The Stacks project

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

  1. the localization of $\varphi $ at $\mathfrak m_ i$ is isomorphic to $\varphi _ i$, and

  2. $\mathop{\mathrm{Ker}}(\varphi )$ 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 15.89.17 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 15.89.20 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 $\mathop{\mathrm{Coker}}(\varphi )_{\mathfrak m_ i} \cong \mathop{\mathrm{Coker}}(\varphi _ i)$. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 42.4: Preparation for tame symbols

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EAE. Beware of the difference between the letter 'O' and the digit '0'.