The Stacks project

Lemma 16.2.4. Let $R$ be a ring. Let $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and write $I = (f_1, \ldots , f_ m)$. Let $a \in A$. Then (16.2.3.3) implies there exists an $A$-linear map $\psi : \bigoplus \nolimits _{i = 1, \ldots , n} A \text{d}x_ i \to A^{\oplus c}$ such that the composition

\[ A^{\oplus c} \xrightarrow {(f_1, \ldots , f_ c)} I/I^2 \xrightarrow {f \mapsto \text{d}f} \bigoplus \nolimits _{i = 1, \ldots , n} A \text{d}x_ i \xrightarrow {\psi } A^{\oplus c} \]

is multiplication by $a$. Conversely, if such a $\psi $ exists, then $a^ c$ satisfies (16.2.3.3).

Proof. This is a special case of Algebra, Lemma 10.15.5. $\square$

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