Proposition 6

Let $A$ be a Noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$ and $I = \mathrm{Ass}(M/N)$. Let $S$ be a multiplicative subset of $A$ and $J$ the subset of $I$ consisting of the indices i such that $S \cap p_i = \varnothing$. Let $N'$ be the saturation of $N$ with respect to $S$ in $M$. Then:

(i) If $(Q_i)_{i \in I}$ is an element of $D_I(M/N)$, the family $(Q_i)_{i \in J}$ is an element of $D_J(M/N')$ and the family $(S^{-1}Q_i)_{i \in J}$ is an element of $D_J(S^{-1}M/S^{-1}N)$.

(ii) The mapping $(Q_i)_{i \in J} \to (S^{-1}Q_i)_{i \in J}$ is a bijection of $D_J(M/N')$ onto $D_J(S^{-1}M/S^{-1}N)$.

(iii) If $(Q_i)_{i \in J}$ is an element of $D_J(M/N')$ and $(R_i)_{i \in I}$ an element of $D_I(M/N)$, the family $(T_i)_{i \in I}$ such that $T_i = Q_i$ for $i \in J$ and $T_i = R_i$ for $i \in I - J$ is an element of $D_I(M/N)$.

(i) We know (no. 1, Proposition 3) that for $i \in J, S^{-1}Q_i$ is primary for $S^{-1}p_i$ and that for $i \in I - J, S^{-1}Q_i = S^{-1}M$; as $S^{-1}N = \bigcap S^{-1}Q_i$ (Chapter II, § 2, no. 4), then also $S^{-1}N = \bigcap_{i \in J} S^{-1}Q_i$. The $S^{-1}p_i$ for $i \in J$ are distinct and their set is $\operatorname{Ass}(S^{-1}M/S^{-1}N)$ ($§ 1$, no. 2, Corollary to Proposition 5); then (Proposition 4) $(S^{-1}Q_i)_{i \in J}$ is a reduced primary decomposition of $S^{-1}N$. Moreover, $Q_i = (i_M^S)^{-1}(S^{-1}Q_i)$ (no. 1 Proposition 3), hence $N' = (i_M^S)^{-1}(S^{-1}N) = \bigcap_{i \in J} Q_i$ and $(Q_i)_{i \in J}$ is obviously a reduced primary decomposition of $N'$ in $M$.

(ii) As $S^{-1}N' = S^{-1}N$, we may replace $N$ by $N'$, that is suppose that $J = I$. Let $(P_i)_{i \in I}$ be a reduced primary decomposition of $S^{-1}N$ in $S^{-1}M$ and let us write $Q_i = (i_M^S)^{-1}(P_i)$; it follows from no. 1, Proposition 3 that $Q_i$ is primary for $p_i$ ($i \in I$) and $(Q_i)_{i \in I}$ is then a reduced primary decomposition of $N$ in $M$ by virtue of no. 3, Corollary to Proposition 4. Finally, as, for all $i \in I$ and every submodule $Q'_i$ of which $M$ is $p_i$-primary with respect to $M, Q'_i = (i_M^S)^{-1}(S^{-1}Q'_i)$ by virtue of no. 1, Proposition 3 and the hypothesis $J = I$, we see that two mappings $D_I(M/N) \to D_I(S^{-1}M/S^{-1}N)$ and $D_I(S^{-1}M/S^{-1}N) \to D_I(M/N)$ have been defined whose compositions are the identities on $D_I(M/N)$ and $D_I(S^{-1}M/S^{-1}N)$, which proves (ii).

(iii) By virtue of (i), $N' = \bigcap_{i \in J} R_i$, whence

and it follows immediately from no. 3, Corollary to Proposition 4 that this primary decomposition is reduced.