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Finite one-sided exactness is equivalent to the existence of the corresponding adjoint

Statement

Let $A, B$ be finite-dimensional unital algebras over a field $k$ and let $F : A - m o d \to B - m o d$ be a $k$ -linear functor between the categories of finite-dimensional left modules. Then: (i) $F$ is right exact if and only if $F$ has a right adjoint; more precisely, if $F$ is right exact then $F ≅ T_{F (A)}$ and ${Hom}_{B} (F (A), -)$ is a right adjoint taking finite-dimensional modules to finite-dimensional modules, while a functor with a right adjoint preserves every finite colimit that exists in $A - m o d$ and hence is right exact. (ii) $F$ is left exact if and only if $F$ has a left adjoint; if $F$ is left exact then $F ≅ {Hom}_{A} (M^{*}, -)$ with $M = F (A^{*})$ and $M^{*} \otimes_{B} -$ is a left adjoint, while a functor with a left adjoint preserves every finite limit that exists and hence is left exact. No commutativity and no choice are used.

Facts & Assumptions

Given: A field $k$ , finite-dimensional unital $k$ -algebras $A, B$ , and a $k$ -linear functor $F : A - m o d \to B - m o d$ between the categories of finite-dimensional left modules.

[F1]

The categories $A - m o d$ and $B - m o d$ are finite $k$ -linear abelian categories, hence have all finite limits and all finite colimits (Finite-dimensional module categories satisfy the intrinsic finiteness conditions, Abelian category, An abelian category has all finite limits and all finite colimits).

[F2]

A $k$ -linear right exact $F$ is naturally isomorphic to $T_{F (A)}$ with $F (A)$ a finite-dimensional $(B, A)$ -bimodule, and a $k$ -linear left exact $F$ is naturally isomorphic to ${Hom}_{A} (M^{*}, -)$ with $M = F (A^{*})$ a finite-dimensional $(B, A)$ -bimodule (Finite Eilenberg–Watts for right exact linear functors, Finite left exact functors are Hom functors with dual bimodule kernels).

[F3]

For a $(B, A)$ -bimodule $N$ the functor $T_{N} = N \otimes_{A} -$ is left adjoint to ${Hom}_{B} (N, -)$ , with unit and counit satisfying the triangle identities; every module occurring is finite-dimensional when $N$ and the arguments are, since tensor products and Hom-spaces of finite-dimensional modules are finite-dimensional (Tensor-Hom adjunction for bimodules over arbitrary unital rings, Adjunction by unit, counit, and the triangle identities, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, The abelian group ${Hom}_{R} (M, N)$ and maps induced by pre- and postcomposition, $(S, R)$ -bimodules and commuting left and right scalar actions).

[F4]

Left adjoints preserve every colimit that exists, and right adjoints preserve every limit that exists (Left adjoints preserve every colimit that exists, Right adjoints preserve every limit that exists).

[F5]

A functor is right exact when it preserves every finite colimit that exists, and left exact when it preserves every finite limit that exists; in particular a functor preserving all finite colimits (or limits) of the abelian source is right (respectively left) exact (Left exact and right exact functors, Module homomorphism and isomorphism, kernel, image and cokernel, Exact sequences and short exact sequences of modules).

Proof

technique · direct

1.1F2F3given

(i), forward direction. Assume $F$ right exact. By [F2] $F ≅ T_{F (A)}$ with $F (A)$ a finite-dimensional $(B, A)$ -bimodule, and by [F3] the functor $T_{F (A)}$ is left adjoint to ${Hom}_{B} (F (A), -)$ , which sends a finite-dimensional left $B$ -module $Y$ to the finite-dimensional space ${Hom}_{B} (F (A), Y)$ ; so ${Hom}_{B} (F (A), -)$ is a right adjoint of $F$ that stays in the finite module categories.

1.2F1F4F5

(i), converse direction. Assume $F$ has a right adjoint. Then $F$ is a left adjoint and preserves every colimit that exists by [F4]; since $A - m o d$ has all finite colimits by [F1], $F$ preserves them and is right exact by [F5].

1.3F2F3given

(ii), forward direction. Assume $F$ left exact. By [F2] $F ≅ {Hom}_{A} (M^{*}, -)$ with $M = F (A^{*})$ a finite-dimensional $(B, A)$ -bimodule, so $M^{*}$ is an $(A, B)$ -bimodule and by [F3] the functor $M^{*} \otimes_{B} -$ is left adjoint to ${Hom}_{A} (M^{*}, -)$ , taking finite-dimensional left $B$ -modules to finite-dimensional left $A$ -modules because $M^{*} \otimes_{B} Y$ is a quotient of the finite-dimensional $M^{*} \otimes_{k} Y$ . Hence $F$ has a left adjoint.

1.4F1F4F5

(ii), converse direction. Assume $F$ has a left adjoint. Then $F$ is a right adjoint and preserves every limit that exists by [F4]; since $A - m o d$ has all finite limits by [F1], $F$ preserves them and is left exact by [F5].

2.1step 1.1step 1.2step 1.3step 1.4given∎

Steps 1.1 and 1.2 prove (i), and steps 1.3 and 1.4 prove (ii). All adjoints exhibited stay inside the finite module categories, no commutativity of $A$ or $B$ was used, and all tensor products, Hom-spaces and adjunction data involved are finite-dimensional, so no choice is used.

Depends on

Used by

Dependency tree · two levels

89 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, §1.8 (Definitions 1.8.1–1.8.6, Proposition 1.8.10, Corollary 1.8.11, Remark 1.8.7), printed pp.9–11 (standard reference, not scraped)
Fuchs, Schaumann, Schweigert, Eilenberg–Watts calculus for finite categories and a bimodule Radford S^4 theorem, arXiv:1612.04561v3, §2.1 (Lemma 2.1, Lemma 2.2, equation (2.1)) (standard reference, not scraped)