----- > [!proposition] Proposition. ([[adjointness and exactness]]) > Let $R$ and $S$ be [[commutative ring|(commutative)]] [[ring|rings]]. > - [[adjoint functor|Right-adjoint]] [[additive functor|additive]] [[covariant functor|functors]] $R$-$\mathsf{Mod} \to S\text{-}\mathsf{Mod}$ are [[exact functor|left-exact]]. >- [[adjoint functor|Left-adjoint]] [[additive functor|additive functors]] $R$-$\mathsf{Mod} \to S\text{-}\mathsf{Mod}$ are [[exact functor|right-exact]]. - [ ] more general setting where this holds > [!basicexample] > Because the [[tensor functor]] $\_ \otimes_{R} N$ [[the tensor product is left-adjoint to hom|is left-adjoint]] to the [[hom functor|hom functor]] $\text{Hom}(N, -)$, it is right-exact. ^basic-example > [!proof]+ Proof. ([[adjointness and exactness]]) > This has to do with the fact that [[adjoint functor|right-adjoint]] functors [[right-adjoint functors commute with limits|preserve]] [[categorical limit|limits]], and therefore [[categorical kernel|kernels]]. > > The first remark is that [[exact sequence|exactness]] of a sequence $0 \to A \xrightarrow{\varphi} B \xrightarrow{\psi} C$ > amounts to the fact that $\varphi:A \hookrightarrow B$ identifies $A$ with $\ker \psi$. Applying our right-adjoint functor $\mathscr{F}$, we get $0 \to \mathscr{F}(A) \xrightarrow{\mathscr{F}(\varphi)} \mathscr{F}(B) \xrightarrow{\mathscr{F}(\psi)} \mathscr{F}(C).$ > (Note that additive functors preserve [[terminal object|zero objects]].) Now, $\mathscr{F}(A)$ may be identified with $\mathscr{F}(\ker \psi)$ and in turn with $\ker \mathscr{F}(\psi)$. Similarly, $\ker \mathscr{F}(\varphi) \cong \mathscr{F}(\ker \varphi)\cong\mathscr{F}(0)=0$. So $\mathscr{F}(\varphi)$ is an embedding, as required. And its image is isomorphic to $\ker \mathscr{F}(\psi)$, finishing the argument. The costatement proceeds dually, noting that [[cokernel of a module homomorphism|cokernels]] are [[categorical colimit|colimits]]. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```