Lemma 10.13.2. Let $R$ be a ring. Let $M_2 \to M_1 \to M \to 0$ be an exact sequence of $R$-modules. There are exact sequences

$M_2 \otimes _ R \text{Sym}^{n - 1}(M_1) \to \text{Sym}^ n(M_1) \to \text{Sym}^ n(M) \to 0$

and similarly

$M_2 \otimes _ R \wedge ^{n - 1}(M_1) \to \wedge ^ n(M_1) \to \wedge ^ n(M) \to 0$

Proof. Omitted. $\square$

There are also:

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