Lemma 19.14.2. Let $f : M \to G(A)$ be an injective map in $\text{Mod}_ R$. Then the adjoint map $f' : F(M) \to A$ is injective too.
Proof. Choose a map $R^{\oplus n} \to M$ and consider the corresponding map $U^{\oplus n} \to F(M)$. Consider a map $v : U \to U^{\oplus n}$ such that the composition $U \to U^{\oplus n} \to F(M) \to A$ is $0$. Then this arrow $v : U \to U^{\oplus n}$ is an element $v$ of $R^{\oplus n}$ mapping to zero in $G(A)$. Since $f$ is injective, we conclude that $v$ maps to zero in $M$ which means that $U \to U^{\oplus n} \to F(M)$ is zero by construction of $F(M)$ in the proof of Lemma 19.14.1. Since $U$ is a generator we conclude that
\[ \mathop{\mathrm{Ker}}(U^{\oplus n} \to F(M) \to A) = \mathop{\mathrm{Ker}}(U^{\oplus n} \to F(M)) \]
To finish the proof we choose a surjection $\bigoplus _{i \in I} R \to M$ and we consider the corresponding surjection
\[ \pi : \bigoplus \nolimits _{i \in I} U \longrightarrow F(M) \]
To prove $f'$ is injective it suffices to show that $\mathop{\mathrm{Ker}}(\pi ) = \mathop{\mathrm{Ker}}(f' \circ \pi )$ as subobjects of $\bigoplus _{i \in I} U$. However, now we can write $\bigoplus _{i \in I} U$ as the filtered colimit of its subobjects $\bigoplus _{i \in I'} U$ where $I' \subset I$ ranges over the finite subsets. Since filtered colimits are exact by AB5 for $\mathcal{A}$, we see that
\[ \mathop{\mathrm{Ker}}(\pi ) = \mathop{\mathrm{colim}}\nolimits _{I' \subset I\text{ finite}} \left(\bigoplus \nolimits _{i \in I'} U\right) \bigcap \mathop{\mathrm{Ker}}(\pi ) \]
and
\[ \mathop{\mathrm{Ker}}(f' \circ \pi ) = \mathop{\mathrm{colim}}\nolimits _{I' \subset I\text{ finite}} \left(\bigoplus \nolimits _{i \in I'} U\right) \bigcap \mathop{\mathrm{Ker}}(f' \circ \pi ) \]
and we get equality because the same is true for each $I'$ by the first displayed equality above. $\square$
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Your email address will not be published.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)
There are also: