Lemma 15.96.7. Let $A$ be a ring. Let $M$, $N_1$, $N_2$ be finite projective $A$-modules. Let $s : M \to N_1 \oplus N_2$ be a split injection. There exists a finitely generated ideal $J \subset A$ with the following property: a ring map $A \to B$ factors through $A/J$ if and only if $s \otimes \text{id}_ B$ identifies $M \otimes _ A B$ with a direct sum of submodules of $N_1 \otimes _ A B \oplus N_2 \otimes _ A B$.
Proof. Choose a splitting $\pi : N_1 \oplus N_2 \to M$ of $s$. Denote $q_ i : N_1 \oplus N_2 \to N_1 \oplus N_2$ the projector onto $N_ i$. Set $p_ i = \pi \circ q_ i \circ s$. Observe that $p_1 + p_2 = \text{id}_ M$. We claim $M$ is a direct sum of submodules of $N_1 \oplus N_2$ if and only if $p_1$ and $p_2$ are orthogonal projectors. Thus $J$ is the smallest ideal of $A$ such that $p_1 \circ p_1 - p_1$, $p_2 \circ p_2 - p_2$, $p_1 \circ p_2$, and $p_2 \circ p_1$ are contained in $J \otimes _ A \text{End}_ A(M)$. Some details omitted. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)