Remark 92.13.4 (09D7): Functoriality of the Lichtenbaum-Schlessinger complex

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

\[ \xymatrix{ L_{B'/A'} \ar[r] & L' \\ L_{B/A} \ar[r] \ar[u] & L \ar[u] } \]

The Stacks project

Comments (0)

Post a comment