Finite Eilenberg–Watts for right exact linear functors

(i) Let $M$ be a finite-dimensional $(B,A)$-bimodule. For a finite-dimensional left $A$-module $X$ the tensor product $M{\otimes }_{A}X$ is a quotient of $M{\otimes }_{k}X$, hence finite-dimensional: if $(m_i)$ and $(x_j)$ are finite $k$-bases then the tensors $m_i\otimes x_j$ span, by expansion in both factors and the agreeing scalar actions, and by [F2] it is a left $B$-module with $b(m\otimes x)=(bm)\otimes x$; for a left $A$-linear $u:X\to Y$ the map $T_M(u)=1_M\otimes u$ is $B$-linear, preserves identities and composition, and is additive and $k$-homogeneous in $u$ by [F2]. Hence $T_M$ is a well-defined $k$-linear functor $A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\to B\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$.

(ii, the comparison.) Let $F$ be $k$-linear and right exact. By [F3] the finite-dimensional left $B$-module $F(A)$ carries a $(B,A)$-bimodule structure commuting with the $B$-action, and for $x\in X$ the map ${\mathrm{\ell }}_x:A\to X$, ${\mathrm{\ell }}_x(a)=ax$, is left $A$-linear between finite-dimensional modules, so $F({\mathrm{\ell }}_x)$ is defined. The balanced-map computation of [F4] uses only additivity and functoriality of $F$ on these maps and the tensor universal property, so it applies verbatim and yields a $B$-linear map ${\tau }_X:F(A){\otimes }_{A}X\to F(X)$, ${\tau }_X(m\otimes x)=F({\mathrm{\ell }}_x)(m)$, natural in $X$. The scalar actions on $F(A)$ agree because $r_{c{1}_A}=c{1}_A$ as endomorphisms and $F(r_{c{1}_A})=c{1}_{F(A)}$ by $k$-linearity.

(i, right exactness.) Let $X\stackrel{u}{\to }Y\stackrel{v}{\to }Z\to 0$ be exact in $A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$. Applying [F1] gives the exact sequence $M{\otimes }_{A}X\stackrel{1\otimes u}{\to }M{\otimes }_{A}Y\stackrel{1\otimes v}{\to }M{\otimes }_{A}Z\to 0$: the functor $M{\otimes }_A-$ preserves cokernels, and every module occurring is finite-dimensional because each is a quotient of a finite tensor product, so the computation takes place entirely inside the finite categories. Hence $T_M$ is right exact.

(ii, the comparison is an isomorphism on free modules.) For $X=A$ the map ${\tau }_A:F(A){\otimes }_{A}A\to F(A)$ sends $m\otimes a$ to $F({\mathrm{\ell }}_a)(m)=F(r_a)(m)=ma$ by [F3], so it is the unit isomorphism ${\rho }_{F(A)}$ of [F6], an isomorphism. Both $F$ and $T_{F(A)}$ are additive and therefore preserve finite biproducts by [F5], and $\tau$ is natural; hence for every $r\ge 0$ the map ${\tau }_{A^r}$ is the direct sum of $r$ copies of ${\tau }_A$ and is an isomorphism.

(iii) Let $M,M^{\mathrm{\prime }}$ be finite-dimensional $(B,A)$-bimodules. Every natural transformation $\eta :T_M\Rightarrow T_{M^{\mathrm{\prime }}}$ has, by [F8], the form ${\eta }_X=f\otimes 1_X$ for the bimodule map $f={\rho }_{M^{\mathrm{\prime }}}{\eta }_A{\rho }_M^{-1}:M\to M^{\mathrm{\prime }}$, and conversely every bimodule map $f$ yields such a natural transformation; the two assignments are inverse bijections compatible with addition, identities and vertical composition. Because every object and every map occurring in the computation ($A$, $X$, the maps ${\mathrm{\ell }}_x$, and the unit isomorphisms) lies in the finite module categories, the classification restricts verbatim from all modules to $A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$.

(ii, isomorphism for all finite-dimensional $X$.) Let $X\in A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ and choose a finite presentation $A^s\stackrel{\alpha }{\to }{A}^r\stackrel{\beta }{\to }X\to 0$ by [F7]. By naturality of $\tau$ and right exactness of $F$ and of $T_{F(A)}$ (step 2.1) there is a commutative diagram with exact rows comparing $\tau$ on $A^s\to A^r\to X\to 0$, and the first two vertical maps are isomorphisms by step 2.2. The induced map on cokernels is therefore an isomorphism: if $q$ and $q^{\mathrm{\prime }}$ are the cokernel maps of $T(\alpha )$ and $F(\alpha )$, then ${\tau }_X$ is characterized by ${\tau }_{X}q=q^{\mathrm{\prime }}{\tau }_r$, and the map $s$ defined by $s{q}^{\mathrm{\prime }}=q{\tau }_r^{-1}$ satisfies $s{\tau }_X=1$ and ${\tau }_{X}s=1$ after composing with the epimorphisms $q,q^{\mathrm{\prime }}$. This is the finite-presentation form of the cokernel-universality argument of [F9]: the coproduct-preservation hypothesis of [F9] is not available for $F$ on $A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$, so [F9] is not applied as a statement, but its argument is reproduced here with finite presentations. Hence ${\tau }_X$ is an isomorphism, so $F\cong T_{F(A)}$ naturally.

(iv) Define $\mathrm{\Phi }$ on finite-dimensional $(B,A)$-bimodules by $\mathrm{\Phi }(M)=T_M$ and on bimodule maps by $f\mapsto (f\otimes 1_X{)}_X$; by step 1.1 this is a functor into the category of $k$-linear right exact functors, and by step 2.3 it is full and faithful. It is essentially surjective: for a $k$-linear right exact $F$ the comparison of step 3.1 is a natural isomorphism $T_{F(A)}\cong F$, and $F(A)$ is a finite-dimensional $(B,A)$-bimodule by step 1.2. More explicitly, the assignments $M\mapsto T_M$ and $F\mapsto F(A)$ are inverse up to natural isomorphism: $F(A){\otimes }_{A}A\cong F(A)$ by [F6] and $T_{F(A)}\cong F$ by step 3.1, so $\mathrm{\Phi }$ is an equivalence of categories with quasi-inverse $F\mapsto F(A)$ (which sends a natural transformation to its component at $A$). The comparison is natural also in $F$: for $\eta :F\Rightarrow G$, naturality at ${\mathrm{\ell }}_x$ gives ${\eta }_X{\tau }_X^F(m\otimes x)=G({\mathrm{\ell }}_x)({\eta }_A(m))={\tau }_X^G({\eta }_A(m)\otimes x)$. The tensor-unit isomorphisms are natural in $M$ by [F6].

Steps 1.1, 2.1, 1.2, 3.1, 2.3 and 4.1 prove (i), (ii), (iii) and (iv). No commutativity of $A$ or $B$ was used, and all presentations, biproducts and bases occurring above are finite data inside finite-dimensional modules, so no choice is used.