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