J_0 = \mathop{\mathrm{Ker}}(P \to B) = I$. Calculating the Tor group using the short exact sequences $0 \to I \to P \to B \to 0$ and $0 \to Rel \to F \to I \to 0$ we find that $\text{Tor}_2^ P(B, B) = \mathop{\mathrm{Ker}}(Rel \otimes B \to F \otimes B)$. The image of the map $\wedge ^2_ B(I/I^2) \to \text{Tor}_2^ P(B, B)$ under this identification is exactly the image of $TrivRel \otimes B$. Thus we see that $H_2(L_{B/A}) \cong H_2(L)$.
Finally, we have to check that our map $L_{B/A} \to L$ actually induces this isomorphism. We will use the notation and results discussed in Example 92.5.9 and Remarks 92.12.6 and 92.11.5 without further mention. Pick an element $\xi $ of $\text{Tor}_2^{P_0}(B, B) = \mathop{\mathrm{Ker}}(I \otimes _ P I \to I^2)$. Write $\xi = \sum h_{t', t}f_{t'} \otimes f_ t$ for some $h_{t', t} \in P$. Tracing through the exact sequences above we find that $\xi $ corresponds to the image in $Rel \otimes B$ of the element $r \in Rel \subset F = \bigoplus _{t \in T} P$ with $t$th coordinate $r_ t = \sum _{t' \in T} h_{t', t}f_{t'}$. On the other hand, $\xi $ corresponds to the element of $H_2(L_{B/A}) = H_2(\Omega )$ which is the image via $\text{d} : H_2(\mathcal{J}/\mathcal{J}^2) \to H_2(\Omega )$ of the boundary of $\xi $ under the $2$-extension
\[ 0 \to \text{Tor}_2^\mathcal {O}(\underline{B}, \underline{B}) \to \mathcal{J} \otimes _\mathcal {O} \mathcal{J} \to \mathcal{J} \to \mathcal{J}/\mathcal{J}^2 \to 0 \]
We compute the successive transgressions of our element. First we have
\[ \xi = (d_0 - d_1)(- \sum s_0(h_{t', t} f_{t'}) \otimes x_ t) \]
and next we have
\[ \sum s_0(h_{t', t} f_{t'}) x_ t = d_0(v_ r) - d_1(v_ r) + d_2(v_ r) \]
by our choice of the variables $v$ in Example 92.5.9. We may choose our map $\lambda $ above such that $\lambda (u_ i) = 0$ and $\lambda (x_ t) = - e_ t$ where $e_ t \in F$ denotes the basis vector corresponding to $t \in T$. Hence the construction of our map $q$ above sends $\text{d}v_ r$ to
\[ \lambda (\sum s_0(h_{t', t} f_{t'}) x_ t) = \sum \nolimits _ t \left(\sum \nolimits _{t'} h_{t', t}f_{t'}\right) e_ t \]
matching the image of $\xi $ in $Rel \otimes B$ (the two minus signs we found above cancel out). This agreement finishes the proof. $\square$
Remark 92.13.4 (Functoriality of the Lichtenbaum-Schlessinger complex). Consider a commutative square
\[ \xymatrix{ A' \ar[r] & B' \\ A \ar[u] \ar[r] & B \ar[u] } \]
of ring maps. Choose a factorization
\[ \xymatrix{ A' \ar[r] & P' \ar[r] & B' \\ A \ar[u] \ar[r] & P \ar[u] \ar[r] & B \ar[u] } \]
with $P$ a polynomial algebra over $A$ and $P'$ a polynomial algebra over $A'$. Choose generators $f_ t$, $t \in T$ for $\mathop{\mathrm{Ker}}(P \to B)$. For $t \in T$ denote $f'_ t$ the image of $f_ t$ in $P'$. Choose $f'_ s \in P'$ such that the elements $f'_ t$ for $t \in T' = T \amalg S$ generate the kernel of $P' \to B'$. Set $F = \bigoplus _{t \in T} P$ and $F' = \bigoplus _{t' \in T'} P'$. Let $Rel = \mathop{\mathrm{Ker}}(F \to P)$ and $Rel' = \mathop{\mathrm{Ker}}(F' \to P')$ where the maps are given by multiplication by $f_ t$, resp. $f'_ t$ on the coordinates. Finally, set $TrivRel$, resp. $TrivRel'$ equal to the submodule of $Rel$, resp. $TrivRel$ generated by the elements $(\ldots , f_{t'}, 0, \ldots , 0, -f_ t, 0, \ldots )$ for $t, t' \in T$, resp. $T'$. Having made these choices we obtain a canonical commutative diagram
\[ \xymatrix{ L' : & Rel'/TrivRel' \ar[r] & F' \otimes _{P'} B' \ar[r] & \Omega _{P'/A'} \otimes _{P'} B' \\ L : \ar[u] & Rel/TrivRel \ar[r] \ar[u] & F \otimes _ P B \ar[r] \ar[u] & \Omega _{P/A} \otimes _ P B \ar[u] } \]
Moreover, tracing through the choices made in the proof of Lemma 92.13.3 the reader sees that one obtains a commutative diagram
\[ \xymatrix{ L_{B'/A'} \ar[r] & L' \\ L_{B/A} \ar[r] \ar[u] & L \ar[u] } \]
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