Exact finite tensor functors have projective right-module kernels

Given: The agreeing scalar convention above, a field $k$, finite-dimensional unital $k$-algebras $A,B$, and a finite-dimensional $(B,A)$-bimodule $M$, with $T_M=M{\otimes }_A-:A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\to B\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$.

A right $A$-module $N$ is flat exactly when $N{\otimes }_A-$ is exact on left $A$-modules; every projective right $A$-module is flat, without choice (Left and right flat modules over an arbitrary ring, Projective left and right modules are flat over an arbitrary ring).

The choice-free direction $1\Rightarrow 4$ of the projective-module characterizations produces, from the canonical free cover, an identification of a projective module with a direct summand of a free module; for a finitely generated module the cover may be taken over a finite generating set, so the free module is finite; conversely a direct summand of a free module whose basis is finite is projective, since lifts of the finitely many basis elements can be chosen (Equivalent characterizations of projective modules, Projective modules and the lifting property, Every module is a quotient of a free module, Generated submodule, cyclic and finitely generated modules, module basis and free module).

For a $(B,A)$-bimodule $N$ and finite-dimensional left $A$-modules $X$, duality and the tensor-hom adjunction give a natural isomorphism $(N{\otimes }_{A}X{)}^{\ast }\cong {\mathrm{Hom}}_{A^{\mathrm{o}\mathrm{p}}}(N,X^{\ast })$ of left $B^{\mathrm{o}\mathrm{p}}$-modules: a functional $\omega$ corresponds to $u\mapsto (x\mapsto \omega (u\otimes x))$, and the balancing relation for $\omega$ is exactly $A^{\mathrm{o}\mathrm{p}}$-linearity of that map, because $u\cdot a$ denotes the right $A$-action on $N$ and $\lambda \cdot a$ the right $A$-action on $X^{\ast }$ (Finite left exact functors are Hom functors with dual bimodule kernels, Finite module duality is exact with commuting bimodule actions, Universal property of the tensor product for balanced maps into abelian groups, Tensor-Hom adjunction for bimodules over arbitrary unital rings).

Duality $X\mapsto X^{\ast }$ is an exact contravariant equivalence between finite-dimensional left $A$-modules and finite-dimensional left $A^{\mathrm{o}\mathrm{p}}$-modules, so a functor on one side is exact exactly when the corresponding functor on the other side is (Finite module duality is exact with commuting bimodule actions).

The category $A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is abelian; a finite-dimensional left $A$-module is finitely generated, and a finite-dimensional right $A$-module has a finite free cover $A^n\twoheadrightarrow N$ built from a finite $k$-basis (Modules over a ring form an abelian category, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Every module is a quotient of a free module, Universal property of a direct sum of modules, The direct sum of an indexed family of modules).

If $N$ is a direct summand of the free right module $A^n$ with inclusion $h:N\to A^n$ and retraction $p:A^n\twoheadrightarrow N$, then $N$ is projective: given a surjection $e:E\twoheadrightarrow N^{\mathrm{\prime }}$ and $f:N\to N^{\mathrm{\prime }}$, lift $fp:A^n\to N^{\mathrm{\prime }}$ by choosing preimages under $e$ of the images of its finitely many basis vectors; precomposing the resulting lift with $h$ lifts $f$ (Equivalent characterizations of projective modules, Projective modules and the lifting property, The splitting lemma for short exact sequences of modules, Split monomorphism, split epimorphism, retraction, and section).