The Stacks project

Remark 91.6.3. Suppose that we are given a square ( such that there exists an arrow $\kappa : B \to A'$ making the diagram commute:

\[ \xymatrix{ B \ar[r]_\beta \ar[rd]_\kappa & B' \\ A \ar[u] \ar[r]^\alpha & A' \ar[u] } \]

In this case we claim the functoriality map $P_\bullet \to P'_\bullet $ is homotopic to the composition $P_\bullet \to B \to A' \to P'_\bullet $. Namely, using $\kappa $ the functoriality map factors as

\[ P_\bullet \to P_{A'/A', \bullet } \to P'_\bullet \]

where $P_{A'/A', \bullet }$ is the standard resolution of $A'$ over $A'$. Since $A'$ is the polynomial algebra on the empty set over $A'$ we see from Simplicial, Lemma 14.34.3 that the augmentation $\epsilon _{A'/A'} : P_{A'/A', \bullet } \to A'$ is a homotopy equivalence of simplicial rings. Observe that the homotopy inverse map $c : A' \to P_{A'/A', \bullet }$ constructed in the proof of that lemma is just the structure morphism, hence we conclude what we want because the two compositions

\[ \xymatrix{ P_\bullet \ar[r] & P_{A'/A', \bullet } \ar@<1ex>[rr]^{\text{id}} \ar@<-1ex>[rr]_{c \circ \epsilon _{A'/A'}} & & P_{A'/A', \bullet } \ar[r] & P'_\bullet } \]

are the two maps discussed above and these are homotopic (Simplicial, Remark 14.26.5). Since the second map $P_\bullet \to P'_\bullet $ induces the zero map $\Omega _{P_\bullet /A} \to \Omega _{P'_\bullet /A'}$ we conclude that the functoriality map $L_{B/A} \to L_{B'/A'}$ is homotopic to zero in this case.

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