Lemma 42.4.1. Let $A$ be a Noetherian ring. Let $\mathfrak m_1, \ldots , \mathfrak m_ r$ be pairwise distinct maximal ideals of $A$. For $i = 1, \ldots , r$ let $\varphi _ i : A_{\mathfrak m_ i} \to B_ i$ be a ring map whose kernel and cokernel are annihilated by a power of $\mathfrak m_ i$. Then there exists a ring map $\varphi : A \to B$ such that
the localization of $\varphi $ at $\mathfrak m_ i$ is isomorphic to $\varphi _ i$, and
$\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are annihilated by a power of $\mathfrak m_1 \cap \ldots \cap \mathfrak m_ r$.
Moreover, if each $\varphi _ i$ is finite, injective, or surjective then so is $\varphi $.
Proof.
Set $I = \mathfrak m_1 \cap \ldots \cap \mathfrak m_ r$. Set $A_ i = A_{\mathfrak m_ i}$ and $A' = \prod A_ i$. Then $IA' = \prod \mathfrak m_ i A_ i$ and $A \to A'$ is a flat ring map such that $A/I \cong A'/IA'$. Thus we may use More on Algebra, Lemma 15.89.16 to see that there exists an $A$-module map $\varphi : A \to B$ with $\varphi _ i$ isomorphic to the localization of $\varphi $ at $\mathfrak m_ i$. Then we can use the discussion in More on Algebra, Remark 15.89.19 to endow $B$ with an $A$-algebra structure matching the given $A$-algebra structure on $B_ i$. The final statement of the lemma follows easily from the fact that $\mathop{\mathrm{Ker}}(\varphi )_{\mathfrak m_ i} \cong \mathop{\mathrm{Ker}}(\varphi _ i)$ and $\mathop{\mathrm{Coker}}(\varphi )_{\mathfrak m_ i} \cong \mathop{\mathrm{Coker}}(\varphi _ i)$.
$\square$
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