The Stacks project

Lemma 91.13.2. 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. $\square$

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