Associative and graded bimodule tensor–Hom adjunction

Let $F:M{\otimes }_{A}X\to Y$ be a degree-zero $B$-linear map and define $f:=\mathrm{\Theta }(F)$ by $f(x)(m):=F(m\otimes x)$. For fixed $x$ the map $f(x)$ is additive and $B$-linear, because $F$ is additive and $F(b(m\otimes x))=(bm)\otimes x$ is mapped to $bF(m\otimes x)$ by [L1]; it is homogeneous of degree $j$ when $x\in X_j$, since $m\in M_i$ makes $m\otimes x$ of degree $i+j$ and $F$ degree-zero, so $f(x)\in {\mathrm{Hom}}_{B,j}(M,Y)\subseteq {\mathrm{HOM}}_B(M,Y)$, and a general $x$ has finitely many nonzero components. Moreover $f$ is $A$-linear and degree-zero: $f(ax)(m)=F(m\otimes ax)=F(ma\otimes x)=f(x)(ma)=(a\cdot f(x))(m)$ for $a\in A$, so $f(ax)=a\cdot f(x)$, and $f(X_j)\subseteq {\mathrm{Hom}}_{B,j}(M,Y)$ shows that $f$ preserves degrees. Hence $\mathrm{\Theta }(F)\in {\mathrm{Hom}}_{A,0}(X,{\mathrm{HOM}}_B(M,Y))$.

Conversely, let $f:X\to {\mathrm{HOM}}_B(M,Y)$ be a degree-zero $A$-linear map and define $F(m\otimes x):=f(x)(m)$ on elementary tensors. The pairing $(m,x)\mapsto f(x)(m)$ is additive in each variable and balanced: for $a\in A$ the $A$-linearity of $f$ and the action of [L1] give $f(ax)(m)=(a\cdot f(x))(m)=f(x)(ma)$, so the two images of $(ma,x)$ and $(m,ax)$ agree. By [L3] there is a unique additive map $F:M{\otimes }_{A}X\to Y$ with that value on elementary tensors; it is $B$-linear because each $f(x)$ is, and degree-zero because $f(x)$ is homogeneous of degree $j$ for $x$ homogeneous of degree $j$, so $m\in M_i$ gives $F(m\otimes x)\in Y_{i+j}$. Hence $F\in {\mathrm{Hom}}_{B,0}(M{\otimes }_{A}X,Y)$.

The two constructions are inverse. For $F$ as in step 1.1, the map $\mathrm{\Psi }(\mathrm{\Theta }(F))$ sends $m\otimes x$ to $\mathrm{\Theta }(F)(x)(m)=F(m\otimes x)$, so it agrees with $F$ on elementary tensors and hence, by [L3], everywhere. For $f$ as in step 2.1, $\mathrm{\Theta }(\mathrm{\Psi }(f))(x)(m)=\mathrm{\Psi }(f)(m\otimes x)=f(x)(m)$ for all $x,m$, so $\mathrm{\Theta }(\mathrm{\Psi }(f))=f$. Thus $\mathrm{\Theta }$ is a bijection with inverse $\mathrm{\Psi }$.

The bijection is natural in $X$ and $Y$: for degree-zero $A$-linear $g:X^{\mathrm{\prime }}\to X$ and degree-zero $B$-linear $h:Y\to Y^{\mathrm{\prime }}$, and $F\in {\mathrm{Hom}}_{B,0}(M{\otimes }_{A}X,Y)$, both $hF(1\otimes g)$ and the map induced by $h$ and $g$ on the right-hand side are $B$-linear and $A$-linear and both send a pair $(x^{\mathrm{\prime }},m)$ to $h(F(m\otimes g(x^{\mathrm{\prime }})))$, since $(1\otimes g)(m\otimes x^{\mathrm{\prime }})=m\otimes g(x^{\mathrm{\prime }})$; because these values agree for all $x^{\mathrm{\prime }}$ and $m$, the two curried maps are equal.

Dropping every degree condition, the formulas of steps 1.1 to 3.1 define mutually inverse bijections between ${\mathrm{Hom}}_B^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(M{\otimes }_{A}X,Y)$ and ${\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(X,{\mathrm{Hom}}_B^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(M,Y))$: well-definedness, $B$-linearity and $A$-linearity were the only properties used, and they do not require homogeneous elements. Consequently ${\mathrm{Hom}}_{A,0}(X,{\mathrm{HOM}}_B(M,Y))$ is exactly the set of ungraded $A$-linear maps whose values are finite sums of homogeneous maps and that preserve degrees, and by [L1] this set can be strictly smaller than ${\mathrm{Hom}}_A^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(X,{\mathrm{Hom}}_B^{\mathrm{u}\mathrm{n}\mathrm{g}\mathrm{r}}(M,Y))$.

Steps 3.1 and 4.1 give the natural bijection $\mathrm{\Theta }$ with the displayed formulas and the associative left action $(a\cdot f)(m)=f(ma)$, and step 4.2 gives the separate ungraded bijection while recording that ${\mathrm{HOM}}_B(M,Y)$ need not contain all ungraded $B$-linear maps. ∎