I will assume that everything in sight is Noetherian and finitely generated. Then the primary decomposition amounts to a description of the geometric support of the module $M$ if you view it as a sheaf on $\text{Spec}(A)$. The scheme $\text{Spec}(A/\text{Ann}(M))$ is a subscheme of $\text{Spec}(A)$ that, by definition, supports $M$. If $M_i$ N_i$ is minimal, then $\text{Spec}(A/P_i)$ is an irreducible component of $\text{Spec}(A/\text{Ann}(M))$. In a reduced decomposition, the The module $M/N_i$ is also a quotient of $M \otimes (A/P_i^n)$ for $n$ large enough (and in a reduced decomposition I think)think they are equal), so it is the fiber part of $M$ on that irreducible component of its support. If $N_i$ is not minimal, then $\text{Spec}(A/P_i)$ is an irreducible scheme inside of a component of $\text{Spec}(A/\text{Ann}(M))$.
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I will assume that everything in sight is Noetherian and finitely generated. Then the primary decomposition amounts to a description of the geometric support of the module $M$ if you view it as a sheaf on $\text{Spec}(A)$. The scheme $\text{Spec}(A/\text{Ann}(M))$ is a subscheme of $\text{Spec}(A)$ that, by definition, supports $M$. If $M_i$ is minimal, then $\text{Spec}(A/P_i)$ is an irreducible component of $\text{Spec}(A/\text{Ann}(M))$. In a reduced decomposition, the module $M/N_i$ is also $M \otimes (A/P_i^n)$ for $n$ large enough (I think), so it is the fiber of $M$ on that irreducible component of its support. If $N_i$ is not minimal, then $\text{Spec}(A/P_i)$ is an irreducible scheme inside of a component of $\text{Spec}(A/\text{Ann}(M))$. |
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