Lemma 10.88.5. Let $f: M \to N$ and $g: M \to M'$ be maps of $R$-modules. Suppose $\mathop{\mathrm{Coker}}(f)$ is of finite presentation. Then $g$ dominates $f$ if and only if $g$ factors through $f$, i.e. there exists a module map $h: N \to M'$ such that $g = h \circ f$.
Proof. Consider the pushout of $f$ and $g$ as in the statement of Lemma 10.88.4. From the construction of the pushout it follows that $\mathop{\mathrm{Coker}}(f') = \mathop{\mathrm{Coker}}(f)$, so $\mathop{\mathrm{Coker}}(f')$ is of finite presentation. Then by Lemma 10.82.4, $f'$ is universally injective if and only if
\[ 0 \to M' \xrightarrow {f'} N' \to \mathop{\mathrm{Coker}}(f') \to 0 \]
splits. This is the case if and only if there is a map $h' : N' \to M'$ such that $h' \circ f' = \text{id}_{M'}$. From the universal property of the pushout, the existence of such an $h'$ is equivalent to $g$ factoring through $f$. $\square$
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Your email address will not be published.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)