----- > [!proposition] Proposition. ([[the tensor product is left-adjoint to hom]]) > Let $R$ be a [[ring|(say, commutative)]] [[ring]] and $N$ be an $R$-[[module]]. The [[tensor functor]] $\_ \otimes_{R} N$ is [[adjoint functor|left-adjoint]] to the [[hom functor]] $\text{Hom}(N, -)$.[^1] > > Explicitly, this is the assertion that for all $R$-[[module|modules]] $M,N,P$ there is a [[natural transformation|natural]] [[bijection]] $\text{Hom}\big(M, \text{Hom}(N,P)\big) \simeq \text{Hom} \big( M \otimes _{R} N, P \big) .$ > This is certainly true: recall that the [[universal property]] defining the [[tensor product of modules|tensor product]] allows us to uniquely identify any $R$-[[linear map|linear map]] $M \otimes_{R} N \xrightarrow{} P$ with a [[bilinear map]] $M \times N \xrightarrow{\varphi} P$. Now, > - $R$-linearity in the second argument of $\varphi$ just means that the map $\varphi(m,-):N \to P$ is linear for all $m \in M$. > - $R$-linearity additionally in the first argument means that the map $m \mapsto \varphi(m,-)$ from $M$ to $\text{Hom}(N,P)$ is $R$-linear. > > Thus $\varphi$ is an element of $\text{Hom}(M, \text{Hom}(N,P))$. > > > [^1]:Although in general the [[hom functor]] is valued in $\mathsf{Set}$, recall that [[homsets in R-mod are R-modules]], so we may upgrade in this case to a functor $\text{Hom}(N,-):R\text{-}\mathsf{Mod} \to R\text{-}\mathsf{Mod}$. ----- #### ---- #### 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 > ```