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Finite Abelian Categories and Eilenberg–Watts

1 · Prerequisites

2 · Summary

This page develops the finite-module theory behind the intrinsic definition of a finite $k$ -linear abelian category. It begins with a general definition of superfluous subobjects and projective covers in an abelian category, matching the module notion already in the library, and with the exact contravariant duality of finite-dimensional modules and their commuting bimodule actions.

From those tools it builds the finite categorical machinery: the category of finite-support families of finite-dimensional vector spaces is locally finite with enough projectives but has infinitely many simples and no generator, so local finiteness plus projective covers is strictly weaker than finiteness; a projective epimorphism onto each simple generates every finite-length object; and the finite-dimensional module categories realise the intrinsic finiteness conditions. The central realization theorem then shows that the intrinsic hypotheses produce a finite projective generator $P$ with $A = {End}_{C} (P)^{o p}$ and an exact, fully faithful, essentially surjective module-model functor $C (P, -)$ to $A - m o d$ . A specified quasi-inverse requires supplied splitting data for essential surjectivity; the objectwise finite-presentation construction does not select those data simultaneously.

On the functor side the page proves the finite Eilenberg–Watts classification: right exact $k$ -linear functors are exactly the tensor functors of their bimodule kernels, left exact functors are the Hom functors of the dual kernels, one-sided exactness is equivalent to the existence of the corresponding adjoint, exact tensor functors have projective kernels, and the whole classification is a biequivalence of bicategories. Bimodules have agreeing $k$ -scalar actions, and assertions forming functor categories use the explicit set-sized conventions of the items.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Superfluous subobjects and projective covers in an abelian category

Definition

Let $C$ be an abelian category (Abelian category) and let $n : N \to P$ be a monomorphism, regarded as the subobject $[n]$ of $P$ (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms). The subobject $[n]$ is superfluous when for every subobject $[m] : M \to P$ whose join with $[n]$ satisfies

$[n] \lor [m] = [1_{P}]$

one already has $[m] = [1_{P}]$ (The join of two subobjects in an abelian category). Here $[1_{P}]$ is the subobject represented by the identity of $P$ . An essential epimorphism $π : Q \to X$ is an epimorphism whose kernel, taken as a morphism $\ker π \to Q$ (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers, Monomorphism and epimorphism by left and right cancellation), represents a superfluous subobject of $Q$ . A projective cover of $X$ is an essential epimorphism $π : Q \to X$ with $Q$ projective in the sense of Projective object.

As elsewhere on this page, the bracket notation abbreviates statements about representatives: $[m] = [1_{P}]$ says that $m$ and $1_{P}$ mutually factor (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms), and such a factorisation of $1_{P}$ through the monomorphism $m$ exhibits $m$ as an isomorphism onto $P$ . Thus in a module category the condition " $[n] \lor [m] = [1_{P}]$ implies $[m] = [1_{P}]$ " reads " $N + M = P$ implies $M = P$ ", where the join of subobjects of a module is the sum of the corresponding submodules, and this is precisely the superfluous-kernel condition of the module notion of An essential epimorphism is a surjection with superfluous kernel, and a projective cover is a projective source with such a map, with the same projective-source requirement. The class-and-size conventions used by the bracket notation are those of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $C A T$ is not formed.

This is the general form of the projective-cover clause of Finite k-linear abelian categories, whose phrasing "every simple object has a projective cover" is the module-scoped language of An essential epimorphism is a surjection with superfluous kernel, and a projective cover is a projective source with such a map read in an abstract abelian category. The definition asserts no existence of covers, selects no object, and uses no choice; each later existence statement is an explicit hypothesis.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Finite module duality is exact with commuting bimodule actions

Statement

Throughout, a bimodule over $k$ -algebras means a $k$ -vector space with $k$ -bilinear commuting actions and agreeing scalar actions: $(c 1_{B}) m = m (c 1_{A}) = c m$ for $c \in k$ in a $(B, A)$ -bimodule. This compatibility is an additional requirement beyond the ring-bimodule definition $(S, R)$ -bimodules and commuting left and right scalar actions.

Let $A$ be a finite-dimensional unital algebra over a field $k$ , and let $A - m o d$ denote the category of finite-dimensional left $A$ -modules with $A$ -linear maps. For such a module $X$ put $X^{*} = {Hom}_{k} (X, k) = L (X, k)$ with the right $A$ -action $(λ \cdot a) (x) = λ (a x)$ , equivalently the left $A^{o p}$ -action $a \cdot λ : = λ \cdot a$ ; for an $A$ -linear $u : X \to Y$ put $u^{*} (λ) = λ \circ u$ for $λ \in Y^{*}$ . Then:

(i) $(-)^{*}$ is a contravariant $k$ -linear functor from $A - m o d$ to the category of finite-dimensional left $A^{o p}$ -modules, and the evaluation ${ev}_{X} : X \to X^{* *}$ , ${ev}_{X} (x) (λ) = λ (x)$ , is a natural isomorphism, so $(-)^{*}$ is a contravariant equivalence;

(ii) $(-)^{*}$ is exact, carrying every short exact sequence $0 \to X \overset{i}{\to} Y \overset{q}{\to} Z \to 0$ of finite-dimensional left $A$ -modules to the short exact sequence $0 \to Z^{*} \overset{q^{*}}{\to} Y^{*} \overset{i^{*}}{\to} X^{*} \to 0$ ;

(iii) if $B$ is a unital $k$ -algebra and $X$ is a finite-dimensional $(A, B)$ -bimodule, then $X^{*}$ is a $(B, A)$ -bimodule under the commuting actions $(b \cdot λ) (x) = λ (x b)$ and $(λ \cdot a) (x) = λ (a x)$ , and duality is a contravariant equivalence between the categories of finite-dimensional $(A, B)$ -bimodules and finite-dimensional $(B, A)$ -bimodules, with the $k$ -linear maps that are simultaneously $A$ -linear and $B$ -linear as morphisms.

No choice is used.

Facts & Assumptions

Given: The agreeing scalar convention above, a field $k$ , a finite-dimensional unital $k$ -algebra $A$ , and the category $A - m o d$ of finite-dimensional left $A$ -modules. For part (iii), a unital $k$ -algebra $B$ and a finite-dimensional $(A, B)$ -bimodule $X$ .

[F1]

For a $k$ -vector space $V$ , the algebraic dual $V^{*} = L (V, k)$ is the space of linear functionals with pointwise addition and scalar multiplication (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ , The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication).

[F2]

If $(v_{1}, \dots, v_{n})$ is an ordered basis of a finite-dimensional $k$ -vector space $V$ , the coordinate functionals $v_{i}^{*} (v_{j}) = δ_{i j}$ form a basis of $V^{*}$ , so $\dim_{k} V^{*} = \dim_{k} V$ (The dual family $(b^{*})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{*} (c) = δ_{b c}$ , The dual family of a finite basis is a basis of the dual space, with the same dimension).

[F3]

A right $R$ -module is the same data as a left $R^{o p}$ -module, and a left $R$ -module is the same data as a right $R^{o p}$ -module (Unital left and right modules over a ring; unqualified module means left module, The opposite ring $R^{o p}$ ).

[L1]

Hom-groups are abelian groups under pointwise operations, composition is additive and $k$ -bilinear, and identity maps are two-sided units (The abelian group ${Hom}_{R} (M, N)$ and maps induced by pre- and postcomposition, k-linear categories and k-linear functors).

[L2]

Rank-nullity: a $k$ -linear map $T : U \to W$ with $U$ finite-dimensional satisfies $\dim_{k} U = \dim_{k} (\ker T) + \dim_{k} (im T)$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[L3]

In a short exact sequence $0 \to X \overset{i}{\to} Y \overset{q}{\to} Z \to 0$ of modules, $i$ is injective, $q$ is surjective, and $im i = \ker q$ (Exact sequences and short exact sequences of modules).

[L4]

Every linearly independent finite family in a finite-dimensional $k$ -vector space is contained in a basis of that space, and no choice principle is used (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

[L5]

Two $k$ -linear maps with a common finite-dimensional domain that agree on a basis of that domain are equal (A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis).

[L6]

In an $(S, R)$ -bimodule $N$ the two actions commute: $(s n) r = s (n r)$ for all $s \in S$ , $n \in N$ , $r \in R$ ( $(S, R)$ -bimodules and commuting left and right scalar actions).

Proof

technique · direct

1.1F1F2F3givenalgebra

For $λ \in X^{*}$ and $a \in A$ define $λ \cdot a$ by $(λ \cdot a) (x) = λ (a x)$ ; this is $k$ -linear in $x$ because $x \mapsto a x$ is $k$ -linear and $λ$ is, and the right-module axioms hold: $(λ + μ) \cdot a = λ \cdot a + μ \cdot a$ and $λ \cdot (a + a^{'}) = λ \cdot a + λ \cdot a^{'}$ by linearity of $λ$ and additivity of the action of $A$ , while $(λ \cdot a) \cdot a^{'} = λ \cdot (a a^{'})$ because both sides send $x$ to $λ (a (a^{'} x))$ , and $λ \cdot 1 = λ$ . Hence $X^{*}$ is a right $A$ -module, equivalently by [F3] a left $A^{o p}$ -module, and [F2] gives $\dim_{k} X^{*} = \dim_{k} X < \infty$ , so $X^{*}$ is a finite-dimensional left $A^{o p}$ -module.

1.2F1L1algebra

For $A$ -linear $u : X \to Y$ the map $u^{*} : Y^{*} \to X^{*}$ , $u^{*} (λ) = λ \circ u$ , is $k$ -linear, since composition with the $k$ -linear $u$ is additive and $k$ -homogeneous by [L1]; moreover $(1_{X})^{*} = 1_{X^{*}}$ and $(v \circ u)^{*} = u^{*} \circ v^{*}$ for composable $A$ -linear maps, while $u \mapsto u^{*}$ is additive and $k$ -homogeneous, because $(u + v)^{*} (λ) = λ \circ (u + v) = λ \circ u + λ \circ v$ and $(c u)^{*} = c u^{*}$ for $c \in k$ by [L1].

1.3F1L3algebra

Let $0 \to X \overset{i}{\to} Y \overset{q}{\to} Z \to 0$ be a short exact sequence of finite-dimensional left $A$ -modules. Then $i^{*} \circ q^{*} = 0$ , since $(i^{*} \circ q^{*}) (λ) = i^{*} (λ \circ q) = λ \circ q \circ i$ and $q \circ i = 0$ by [L3]; and $q^{*}$ is injective, since if $q^{*} (λ) = λ \circ q = 0$ then $λ$ vanishes on $im q = Z$ by surjectivity of $q$ from [L3], so $λ = 0$ . Thus $0 \to Z^{*} \overset{q^{*}}{\to} Y^{*} \overset{i^{*}}{\to} X^{*}$ is exact at $Z^{*}$ and a complex at $Y^{*}$ .

1.4F2L3L4L5chooseconstructalgebra

The map $i^{*}$ is surjective: let $(x_{1}, \dots, x_{m})$ be an ordered basis of the finite-dimensional space $X$ ; since $i$ is injective by [L3], the images $i (x_{1}), \dots, i (x_{m})$ are linearly independent and by [L4] are contained in an ordered basis $(y_{1}, \dots, y_{n})$ of $Y$ with $y_{j} = i (x_{j})$ for $j \leq m$ ; let $(y_{1}^{*}, \dots, y_{n}^{*})$ be its dual basis of $Y^{*}$ by [F2]. Given $λ = \sum_{j \leq m} c_{j} x_{j}^{*}$ in $X^{*}$ with the dual basis $(x_{j}^{*})$ of [F2], put $μ = \sum_{j \leq m} c_{j} y_{j}^{*} \in Y^{*}$ ; then $i^{*} (μ) (x_{r}) = μ (i (x_{r})) = μ (y_{r}) = c_{r} = λ (x_{r})$ for every $r \leq m$ , so $i^{*} (μ) = λ$ by [L5]. Hence $im i^{*} = X^{*}$ .

2.1F1F3L6algebra

For an $(A, B)$ -bimodule $X$ define $(b \cdot λ) (x) = λ (x b)$ and $(λ \cdot a) (x) = λ (a x)$ for $a \in A$ , $b \in B$ , $λ \in X^{*}$ , $x \in X$ ; as in step 1.1 both are $k$ -linear functionals and the module axioms hold for the left $B$ - and right $A$ -actions, so $X^{*}$ is at once a left $B$ -module and a right $A$ -module. The two actions commute: $((b \cdot λ) \cdot a) (x) = (b \cdot λ) (a x) = λ ((a x) b) = λ (a (x b)) = (λ \cdot a) (x b) = (b \cdot (λ \cdot a)) (x)$ , using the bimodule identity $(a x) b = a (x b)$ of [L6]; hence $X^{*}$ is a $(B, A)$ -bimodule.

2.2L2L3algebra

In the situation of step 1.3, $\dim_{k} Y = \dim_{k} X + \dim_{k} Z$ : rank-nullity [L2] applied to the $k$ -linear $q$ gives $\dim_{k} Y = \dim_{k} (\ker q) + \dim_{k} (im q)$ , and $\ker q = im i$ with $i$ injective and $q$ surjective by [L3], so $\dim_{k} (\ker q) = \dim_{k} X$ and $\dim_{k} (im q) = \dim_{k} Z$ .

2.3step 1.1step 1.2algebra

The map $u^{*}$ is $A^{o p}$ -linear: for $λ \in Y^{*}$ and $a \in A$ , $u^{*} (λ \cdot a) (x) = (λ \cdot a) (u (x)) = λ (a u (x)) = λ (u (a x)) = (u^{*} λ) (a x) = (u^{*} (λ) \cdot a) (x)$ for all $x$ , using the action of step 1.1 and $A$ -linearity of $u$ . Consequently, with step 1.1 for objects and step 1.2 for the morphism assignment, identities, composition and $k$ -linearity on hom-spaces, $(-)^{*}$ is a contravariant $k$ -linear functor from $A - m o d$ to finite-dimensional left $A^{o p}$ -modules.

2.4F2step 1.1choosealgebra

Applying step 1.1 with the unital algebra $A^{o p}$ in place of $A$ , the dual $X^{* *}$ of the finite-dimensional left $A^{o p}$ -module $X^{*}$ is a finite-dimensional left $(A^{o p})^{o p} = A$ -module, and ${ev}_{X} : X \to X^{* *}$ , ${ev}_{X} (x) (λ) = λ (x)$ , is $k$ -linear; it is $A$ -linear because ${ev}_{X} (a x) (λ) = λ (a x) = (λ \cdot a) (x) = {ev}_{X} (x) (λ \cdot a) = (a \cdot {ev}_{X} (x)) (λ)$ for the left action on $X^{* *}$ induced by step 1.1. Choosing an ordered basis $(x_{1}, \dots, x_{m})$ of $X$ with dual basis $(x_{1}^{*}, \dots, x_{m}^{*})$ of $X^{*}$ , the dual family $(ε_{1}, \dots, ε_{m})$ of $(x_{1}^{*}, \dots, x_{m}^{*})$ is a basis of $X^{* *}$ by [F2], and ${ev}_{X} (x_{i}) (x_{j}^{*}) = δ_{i j} = ε_{i} (x_{j}^{*})$ for all $i, j$ ; hence ${ev}_{X} (\sum_{i} a_{i} x_{i}) = \sum_{i} a_{i} ε_{i}$ , which is zero only for the zero combination and realizes every element of $X^{* *}$ , so ${ev}_{X}$ is a $k$ -linear isomorphism.

3.1F2L2step 1.3step 1.4step 2.2algebra

By rank-nullity [L2] applied to $i^{*}$ and $q^{*}$ , and [F2] together with steps 1.3, 1.4 and 2.2: $\dim_{k} (\ker i^{*}) = \dim_{k} Y^{*} - \dim_{k} (im i^{*}) = \dim_{k} Y - \dim_{k} X$ , while $\dim_{k} (im q^{*}) = \dim_{k} Z^{*} - \dim_{k} (\ker q^{*}) = \dim_{k} Z$ ; hence by step 2.2 both quantities equal $\dim_{k} Z$ .

3.2step 1.2step 2.1algebra

A map $u : X \to Y$ of $(A, B)$ -bimodules, that is $u (a x b) = a u (x) b$ for all $a \in A$ , $b \in B$ , $x \in X$ , has $u^{*}$ a map of $(B, A)$ -bimodules: $u^{*} (b \cdot λ) (x) = (b \cdot λ) (u (x)) = λ (u (x) b) = λ (u (x b)) = (u^{*} λ) (x b) = (b \cdot (u^{*} λ)) (x)$ and $u^{*} (λ \cdot a) (x) = (λ \cdot a) (u (x)) = λ (a u (x)) = λ (u (a x)) = (u^{*} λ) (a x) = ((u^{*} λ) \cdot a) (x)$ for all $x$ , using the actions of step 2.1; with step 1.2 the assignment is functorial on the bimodule categories.

3.3step 1.2step 2.3step 2.4algebra

For $A$ -linear $u : X \to Y$ , $x \in X$ and $λ \in Y^{*}$ one has $(u^{* *} \circ {ev}_{X}) (x) (λ) = {ev}_{X} (x) (u^{*} λ) = u^{*} (λ) (x) = λ (u (x)) = {ev}_{Y} (u (x)) (λ)$ , so ${ev}_{Y} \circ u = u^{* *} \circ {ev}_{X}$ , where $u^{* *} = (u^{*})^{*}$ is the map of step 2.3; hence the isomorphisms ${ev}_{X}$ of step 2.4 form a natural isomorphism $1 \Rightarrow (-)^{* *}$ . Therefore $(-)^{*}$ is a contravariant equivalence of categories with quasi-inverse $(-)^{*}$ , since both composites are $(-)^{* *}$ and are naturally isomorphic to the identities, which proves (i).

4.1step 1.3step 3.1algebra

In the situation of step 1.3, $im q^{*} \subseteq \ker i^{*}$ because $i^{*} \circ q^{*} = 0$ , and by step 3.1 both are $k$ -subspaces of $Y^{*}$ of dimension $\dim_{k} Z$ ; since a subspace of the same finite dimension equals the whole space, $im q^{*} = \ker i^{*}$ . With step 1.3 the dual sequence $0 \to Z^{*} \overset{q^{*}}{\to} Y^{*} \overset{i^{*}}{\to} X^{*} \to 0$ is exact, which proves (ii).

4.2step 2.1step 2.4step 3.2step 3.3algebra

For an $(A, B)$ -bimodule $X$ the evaluation of step 2.4 is also $B$ -linear on the right: ${ev}_{X} (x b) (λ) = λ (x b) = (b \cdot λ) (x) = {ev}_{X} (x) (b \cdot λ) = ({ev}_{X} (x) \cdot b) (λ)$ for all $b \in B$ , $λ \in X^{*}$ , where the right $B$ -action on $X^{* *}$ is the one induced by the left $B$ -action on $X^{*}$ of step 2.1; so ${ev}_{X}$ is a map of $(A, B)$ -bimodules, and it is natural in the bimodule variable by the computation of step 3.3 applied to bimodule maps. By steps 3.2 and 3.3 the restriction of $(-)^{*}$ to finite-dimensional bimodules is a contravariant equivalence between finite-dimensional $(A, B)$ -bimodules and finite-dimensional $(B, A)$ -bimodules with quasi-inverse $(-)^{*}$ , which proves (iii).

5.1step 1.4step 2.4step 4.1step 3.3step 4.2given∎

Steps 3.3, 4.1 and 4.2 prove (i), (ii) and (iii) respectively. The only choices made are finite bases, dual bases and basis extensions in finite-dimensional spaces, supplied without any choice principle by [F2] and [L4], so no choice is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Finite-support families of finite-dimensional vector spaces are locally finite but not finite

Statement

Let $k$ be a field and let $C$ be the category whose objects are the families $(V_{n})_{n \in N}$ of finite-dimensional $k$ -vector spaces with $V_{n} = 0$ for all but finitely many $n$ , and whose morphisms $(V_{n}) \to (W_{n})$ are the families $(f_{n} : V_{n} \to W_{n})$ of $k$ -linear maps, with componentwise identities and composition. Then $C$ is a $k$ -linear abelian category in which kernels, cokernels and finite biproducts are computed componentwise; every hom-space is finite-dimensional over $k$ ; every object has finite length; every object is projective, hence every simple object has a projective cover; and the objects $S_{m}$ with $(S_{m})_{m} = k$ and $(S_{m})_{n} = 0$ for $n \neq m$ are pairwise non-isomorphic simple objects. Consequently $C$ is locally finite and has enough projectives, but it has infinitely many isomorphism classes of simple objects and no object of $C$ is a generator, so $C$ is not a finite $k$ -linear abelian category. No choice is used.

Facts & Assumptions

Given: A field $k$ , the category $k - M o d$ of $k$ -vector spaces, the product category $P = \prod_{n \in N} k - M o d$ (Product category and its projection functors), and its full subcategory $C$ on the families $(V_{n})$ with every $V_{n}$ finite-dimensional and $V_{n} = 0$ for all but finitely many $n$ . For an object $V$ write $supp V = {n : V_{n} \neq 0}$ , a finite set by hypothesis, and write $d (V) = \sum_{n} \dim_{k} V_{n}$ .

[F1]

$k - M o d$ is the category of modules over the field $k$ and is abelian (Modules over a ring form an abelian category).

[F2]

Every set-indexed product of abelian categories is abelian, with the zero object, finite biproducts, kernels and cokernels computed componentwise (A small product of abelian categories is abelian, Product category and its projection functors).

[F3]

For a homomorphism $f : M \to N$ of $k$ -modules, $\ker f$ is a submodule of $M$ , $im f$ is a submodule of $N$ , and $f$ is injective if and only if $\ker f = {0_{M}}$ ; the cokernel is $coker f = N / im f$ (Module homomorphism and isomorphism, kernel, image and cokernel, Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel).

[F4]

In an abelian category a morphism is monic exactly when its kernel is zero, and epic exactly when its cokernel is zero (In an abelian category, monic means zero kernel and epic means zero cokernel).

[F5]

A linear subspace of a finite-dimensional space is finite-dimensional, and $\dim_{F} V = \dim_{F} (\ker T) + \dim_{F} (im T)$ for a linear $T$ on a finite-dimensional $V$ (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ , Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F6]

A finite-dimensional $k$ -vector space has a finite ordered basis, and a $k$ -linear map on such a space is uniquely determined by, and may be prescribed arbitrarily on, the elements of an ordered basis (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis).

[F7]

For finite-dimensional $k$ -vector spaces $V, W$ one has $\dim_{k} L (V, W) = (\dim_{k} V) (\dim_{k} W)$ , and the dimension of a finite internal direct sum is the sum of the dimensions of its summands ( $\dim_{F} M_{m \times n} (F) = m n$ and $\dim_{F} L (V, W) = (\dim_{F} V) (\dim_{F} W)$ for finite-dimensional $V, W$ , If $V = ⨁_{i < n} U_{i}$ with every $U_{i}$ finite-dimensional, then $V$ is finite-dimensional and $\dim_{F} V = \sum_{i < n} \dim_{F} U_{i}$ ; in particular $\dim_{F} (U \oplus W) = \dim_{F} U + \dim_{F} W$ ).

[F8]

A subcategory is full when it contains every morphism between its objects that exists in the ambient category (Subcategory and full subcategory).

Proof

technique · direct

1.1F1F2F8given

The category $k - M o d$ is abelian by [F1], so the product category $P = \prod_{n \in N} k - M o d$ is abelian by [F2], with zero object, finite biproducts, kernels and cokernels computed componentwise; $C$ is by definition the full subcategory of $P$ on the families that are finite-dimensional in every degree and zero in all but finitely many degrees.

1.2F7givenalgebra

The family of zero spaces is an object of $C$ and is the zero object of $P$ , and if $V, W \in C$ then the componentwise family $(V_{n} \oplus W_{n})$ lies in $C$ , because $supp (V \oplus W) \subseteq supp V \cup supp W$ is finite and $\dim_{k} (V_{n} \oplus W_{n}) = \dim_{k} V_{n} + \dim_{k} W_{n}$ by [F7] is finite in every degree; the biproduct morphisms of $P$ are the componentwise ones, so $C$ contains the zero object and is closed under finite biproducts computed in $P$ .

1.3F2F3F5givenalgebra

Let $f : V \to W$ be a morphism in $C$ . Its kernel in $P$ is the family $(\ker f_{n})$ of [F3] with the canonical inclusions, whose support lies in the finite set $supp V$ , and $\ker f_{n}$ is a subspace of the finite-dimensional space $V_{n}$ , hence finite-dimensional by [F5]; its cokernel in $P$ is the family $(coker f_{n}) = (W_{n} / im f_{n})$ of [F3], whose support lies in the finite set $supp W$ , and $\dim_{k} (W_{n} / im f_{n}) = \dim_{k} W_{n} - \dim_{k} (im f_{n}) < \infty$ by rank-nullity [F5]. Hence both the kernel and the cokernel in $P$ of a morphism of $C$ are objects of $C$ with their canonical maps, so $C$ is closed under kernels and cokernels of its morphisms computed in $P$ .

1.4F7given

For $V, W \in C$ the set ${Hom}_{C} (V, W)$ is the product $\prod_{n \in N} L (V_{n}, W_{n})$ , which is a finite product over the finite set $supp V \cup supp W$ because $L (0, X)$ and $L (X, 0)$ are the zero space; identified with the finite direct sum $⨁_{n \in supp V \cup supp W} L (V_{n}, W_{n})$ it is a finite-dimensional $k$ -vector space with $\dim_{k} {Hom}_{C} (V, W) = \sum_{n} (\dim_{k} V_{n}) (\dim_{k} W_{n})$ by [F7]. Composition is componentwise, hence $k$ -bilinear, so $C$ is a locally small $k$ -linear category in which every hom-space is finite-dimensional over $k$ .

1.5F8givenconstruct

A family $(u_{n}) : (V_{n}) \to (W_{n})$ of linear maps is an isomorphism in $C$ exactly when every $u_{n}$ is a linear isomorphism, and then $(u_{n}^{- 1})$ is its inverse; for $m \in N$ let $S_{m}$ be the object with $(S_{m})_{m} = k$ and $(S_{m})_{n} = 0$ for $n \neq m$ , so that $S_{m} \neq 0$ .

2.1F8step 1.1step 1.2step 1.3algebra

By steps 1.1, 1.2 and 1.3 the full subcategory $C$ of the abelian category $P$ contains the zero object and is closed under finite biproducts, kernels and cokernels computed in $P$ , so it is an abelian subcategory of $P$ in the sense of Abelian subcategory and exact embedding. Consequently it is itself abelian: hom-sets are abelian groups with bilinear composition inherited from $P$ , the zero object and finite biproducts of $C$ are those of $P$ , every morphism of $C$ has its $P$ -kernel and $P$ -cokernel in $C$ , and its image and coimage, being built from those kernels and cokernels (Image and coimage in a category with kernels and cokernels), are also objects of $C$ , with the canonical comparison an isomorphism in $P$ whose inverse is a morphism of $C$ by fullness [F8]; this is exactly additivity with invertible image-coimage comparison, so $C$ is abelian (Abelian category).

3.1F3F4step 2.1algebra

In the abelian category $C$ the kernel and cokernel of a morphism are computed componentwise, as in step 1.3; by [F4] a morphism $u : V \to W$ of $C$ is monic if and only if $\ker u = 0$ , that is if and only if $\ker u_{n} = 0$ for every $n$ , which by [F3] holds exactly when every $u_{n}$ is injective; and $u$ is epic if and only if $coker u = 0$ , that is if and only if $W_{n} = im u_{n}$ for every $n$ , which holds exactly when every $u_{n}$ is surjective.

4.1F6step 3.1chooseconstructalgebra

Every object $V$ of $C$ is projective (Projective object): let $q : E \to M$ be an epimorphism and $f : V \to M$ a morphism in $C$ ; by step 3.1 each $q_{n} : E_{n} \to M_{n}$ is surjective. For each $n \in supp V$ choose an ordered basis $(v_{n, 1}, \dots, v_{n, d_{n}})$ of $V_{n}$ (possible since $V_{n}$ is finite-dimensional) and for each $j \leq d_{n}$ choose $e_{n, j} \in E_{n}$ with $q_{n} (e_{n, j}) = f_{n} (v_{n, j})$ ; finitely many such choices are made. By [F6] there is for each such $n$ a unique linear $g_{n} : V_{n} \to E_{n}$ with $g_{n} (v_{n, j}) = e_{n, j}$ , and set $g_{n} = 0$ for the remaining $n$ ; then $q_{n} g_{n} = f_{n}$ for $n \in supp V$ because both sides agree on the basis $(v_{n, j})$ , and for the remaining $n$ both sides are zero, so the family $(g_{n})$ is a morphism of $C$ with $q \circ g = f$ . Thus every morphism into $M$ lifts along every epimorphism $q$ , so $V$ is projective.

4.2F5step 1.5step 3.1algebra

For every $m$ the object $S_{m}$ of step 1.5 is simple (Simple object): it is nonzero, and if $u : T \to S_{m}$ is a monomorphism in $C$ then every $u_{n}$ is injective by step 3.1; for $n \neq m$ the target $(S_{m})_{n}$ is zero, so an injective map into it has zero domain and $T_{n} = 0$ , while a nonzero subobject has $T \neq 0$ , hence $T_{m} \neq 0$ ; then $u_{m} : T_{m} \to k$ is an injective linear map with nonzero finite-dimensional domain, so $\dim_{k} T_{m} \leq 1$ and $\dim_{k} T_{m} \geq 1$ , whence $u_{m}$ is an isomorphism and so is $u$ by step 1.5. Therefore the only subobjects of $S_{m}$ are the zero subobject and $1_{S_{m}}$ , so $S_{m}$ is simple.

5.1step 4.2algebra

Conversely, if $V \in C$ is simple, then $V \neq 0$ gives $V_{m} \neq 0$ for some $m$ , and a nonzero vector of $V_{m}$ spans a line $L \subseteq V_{m}$ ; the object $S$ with $S_{m} = L$ and $S_{n} = 0$ for $n \neq m$ is a nonzero subobject of $V$ isomorphic to $S_{m}$ , so by simplicity the subobject $S$ equals $V$ and $V ≅ S_{m}$ . Moreover $S_{m} ≅ S_{n}$ forces $k = (S_{m})_{m} ≅ (S_{n})_{m}$ , so $(S_{n})_{m} \neq 0$ and $m = n$ . Hence the objects $S_{m}$ , $m \in N$ , form a complete set of pairwise non-isomorphic simple objects, so $C$ has infinitely many isomorphism classes of simple objects.

5.2F5step 2.1step 4.2induction

Every object $V$ of $C$ has finite length (Object of finite length): induct on the natural number $d (V) = \sum_{n} \dim_{k} V_{n}$ from step 1.4. If $d (V) = 0$ then every $V_{n} = 0$ , so $V = 0$ and the empty composition series exhibits finite length. If $d (V) > 0$ , choose $n \in supp V$ and a line $L \subseteq V_{n}$ ; the object $S$ with $S_{n} = L$ and $S_{k} = 0$ for $k \neq n$ is a simple subobject of $V$ , and the quotient $V / S$ (The quotient of an object by a subobject) is the family with $(V / S)_{n} = V_{n} / L$ and $(V / S)_{j} = V_{j}$ for $j \neq n$ , of total dimension $d (V) - 1$ ; by the induction hypothesis $V / S$ has finite length, and $S ≅ S_{n}$ has the one-step composition series $0 < S$ because it is simple by step 4.2, so the additivity theorem for lengths along a subobject (Length is additive along a subobject) gives that $V$ has finite length and $ℓ (V) = 1 + ℓ (V / S)$ .

5.3step 4.1algebra

Every simple object of $C$ has a projective cover (Superfluous subobjects and projective covers in an abelian category): if $S$ is simple then its identity $1_{S} : S \to S$ is an epimorphism whose source $S$ is projective by step 4.1, and its kernel is the zero subobject, which is superfluous because $[0] \lor [m] = [m]$ for every subobject $[m]$ , so the condition $[0] \lor [m] = [1_{S}]$ forces $[m] = [1_{S}]$ ; hence $1_{S}$ is essential and is a projective cover of $S$ .

5.4step 1.4step 4.2algebra

No object $P$ of $C$ is a generator (Generator and cogenerator of a category): choose $m \notin supp P$ , possible because $supp P$ is finite; then ${Hom}_{C} (P, S_{m}) = \prod_{n} L (P_{n}, (S_{m})_{n}) = 0$ by step 1.4, since the $m$ -th factor is $L (0, k) = 0$ . The two distinct parallel morphisms $1_{S_{m}}, 0 : S_{m} \to S_{m}$ therefore cannot be separated by any morphism $P \to S_{m}$ , so the singleton ${P}$ is not separating in the sense of Separating and coseparating sets of objects, and $P$ is not a generator.

6.1step 1.4step 2.1step 4.1step 5.1step 5.2step 5.3step 5.4given∎

By step 2.1 the category $C$ is abelian, by step 1.4 it is locally small, $k$ -linear with finite-dimensional hom-spaces, and by step 5.2 every object has finite length; hence $C$ is a locally finite $k$ -linear abelian category (Locally finite k-linear abelian categories). By step 4.1 every object is projective, hence by step 5.3 every simple object has a projective cover, so $C$ has enough projectives; by step 5.1 it has infinitely many isomorphism classes of simple objects and by step 5.4 no object is a generator, so the finiteness conditions of Finite k-linear abelian categories fail and $C$ is not a finite $k$ -linear abelian category. All bases chosen lie in finite-dimensional spaces, the hom-space products and objectwise sums reduce to finite ones; the ambient countable product uses the explicit componentwise module constructions, and no element is selected from an infinite family, so no choice is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Projective epimorphisms onto the simples generate every finite-length object

Statement

Let $C$ be a locally finite $k$ -linear abelian category with finitely many isomorphism classes of simple objects, represented by $S_{1}, \dots, S_{n}$ , and suppose that for each $i$ a projective epimorphism $Q_{i} ↠ S_{i}$ is chosen (for instance the projective cover supplied by Superfluous subobjects and projective covers in an abelian category when $C$ is finite in the intrinsic sense). Then for every object $X$ of finite length there are integers $m_{i} \geq 0$ and an epimorphism $⨁_{i = 1}^{n} Q_{i}^{m_{i}} ↠ X$ ; in particular, with $P = ⨁_{i} Q_{i}$ , every object of $C$ admits an epimorphism $P^{m} ↠ X$ for some $m \geq 0$ . Only the finitely many chosen maps $Q_{i} ↠ S_{i}$ are selected, and no other choice is used.

Facts & Assumptions

Given: A field $k$ , a locally finite $k$ -linear abelian category $C$ , simple representatives $S_{1}, \dots, S_{n}$ for all isomorphism classes of simple objects of $C$ , and chosen projective epimorphisms $φ_{i} : Q_{i} ↠ S_{i}$ , $i = 1, \dots, n$ .

[F1]

An object has finite length exactly when it admits a composition series $0 = X_{0} < X_{1} < \dots < X_{ℓ} = X$ with simple factors $X_{j} / X_{j - 1}$ ; the length $ℓ (X)$ is the number of factors, the truncation $0 = X_{0} < \dots < X_{j}$ is a composition series of $X_{j}$ so that $ℓ (X_{j}) = j$ , and lengths are additive along a subobject (Object of finite length, Composition series and composition factors of an object, Length is additive along a subobject).

[F2]

The quotient of an object $X$ by a subobject represented by $m : M ↣ X$ is $coker (m)$ , written $X / M$ , and its defining map is the cokernel map (The quotient of an object by a subobject, Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms).

[F3]

A cokernel $c : B \to C$ of $f : A \to B$ satisfies $c f = 0$ , and every $h$ with $h f = 0$ factors as $h = \overset{ˉ}{h} c$ for a unique $\overset{ˉ}{h}$ (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers).

[F4]

A projective object $Q$ has the lifting property: for every epimorphism $e : E ↠ M$ and every $f : Q \to M$ there is $\tilde{f} : Q \to E$ with $e \tilde{f} = f$ (Projective object).

[F5]

Identities are epic, composites of epimorphisms are epic, and split epimorphisms are epic (Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic).

[F6]

A biproduct $⨁_{i} A_{i}$ is simultaneously a product and a coproduct with injections and projections satisfying the biproduct identities, and biproducts are associative and commutative up to canonical isomorphism; in particular a morphism out of a biproduct is determined by its components, and an inclusion of a subfamily of summands is split by the corresponding projection (Biproduct, Biproducts are associative, commutative, and unital up to canonical isomorphism, An additive category is an Ab-enriched category with a zero object and finite biproducts).

[F7]

$C$ is abelian, hence additive and preadditive: its hom-sets are abelian groups, composition is bilinear, and there is a zero object (Abelian category, Additive category, Preadditive category).

[F8]

A simple object is nonzero and has exactly two subobjects, the zero subobject and its identity; by hypothesis every simple object of $C$ is isomorphic to one of $S_{1}, \dots, S_{n}$ (Simple object, given).

Proof

technique · induction

1.1baseF1F5F6given

The assertion to be proved by induction on the natural number $p$ is: for every object $Y$ of $C$ of length $p$ there are integers $m_{i} \geq 0$ and an epimorphism $⨁_{i} Q_{i}^{m_{i}} ↠ Y$ . At $p = 0$ a composition series of $Y$ has no factors, so $Y = 0$ by [F1]; taking all $m_{i} = 0$ , the empty biproduct is the zero object and the identity $0 \to Y$ is an epimorphism by [F5], so the assertion holds at $p = 0$ .

1.2ih

Assume the assertion at the natural number $p$ : for every object $Y$ of $C$ with $ℓ (Y) = p$ there are integers $m_{i} \geq 0$ and an epimorphism $⨁_{i} Q_{i}^{m_{i}} ↠ Y$ .

1.3F1F8given

Suppose $X$ has length $p + 1$ and let $0 = X_{0} < \dots < X_{p + 1} = X$ be a composition series of $X$ ; then the last factor $S = X_{p + 1} / X_{p}$ is simple by [F1], hence $S ≅ S_{i}$ for some $i$ by [F8], and $ℓ (X_{p}) = p$ by [F1].

2.1step 1.2step 1.3F2F4F6constructgiven

Successor step. Let $X$ have length $p + 1$ , with composition series, last simple factor $S ≅ S_{i}$ and $Y : = X_{p}$ of length $p$ as in step 1.3. By the induction hypothesis of step 1.2 applied to $Y$ there are integers $m_{j} \geq 0$ and an epimorphism $e : W ↠ Y$ , where $W : = ⨁_{j} Q_{j}^{m_{j}}$ is a finite biproduct of the chosen projectives. The quotient $π : X ↠ X / X_{p} = S$ of [F2] is an epimorphism, and composing the chosen epimorphism $φ_{i} : Q_{i} ↠ S_{i}$ with an isomorphism $S_{i} ≅ S$ gives an epimorphism $φ : Q_{i} ↠ S$ , so by the lifting property [F4] of the projective $Q_{i}$ there is $ψ : Q_{i} \to X$ with $π ψ = φ$ . Let $u : W \oplus Q_{i} \to X$ be the morphism with components the composite $W \overset{e}{\to} Y ↣ X$ and $ψ$ , which exists and is unique by the biproduct property [F6].

3.1step 2.1F3F5F7algebra

The morphism $u$ of step 2.1 is an epimorphism. Let $c : X \to C$ be a morphism with $c u = 0$ . Composing with the first biproduct injection gives $c \circ u \circ {i n j}_{W} = c \circ (inclusion \circ e) = 0$ , and $e$ is epic, so $c \circ inclusion = 0$ for the inclusion $X_{p} ↣ X$ ; since $π$ is a cokernel of that inclusion by [F2], the universal property [F3] gives $c = \overset{ˉ}{c} \circ π$ for some $\overset{ˉ}{c} : S \to C$ . Then $0 = c u$ composed with the second biproduct injection gives $\overset{ˉ}{c} \circ π \circ ψ = \overset{ˉ}{c} \circ φ = 0$ , and $φ$ is epic, so $\overset{ˉ}{c} = 0$ and therefore $c = 0$ . Since $C$ is preadditive [F7], a morphism $u$ with the property that every $c$ satisfying $c u = 0$ is zero is an epimorphism: from $g u = h u$ one gets $(g - h) u = 0$ , hence $g - h = 0$ .

4.1step 3.1F6given

The source of the epimorphism $u$ of step 3.1 is a finite biproduct of the chosen objects $Q_{1}, \dots, Q_{n}$ : regrouping its summands by index, $W \oplus Q_{i} ≅ ⨁_{i = 1}^{n} Q_{i}^{m_{i}^{'}}$ for integers $m_{i}^{'} \geq 0$ by the associativity and commutativity of biproducts [F6]. Hence the successor case holds: $X$ of length $p + 1$ admits an epimorphism $⨁_{i} Q_{i}^{m_{i}^{'}} ↠ X$ .

5.1step 1.1step 4.1F5F6discharge-induction∎

Step 1.1 is the base case and steps 1.3, 2.1, 3.1 and 4.1 pass from $p$ to $p + 1$ using the induction hypothesis of step 1.2, so by induction on $p$ every object $X$ of finite length admits an epimorphism $⨁_{i} Q_{i}^{m_{i}} ↠ X$ for suitable integers $m_{i} \geq 0$ . For the final clause put $P = ⨁_{i} Q_{i}$ and $m = \sum_{i} m_{i}$ ; by [F6] the power $P^{m}$ is the biproduct $⨁_{i} Q_{i}^{m}$ , whose subfamily of summands $Q_{i}^{m_{i}}$ has $⨁_{i} Q_{i}^{m_{i}}$ as a biproduct, and the corresponding projection $P^{m} ↠ ⨁_{i} Q_{i}^{m_{i}}$ is a split epimorphism, hence epic by [F5]; composing it with an epimorphism onto $X$ gives an epimorphism $P^{m} ↠ X$ by [F5]. The proof selects only finite data (a composition series of the object at hand, finitely many summand indices and biproduct structure maps, and the $n$ supplied epimorphisms $φ_{i}$ ), so no choice principle is used.

PropositionStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Finite-dimensional module categories satisfy the intrinsic finiteness conditions

Statement

Let $A$ be a finite-dimensional unital algebra over a field $k$ , and let $A - m o d$ denote the category of finite-dimensional left $A$ -modules with $A$ -linear maps. Then $A - m o d$ is a finite $k$ -linear abelian category in the intrinsic sense of Finite k-linear abelian categories: it is a locally finite $k$ -linear abelian category (Locally finite k-linear abelian categories); every simple object has a projective cover, namely the cover supplied by Every finite-dimensional module has a projective cover, unique up to isomorphism over the target and identified with the general notion by Superfluous subobjects and projective covers in an abelian category; and there are finitely many isomorphism classes of simple objects. Moreover every simple left $A$ -module is isomorphic to a composition factor of the regular module $_{A} A$ , and every finite-dimensional module has length at most its $k$ -dimension. No choice is used.

Facts & Assumptions

Given: A field $k$ and a finite-dimensional unital $k$ -algebra $A$ , with $A - M o d$ the category of all left $A$ -modules and $A - m o d$ the full subcategory of finite-dimensional left $A$ -modules.

[F1]

For every ring $R$ the category $R - M o d$ of left $R$ -modules is abelian, with zero object, finite biproducts, kernels and cokernels given by the usual module constructions (Modules over a ring form an abelian category, Module homomorphism and isomorphism, kernel, image and cokernel).

[F2]

Dimension facts over $k$ : a linear subspace of a finite-dimensional space is finite-dimensional and has dimension at most that of the ambient space, with equality exactly for $U = V$ ; the dimension of a finite direct sum is the sum of the dimensions; and rank-nullity gives $\dim_{k} (M / N) = \dim_{k} M - \dim_{k} N$ for a subspace $N \subseteq M$ of a finite-dimensional space, in particular quotients of finite-dimensional spaces by subspaces are finite-dimensional (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ , If $V = ⨁_{i < n} U_{i}$ with every $U_{i}$ finite-dimensional, then $V$ is finite-dimensional and $\dim_{F} V = \sum_{i < n} \dim_{F} U_{i}$ ; in particular $\dim_{F} (U \oplus W) = \dim_{F} U + \dim_{F} W$ , Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F3]

For left $A$ -modules $M, N$ the set ${Hom}_{A} (M, N)$ is a $k$ -vector subspace of the space $L (M, N)$ of $k$ -linear maps, with pointwise addition and scalar multiplication, and composition of $A$ -linear maps is $k$ -bilinear; moreover, if $M, N$ are finite-dimensional over $k$ , then $\dim_{k} L (M, N) = (\dim_{k} M) (\dim_{k} N)$ (Module homomorphism and isomorphism, kernel, image and cokernel, k-linear categories and k-linear functors, $\dim_{F} M_{m \times n} (F) = m n$ and $\dim_{F} L (V, W) = (\dim_{F} V) (\dim_{F} W)$ for finite-dimensional $V, W$ ).

[F4]

An object has finite length exactly when it admits a composition series with simple factors; the length is the number of factors; and if any two of $N$ , $M$ , $M / N$ (for a submodule $N \leq M$ ) have finite length then so does the third, with $ℓ (M) = ℓ (N) + ℓ (M / N)$ (Composition series and composition factors of an object, Object of finite length, Length is additive along a subobject).

[F5]

Every finite-dimensional left module over a finite-dimensional algebra has a projective cover in the module sense: there is an epimorphism $π : Q ↠ S$ with $Q$ projective and $Q$ finite-dimensional (Every finite-dimensional module has a projective cover, unique up to isomorphism over the target, Projective modules and the lifting property).

[F6]

In a module category the general notion of a projective cover agrees with the module notion: joins of submodules are sums, so the general superfluity condition is the superfluous-kernel condition of the module definition (Superfluous subobjects and projective covers in an abelian category).

[F7]

A module $S$ is simple when $S \neq 0$ and its only submodules are $0$ and $S$ ; if $N \leq M$ is a submodule contained in the kernel of an $A$ -linear map $M \to P$ , then the map factors uniquely through the quotient module $M / N$ (Simple module: a nonzero module with no proper nonzero submodule, Quotient module $M / N$ with scalar multiplication on additive cosets, A module homomorphism vanishing on $N$ factors uniquely through $M / N$ ).

[F8]

A full subcategory of an abelian category containing the zero object and closed under finite biproducts, kernels and cokernels computed in the ambient category is an abelian subcategory there; the ambient abelian structure then makes it abelian, since kernels, cokernels and their images and coimages are those of the ambient category and the inverse of the image-coimage comparison again lies in the full subcategory (Abelian subcategory and exact embedding, Image and coimage in a category with kernels and cokernels, Abelian category, Subcategory and full subcategory).

[L1]

Every nonempty set of natural numbers has a least element (The well-ordering principle).

Proof

technique · direct

1.1F1F2given

The full subcategory $A - m o d$ of $A - M o d$ contains the zero module, and it is closed in $A - M o d$ under finite biproducts, kernels and cokernels: direct sums of finite-dimensional modules are finite-dimensional with dimensions adding, by [F2]; the kernel of an $A$ -linear map $M \to N$ is a $k$ -subspace of the finite-dimensional $M$ , hence finite-dimensional; and the cokernel $N / im f$ is a quotient of the finite-dimensional $N$ by a subspace, hence finite-dimensional with $\dim_{k} (N / im f) = \dim_{k} N - \dim_{k} (im f)$ by [F2].

1.2F5F6given

Every simple object $S$ of $A - m o d$ has a projective cover in the sense of Superfluous subobjects and projective covers in an abelian category: by the cover theorem [F5] there is a finite-dimensional projective module $Q$ and an epimorphism $π : Q ↠ S$ whose kernel is superfluous in the module sense, and by [F6] this is exactly a superfluous subobject of $Q$ in the general sense, so $π$ is an essential epimorphism with projective source, that is, a projective cover of $S$ .

2.1F1F8step 1.1

Because $A - M o d$ is abelian by [F1] and $A - m o d$ is a full subcategory containing the zero object and closed under finite biproducts, kernels and cokernels computed there, step 1.1 makes $A - m o d$ an abelian subcategory of $A - M o d$ ; by [F8] it is therefore itself abelian.

2.2F2F3step 1.1

For finite-dimensional modules $M, N$ the hom-set ${Hom}_{A} (M, N)$ is a $k$ -subspace of $L (M, N)$ by [F3], and $L (M, N)$ is finite-dimensional with $\dim_{k} L (M, N) = (\dim_{k} M) (\dim_{k} N)$ ; a subspace of a finite-dimensional space is finite-dimensional by [F2], and composition is $k$ -bilinear by [F3], so $A - m o d$ is a locally small $k$ -linear category with finite-dimensional hom-spaces.

3.1F2F4L1step 2.1induction

Every object $M$ of $A - m o d$ has finite length and $ℓ (M) \leq \dim_{k} M$ . Induct on $d = \dim_{k} M$ . If $d = 0$ then $M = 0$ and the empty composition series witnesses finite length with $ℓ (M) = 0$ . If $d > 0$ , the set of dimensions of nonzero submodules of $M$ is a nonempty set of natural numbers, so by [L1] it has a least element $d_{0} \geq 1$ , and some nonzero submodule $N \leq M$ has $\dim_{k} N = d_{0}$ ; such an $N$ is simple, because for any nonzero $N^{'} \leq N$ the submodule $N^{'}$ is also a nonzero submodule of $M$ with $\dim_{k} N^{'} \geq d_{0}$ and $\dim_{k} N^{'} \leq \dim_{k} N = d_{0}$ by [F2], so $\dim_{k} N^{'} = d_{0}$ and $N^{'} = N$ by the equality case of [F2]. By [F2] the quotient $M / N$ has $\dim_{k} (M / N) = d - d_{0} < d$ , so by the induction hypothesis $M / N$ has finite length with $ℓ (M / N) \leq \dim_{k} (M / N)$ ; the simple module $N$ has finite length with $ℓ (N) = 1$ , so the additivity theorem [F4] gives that $M$ has finite length and $ℓ (M) = ℓ (N) + ℓ (M / N) = 1 + ℓ (M / N) \leq 1 + (d - d_{0}) \leq d$ .

4.1F4F7step 3.1constructgiven

Every simple left $A$ -module is isomorphic to a composition factor of the regular module $_{A} A$ , and there are finitely many isomorphism classes of simple modules. The algebra $A$ is a finite-dimensional left $A$ -module, so by step 3.1 it has a composition series $0 = A_{0} < A_{1} < \dots < A_{m} = A$ . Let $S$ be a simple left $A$ -module and $0 \neq s \in S$ ; the map $φ : A \to S$ , $φ (a) = a s$ , is $A$ -linear with $φ (1) = s \neq 0$ , so its image is a nonzero submodule of the simple module $S$ and $φ$ is surjective. Let $j \in {1, \dots, m}$ be least with $φ (A_{j}) \neq 0$ , which exists because $φ (A_{m}) = S \neq 0$ and the set is finite; then $φ (A_{j - 1}) = 0$ , and $φ (A_{j})$ is a nonzero submodule of $S$ , hence equals $S$ , so the restriction of $φ$ to $A_{j}$ is surjective with kernel containing $A_{j - 1}$ and therefore factors through the quotient $A_{j} / A_{j - 1}$ by [F7], giving a nonzero surjection $A_{j} / A_{j - 1} \to S$ ; the source is simple, so this surjection is an isomorphism, whence $S ≅ A_{j} / A_{j - 1}$ . Thus every simple module is isomorphic to one of the $m$ composition factors of $_{A} A$ , so there are at most $m$ isomorphism classes of simple modules.

5.1step 2.1step 2.2step 3.1step 4.1step 1.2given∎

Steps 2.1, 2.2 and 3.1 make $A - m o d$ a locally small $k$ -linear abelian category in which every object has finite length and every hom-space is finite-dimensional over $k$ , that is, a locally finite $k$ -linear abelian category; step 1.2 gives every simple object a projective cover, and step 4.1 shows that there are finitely many isomorphism classes of simple objects; hence $A - m o d$ is a finite $k$ -linear abelian category in the intrinsic sense of Finite k-linear abelian categories. The further claims are steps 4.1 and 3.1. All selections in the proof are made inside finite-dimensional objects (a nonzero submodule of least dimension and a composition series of the finite-dimensional algebra), so no choice principle is used.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Intrinsic finite category hypotheses give a finite projective generator

Statement

Let $C$ be a finite $k$ -linear abelian category, with simple representatives $S_{1}, \dots, S_{n}$ and chosen projective covers $Q_{i} ↠ S_{i}$ , and put $P = ⨁_{i = 1}^{n} Q_{i}$ and $A = {End}_{C} (P)^{o p}$ . Then: (i) $P$ is projective; (ii) $P$ is a generator of $C$ , that is, ${P}$ is separating; (iii) $A$ is a finite-dimensional unital $k$ -algebra; (iv) $C (P, -)$ is exact and faithful; and (v) every object of $C$ is a quotient of a finite direct sum of copies of $P$ . The proof uses only that each $Q_{i} ↠ S_{i}$ is a projective epimorphism, never the superfluity of its kernel, so the same conclusions hold if "enough projectives" is read as "every simple object admits a projective epimorphism onto it"; for the finite module categories of Finite-dimensional module categories satisfy the intrinsic finiteness conditions the two readings coincide. Only the finitely many covers $Q_{i}$ are selected; no further choice is used.

Facts & Assumptions

Given: A field $k$ , a finite $k$ -linear abelian category $C$ in the sense of Finite k-linear abelian categories, simple representatives $S_{1}, \dots, S_{n}$ for all isomorphism classes of simple objects, and chosen projective epimorphisms $Q_{i} ↠ S_{i}$ , $i = 1, \dots, n$ . Put $P = ⨁_{i = 1}^{n} Q_{i}$ and $A = {End}_{C} (P)^{o p}$ .

[F1]

For an object $P$ of an abelian category the following are equivalent: $P$ is projective; the functor $C (P, -)$ carries every short exact sequence to a short exact sequence; and every epimorphism onto $P$ splits (Projective object characterisations, Projective object).

[F2]

A biproduct $⨁_{i} Q_{i}$ is a coproduct with injections ${i n j}_{i}$ and a product with projections ${p r}_{i}$ satisfying ${p r}_{i} {i n j}_{i} = 1_{Q_{i}}$ ; the projections are split epimorphisms, hence epimorphisms, and morphisms out of a coproduct are determined by their composites with the injections (Biproduct, Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic).

[F3]

Composites of epimorphisms are epimorphisms; if $π h \neq 0$ then $h \neq 0$ ; a monomorphism composed with a nonzero morphism is nonzero; and if $e$ is epic and $f e = 0$ then $f = 0$ (Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic, Monomorphism and epimorphism by left and right cancellation).

[F4]

Every morphism $f : X \to Y$ of an abelian category factors as $f = m \circ e$ with $e$ an epimorphism and $m$ a monomorphism, with $m$ representing $im f$ ; in particular $f \neq 0$ exactly when $im f \neq 0$ (Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism, Image and coimage in a category with kernels and cokernels).

[F5]

Every nonzero object of finite length has a composition series whose last factor $Z ↠ S$ is a simple quotient, and every simple object of $C$ is isomorphic to one of $S_{1}, \dots, S_{n}$ (Composition series and composition factors of an object, Simple object, given).

[F6]

An object $G$ is a generator when the singleton ${G}$ is separating, that is, when for every pair of distinct parallel morphisms $u \neq v : X \to Y$ there is $g : G \to X$ with $u g \neq v g$ (Separating and coseparating sets of objects, Generator and cogenerator of a category).

[F7]

A functor is faithful when it is injective on every hom-set; for $C (P, -)$ this means that $u \neq v : X \to Y$ yields $u \circ - \neq v \circ -$ , that is, some $g : P \to X$ has $u g \neq v g$ (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[F8]

Under the hypotheses of the statement, every object of finite length, in particular every object of $C$ , admits an epimorphism $P^{m} ↠ X$ for some $m \geq 0$ , using only that the $Q_{i} ↠ S_{i}$ are projective epimorphisms (Projective epimorphisms onto the simples generate every finite-length object, Locally finite k-linear abelian categories).

[F9]

For an object $P$ of a preadditive category, ${End}_{C} (P) = C (P, P)$ with addition from the hom-group and multiplication given by composition is a unital ring with identity $1_{P}$ , composition is bilinear in both variables, and reversing the multiplication gives the opposite ring (Endomorphisms of an object of a preadditive category form a ring, The opposite ring $R^{o p}$ ).

[F10]

A finite $k$ -linear abelian category is locally finite: every object has finite length and every hom-space is finite-dimensional over $k$ (Finite k-linear abelian categories, Locally finite k-linear abelian categories, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis). A $k$ -algebra is a unital ring with a central unital structure map $k \to A$ , equivalently a $k$ -vector space with a bilinear unital multiplication (Algebras over a commutative ring, central structure maps, and algebra homomorphisms, k-linear categories and k-linear functors).

[F11]

For the finite module categories of Finite-dimensional module categories satisfy the intrinsic finiteness conditions the stronger reading holds: the published cover theorem gives every simple module a projective cover (Every finite-dimensional module has a projective cover, unique up to isomorphism over the target).

Proof

technique · direct

1.1F1F2givenchooseconstruct

(i) $P$ is projective. Let $q : E ↠ P$ be an epimorphism. Projectivity of each $Q_{i}$ gives a lift $s_{i} : Q_{i} \to E$ of the injection ${i n j}_{i} : Q_{i} \to P$ , so $q s_{i} = {i n j}_{i}$ . The coproduct property [F2] gives $s : P \to E$ with $s {i n j}_{i} = s_{i}$ . Thus $q s {i n j}_{i} = {i n j}_{i}$ for every $i$ , and equality on all injections implies $q s = 1_{P}$ . Hence every epimorphism onto $P$ splits, so $P$ is projective by [F1].

1.2F1F2F3F4F5F6given

(ii) $P$ is separating. Let $u \neq v : X \to Y$ and put $f = u - v \neq 0$ , using that $C$ is additive. By [F4] $f$ factors as $f = m \circ e$ with $e : X ↠ Z$ epic, $m : Z ↣ Y$ monic and $Z = im f \neq 0$ ; by [F5] the nonzero object $Z$ of finite length has a simple quotient $π : Z ↠ S$ , and $S ≅ S_{i}$ for some $i$ by [F5]. Composing the chosen epimorphism $Q_{i} ↠ S_{i}$ with an isomorphism $S_{i} ≅ S$ gives an epimorphism $Q_{i} ↠ S$ , which by projectivity of $Q_{i}$ and [F1] lifts along $π$ to $h : Q_{i} \to Z$ with $π h$ epic; then $h \neq 0$ because $π h$ is an epimorphism onto the nonzero simple $S$ . Since $e$ is epic and $Q_{i}$ projective, $h$ lifts further along $e$ to $\tilde{h} : Q_{i} \to X$ with $e \tilde{h} = h$ . Then $m h = m e \tilde{h} = f \tilde{h}$ , and $m h \neq 0$ by [F3] because $m$ is monic and $h \neq 0$ ; so $f \tilde{h} \neq 0$ , and composing with the split epimorphism ${p r}_{i} : P ↠ Q_{i}$ of [F2] gives $g : = \tilde{h} {p r}_{i} : P \to X$ with $f g \neq 0$ , again by [F3]. Hence $u g \neq v g$ for the distinct morphisms $u, v$ , so ${P}$ is separating and $P$ is a generator by [F6].

1.3F9F10given

(iii) $A$ is a finite-dimensional unital $k$ -algebra. By [F9] the endomorphism set ${End}_{C} (P) = C (P, P)$ is a unital ring under composition with identity $1_{P}$ , and reversing the multiplication gives the opposite ring $A$ ; by [F10] the hom-space is finite-dimensional over $k$ and composition is $k$ -bilinear, so both ${End}_{C} (P)$ and its opposite $A$ are $k$ -vector spaces with bilinear unital multiplication. The structure map $η : k \to A$ , $η (c) = c \cdot 1_{P}$ , is a unital ring homomorphism whose image is central, because multiplication by scalars commutes with composition by $k$ -bilinearity; hence $A$ is a unital $k$ -algebra by [F10], finite-dimensional over $k$ since $C (P, P)$ is.

1.4F8F10

(v) Every object is a quotient of $P^{m}$ for some $m \geq 0$ : by [F10] every object of $C$ has finite length, so [F8] supplies an epimorphism $P^{m} ↠ X$ for some $m$ , whose target $X$ is therefore a quotient of $P^{m}$ .

2.1F1F7step 1.1step 1.2

(iv) $C (P, -)$ is exact and faithful. Exactness is condition 2 of [F1] applied to the projective object $P$ of step 1.1. For faithfulness, let $u \neq v : X \to Y$ ; by step 1.2 there is $g : P \to X$ with $u g \neq v g$ , so the induced maps on hom-sets differ and $C (P, -)$ is injective on this hom-set; since $u, v$ were arbitrary, $C (P, -)$ is faithful in the sense of [F7].

3.1step 1.1step 1.2step 1.3step 2.1step 1.4F8F11given∎

The claims (i), (ii), (iii), (iv) and (v) are steps 1.1, 1.2, 1.3, 2.1 and 1.4. Inspecting these steps and the covering lemma [F8], the only properties of the maps $Q_{i} ↠ S_{i}$ that were used are that they are epimorphisms and that their sources are projective; the superfluity of their kernels was never used, so replacing the covers by arbitrary projective epimorphisms onto the simples does not change the argument, which proves the stated reading-independence. For the finite module categories of Finite-dimensional module categories satisfy the intrinsic finiteness conditions the stronger reading is available in any case, since by [F11] every simple module there has a projective cover, so the two readings coincide there. The proof selects only the finitely many supplied maps $Q_{i} ↠ S_{i}$ and finitely many biproduct and lifting data inside finite-dimensional hom-spaces, so no choice principle is used beyond them.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Finite abelian categories admit finite-dimensional module models

Statement

Let $C$ be a finite $k$ -linear abelian category, let $S_{1}, \dots, S_{n}$ be representatives of its simple objects with chosen projective covers $Q_{i} ↠ S_{i}$ , and put $P = ⨁_{i} Q_{i}$ and $A = {End}_{C} (P)^{o p}$ . Then $A$ is a finite-dimensional unital $k$ -algebra and $H = C (P, -) : C \to A - m o d$ is a fully faithful, essentially surjective $k$ -linear functor, where $A - m o d$ is the category of finite-dimensional left $A$ -modules: $H$ is faithful, full and exact, and every finite-dimensional left $A$ -module is isomorphic to $H (X)$ for some object $X$ of $C$ , with the preimage exhibited by the finite-presentation construction in the proof. If a splitting of essential surjectivity is additionally supplied — an object $X_{V}$ and an isomorphism $H (X_{V}) \to V$ for every target module $V$ — then $H$ is a $k$ -linear equivalence in the specified-quasi-inverse sense of Equivalence, quasi-inverse, and adjoint equivalence of categories. Conversely any such $k$ -linear equivalence from a finite-dimensional module category transfers the intrinsic finiteness conditions, so the usual equivalence formulation holds when these splitting data are supplied. Fullness, faithfulness, exactness and objectwise essential surjectivity use only finite choices. Selecting finite presentations separately for each $V$ does not itself supply a simultaneous splitting; no choice-free existence of that splitting is asserted.

Facts & Assumptions

Given: A field $k$ , a finite $k$ -linear abelian category $C$ , simple representatives $S_{1}, \dots, S_{n}$ for its simple objects, chosen projective covers $Q_{i} ↠ S_{i}$ , and the objects $P = ⨁_{i} Q_{i}$ , $A = {End}_{C} (P)^{o p}$ and functor $H = C (P, -)$ .

[F1]

$C$ is locally finite: every object has finite length and every hom-space is finite-dimensional over $k$ ; $C$ is additive, so hom-sets are abelian groups with bilinear composition (Finite k-linear abelian categories, Locally finite k-linear abelian categories, k-linear categories and k-linear functors, Abelian category).

[F2]

A finite biproduct is at once a product and a coproduct: morphisms out of $⨁_{j} A_{j}$ are determined by their components, morphisms into it by their components, ${p r}_{i} {i n j}_{j} = δ_{i j}$ , and every morphism $h : A \to ⨁_{j} A_{j}$ equals $\sum_{j} {i n j}_{j} \circ ({p r}_{j} \circ h)$ (Biproduct, The direct sum of an indexed family of modules).

[F3]

$P$ is projective and a generator (separating); consequently $H = C (P, -)$ is exact and faithful, and every object $X$ of $C$ admits an epimorphism $P^{r} ↠ X$ for some $r \geq 0$ (Intrinsic finite category hypotheses give a finite projective generator, Projective object characterisations, Generator and cogenerator of a category, Projective object, Projective epimorphisms onto the simples generate every finite-length object).

[F4]

$A$ is a finite-dimensional unital $k$ -algebra, $H$ takes values in finite-dimensional left $A$ -modules, and the action of $a \in A$ on $h \in H (X)$ is $a \cdot h = h \circ a$ ; the component map $H (P^{m}) \to A^{m}$ , $h \mapsto ({p r}_{j} \circ h)_{j \leq m}$ , is an isomorphism of left $A$ -modules, because the left action of $A$ on ${End}_{C} (P)$ is $(a \cdot b) = b \circ a$ and $A$ acts componentwise on $A^{m}$ (Intrinsic finite category hypotheses give a finite projective generator, Endomorphisms of an object of a preadditive category form a ring, k-linear categories and k-linear functors).

[F5]

A sequence $P^{s} \overset{d}{\to} P^{r} \overset{q}{\to} X \to 0$ in an abelian category is exact exactly when $q$ is a cokernel of $d$ ; the cokernel universal property says that a morphism $g$ with $g d = 0$ factors uniquely as $g = u q$ (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers, Exact sequences and short exact sequences of modules, The quotient of an object by a subobject, Image and coimage in a category with kernels and cokernels).

[F6]

For a left module $M$ over a unital ring, the universal property of the direct sum identifies ${Hom}_{R} (R^{r}, M)$ with $M^{r}$ : an $R$ -linear map $R^{r} \to M$ is determined by, and may be prescribed arbitrarily on, the standard generators, and the correspondence is additive in the family (no pointwise left $R$ -module structure on ${Hom}_{R} (R^{r}, M)$ is asserted) (Universal property of a direct sum of modules, Generated submodule, cyclic and finitely generated modules, module basis and free module, Every module is a quotient of a free module).

[F7]

A finite-dimensional left $A$ -module $V$ is finitely generated: a finite $k$ -basis generates it, giving an epimorphism $A^{t} ↠ V$ ; its kernel is a submodule of the finite-dimensional $k$ -space $A^{t}$ , hence finite-dimensional and again finitely generated, so $V$ admits a finite presentation $A^{u} \to A^{t} \to V \to 0$ (Generated submodule, cyclic and finitely generated modules, module basis and free module, Every module is a quotient of a free module, Module homomorphism and isomorphism, kernel, image and cokernel, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

[F8]

A functor is an equivalence exactly when it is fully faithful and split essentially surjective, the splitting being supplied data (A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice, Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors, Equivalence, quasi-inverse, and adjoint equivalence of categories).

[F9]

An equivalence of categories preserves and reflects every existing limit and colimit, hence the zero object, kernels, cokernels, images and finite biproducts; a fully faithful functor reflects isomorphisms (Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense, Every fully faithful functor reflects isomorphisms, Initial object, terminal object, and zero object, Biproduct).

[F10]

In an abelian category a morphism is monic if and only if its kernel is zero and epic if and only if its cokernel is zero; the subobjects of an object are its monomorphisms modulo mutual factorisation, and the join of subobjects represented by $b : B ↣ A$ and $c : C ↣ A$ is the image inclusion of the map $[b, c] : B \oplus C \to A$ (In an abelian category, monic means zero kernel and epic means zero cokernel, Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms, The join of two subobjects in an abelian category, Image and coimage in a category with kernels and cokernels).

Proof

technique · direct

1.1F1F2F4given

(H is a $k$ -linear functor into $A - m o d$ .) The functor $H = C (P, -)$ sends an object $X$ to the hom-group $C (P, X)$ , which by [F1] is a finite-dimensional $k$ -vector space, and the action $a \cdot h = h \circ a$ of [F4] makes it a left $A$ -module; for a morphism $u : X \to Y$ , $H (u) = u \circ (-)$ is additive and $k$ -linear by [F1] and $A$ -linear because $u \circ (h \circ a) = (u \circ h) \circ a$ . Hence $H$ is a $k$ -linear functor $C \to A - m o d$ . Moreover for each $m$ the component map identifies $H (P^{m}) = C (P, P^{m})$ with $A^{m}$ as a left $A$ -module: it is a bijection by the product universal property of the biproduct [F2], and it is $A$ -linear since $(a \cdot h)_{j} = {p r}_{j} \circ h \circ a = a \cdot ({p r}_{j} \circ h)$ for the componentwise action on $A^{m}$ [F4].

1.2F3given

(Exactness and faithfulness.) By [F3] the object $P$ is projective, so $H = C (P, -)$ preserves kernels and cokernels of short exact sequences, that is, $H$ is exact; and $P$ is separating, which by [F3] says exactly that $H$ is injective on every hom-set, so $H$ is faithful.

1.3F9F10given

(The locally finite conditions transfer along the equivalence.) Let $D$ be a $k$ -linear abelian category and $E : A - m o d \to D$ a $k$ -linear equivalence, with quasi-inverse $E^{'}$ and unit and counit isomorphisms. By [F9] the functor $E$ preserves and reflects kernels, cokernels, images, finite biproducts and the zero object, and being fully faithful it reflects isomorphisms; by [F10] it therefore preserves and reflects monomorphisms and epimorphisms, and it carries a factor $M / N = coker (N ↣ M)$ to $E (M) / E (N) = coker (E (N) ↣ E (M))$ . The subobjects of $M$ are the monomorphisms into $M$ modulo mutual factorisation, so $E$ induces an order-preserving bijection between the subobjects of $M$ and of $E (M)$ , and since simplicity says that the only subobjects are the zero subobject and the identity, simplicity is preserved and reflected as well. Hence for $X \in D$ , written as $X ≅ E (M)$ with $M = E^{'} (X)$ via the counit, a composition series $0 = M_{0} < \dots < M_{ℓ} = M$ of $M$ , which exists because $M$ is a finite-dimensional left $A$ -module and $A - m o d$ is intrinsically finite (Finite-dimensional module categories satisfy the intrinsic finiteness conditions), maps to a strictly increasing chain $0 ≅ E (M_{0}) < \dots < E (M_{ℓ}) ≅ X$ whose successive factors $E (M_{i + 1}) / E (M_{i}) ≅ E (M_{i + 1} / M_{i})$ are simple, so $X$ has finite length in $D$ . Finally ${Hom}_{D} (E (M), E (N)) ≅ {Hom}_{A} (M, N)$ is a $k$ -linear bijection because $E$ is fully faithful, so every hom-space of $D$ is finite-dimensional.

2.1F2F5F6step 1.1step 1.2chooseconstruct

(Fullness.) Let $X, Y \in C$ and let $φ : H (X) \to H (Y)$ be $A$ -linear. By [F3] choose an epimorphism $q : P^{r} ↠ X$ and then an epimorphism $P^{s} ↠ \ker q$ , and let $d : P^{s} \to P^{r}$ be the composite with the inclusion $\ker q ↣ P^{r}$ , so that $P^{s} \overset{d}{\to} P^{r} \overset{q}{\to} X \to 0$ is exact and $q$ is a cokernel of $d$ by [F5]. Since $H$ is exact by step 1.2, $H (q) : H (P^{r}) \to H (X)$ is the cokernel of $H (d)$ and in particular an epimorphism, and $ψ : = φ \circ H (q) : H (P^{r}) \to H (Y)$ is $A$ -linear. Under the identification $H (P^{r}) ≅ A^{r}$ of step 1.1 the free-module universal property [F6] presents $ψ$ by the $r$ -tuple $y_{j} : = ψ ({i n j}_{j}) : P \to Y$ , and by the coproduct universal property of $P^{r} = ⨁_{j} P$ [F2] there is $g : P^{r} \to Y$ with $g \circ {i n j}_{j} = y_{j}$ . Then $ψ = H (g)$ : for $h : P \to P^{r}$ with components $a_{j} : = {p r}_{j} \circ h \in A$ , step 1.1 gives $h = \sum_{j} {i n j}_{j} \circ a_{j} = \sum_{j} a_{j} \cdot {i n j}_{j}$ , so $ψ (h) = \sum_{j} a_{j} \cdot ψ ({i n j}_{j}) = \sum_{j} y_{j} \circ a_{j} = g \circ h$ by $A$ -linearity and additivity of $ψ$ . Now $ψ \circ H (d) = φ \circ H (q) \circ H (d) = φ \circ H (q \circ d) = 0$ because $q \circ d = 0$ , that is $H (g \circ d) = 0$ , and $H$ is faithful by step 1.2, so $g \circ d = 0$ . By the cokernel universal property [F5] for $X = coker (d)$ there is $u : X \to Y$ with $g = u \circ q$ , and then $H (u) \circ H (q) = H (u \circ q) = H (g) = ψ = φ \circ H (q)$ ; since $H (q)$ is an epimorphism, $H (u) = φ$ . Hence every $A$ -linear map $H (X) \to H (Y)$ is $H (u)$ for some $u : X \to Y$ , so $H$ is full.

2.2F9F10step 1.3

(Simple classes and projective covers transfer.) Keep the equivalence $E : A - m o d \to D$ of step 1.3. Since $E$ is full, faithful and essentially surjective it induces a bijection between the isomorphism classes of objects of the two categories, and by step 1.3 it preserves and reflects simplicity, so $D$ has exactly as many isomorphism classes of simple objects as $A - m o d$ , namely finitely many by Finite-dimensional module categories satisfy the intrinsic finiteness conditions. For enough projectives let $S$ be a simple object of $D$ and write $S ≅ E (M)$ with $M$ simple by step 1.3; the finite-dimensional $A$ -module $M$ has a projective cover $π : Q ↠ M$ by Finite-dimensional module categories satisfy the intrinsic finiteness conditions, that is, an essential epimorphism with $Q$ projective in $A - m o d$ . The object $E (Q)$ is projective in $D$ : a lifting problem $f : E (Q) \to Z$ against an epimorphism $q : Y ↠ Z$ transports under the quasi-inverse $E^{'}$ — which preserves epimorphisms by [F9] — to a lifting problem for the projective $Q$ , and the lift transports back along $E$ using the naturality of the counit. Moreover $\ker E (π) ≅ E (\ker π)$ because $E$ preserves kernels, and $E (\ker π)$ is a superfluous subobject of $E (Q)$ : subobjects correspond bijectively under $E$ by step 1.3, $E$ preserves joins because a join is the image of a map out of a finite biproduct [F10], and the superfluity condition of Superfluous subobjects and projective covers in an abelian category is therefore carried across, so $E (π)$ is an essential epimorphism with projective source, a projective cover of $E (M) ≅ S$ .

3.1F5F7step 1.1step 2.1given

(Essential surjectivity.) Let $V$ be a finite-dimensional left $A$ -module. By [F7] $V$ admits a finite presentation $A^{u} \overset{δ}{\to} A^{t} \overset{ε}{\to} V \to 0$ with $V ≅ coker (δ)$ . By step 2.1 the functor $H$ is full and faithful, so it is bijective on hom-sets and, under the identification $H (P^{m}) ≅ A^{m}$ of step 1.1, the map ${Hom}_{C} (P^{u}, P^{t}) \to {Hom}_{A} (A^{u}, A^{t})$ , $d \mapsto H (d)$ , is a bijection; let $d : P^{u} \to P^{t}$ be the morphism with $H (d) = δ$ and put $X : = coker (d)$ , which exists because $C$ is abelian. Exactness of $H$ (step 1.2) gives $H (X) ≅ coker (H (d)) = coker (δ) ≅ V$ , so every finite-dimensional left $A$ -module is isomorphic to $H (X)$ for some $X \in C$ .

4.1F8step 1.1step 1.2step 2.1step 3.1givenconstruct

(Conclusion of the equivalence.) Steps 1.1, 1.2, 2.1 and 3.1 prove that $H$ is $k$ -linear, exact, fully faithful and essentially surjective, with $A$ finite-dimensional. For the further equivalence assertion assume supplied objects $X_{V}$ and isomorphisms $ε_{V} : H (X_{V}) \to V$ for every target module $V$ . Put $G (V) = X_{V}$ ; for $f : V \to W$ , fullness and faithfulness give a unique $G (f)$ satisfying $H (G (f)) = ε_{W}^{- 1} f ε_{V}$ . Uniqueness proves functoriality and $k$ -linearity, and [F8] gives the unit isomorphism and hence the equivalence. Step 3.1 establishes each witness separately; it does not choose this entire family.

5.1step 4.1step 1.3step 2.2F8F9F10given∎

(Conclusion.) Steps 1.3 and 2.2 show that a $k$ -linear equivalence from $A - m o d$ transfers local finiteness, finitely many simple classes and projective covers, hence intrinsic finiteness. In the other direction step 4.1 gives the equivalence when splitting data are supplied, and unconditionally gives the fully faithful, essentially surjective module-model functor. The objectwise arguments choose only finite bases, presentations, covers and lifts. The simultaneous splitting is additional data, not a consequence of finite choice.

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Finite Eilenberg–Watts for right exact linear functors

Statement

For assertions forming categories of functors, fix a set of allowed finite-dimensional $k$ -vector space structures containing $k$ and the underlying spaces of the algebras considered, and closed under finite biproducts, subspaces, quotients, $k$ -tensor products, $k$ -duals and spaces of linear maps. Allow every compatible algebra, module and bimodule structure on these spaces. The resulting module categories are small, so their functors and natural transformations are set-coded as required by Functor category $[C, D]$ . The objectwise formulas apply without this size restriction; no category of proper-class functors is asserted.

Let $A$ and $B$ be finite-dimensional unital algebras over a field $k$ , and let $A - m o d$ and $B - m o d$ be the categories of finite-dimensional left modules.

(i) For every finite-dimensional $(B, A)$ -bimodule $M$ the functor $T_{M} = M \otimes_{A} - : A - m o d \to B - m o d$ is well defined, $k$ -linear and right exact.

(ii) Conversely every $k$ -linear right exact functor $F : A - m o d \to B - m o d$ is naturally isomorphic to $T_{F (A)}$ , where $F (A)$ carries the $(B, A)$ -bimodule structure of $F (A)$ is a $(B, A)$ -bimodule for every additive functor $F$ ; explicitly the canonical comparison $τ_{X} : F (A) \otimes_{A} X \to F (X)$ , $τ_{X} (m \otimes x) = F (ℓ_{x}) (m)$ with $ℓ_{x} (a) = a x$ , is a natural isomorphism.

(iii) For finite-dimensional $(B, A)$ -bimodules $M, M^{'}$ the assignment $f \mapsto (f \otimes 1_{X})_{X}$ is a bijection ${Hom}_{B - A} (M, M^{'}) \to Nat (T_{M}, T_{M^{'}})$ , compatible with addition, identities and vertical composition.

(iv) Hence $M \mapsto T_{M}$ is an equivalence of categories between the category of finite-dimensional $(B, A)$ -bimodules with bimodule maps and the category of $k$ -linear right exact functors $A - m o d \to B - m o d$ with all natural transformations. No commutativity of $A$ or $B$ is assumed and no choice is used.

Facts & Assumptions

Given: The scalar and size conventions above, a field $k$ , finite-dimensional unital $k$ -algebras $A$ and $B$ , a finite-dimensional $(B, A)$ -bimodule $M$ , and a $k$ -linear right exact functor $F : A - m o d \to B - m o d$ on finite-dimensional left modules.

[F1]

For a unital ring $A$ and a right $A$ -module $M$ the functor $M \otimes_{A} -$ is additive, preserves cokernels, and hence is right exact; if $M$ is a $(B, A)$ -bimodule it takes values in left $B$ -modules and all the induced maps are $B$ -linear, with no commutativity and no choice (The functor $M \otimes_{A} -$ is additive, right exact, and preserves direct sums over an arbitrary unital ring).

[F2]

The tensor product $M \otimes_{A} X$ of a right $A$ -module with a left $A$ -module is a quotient of $M \otimes_{k} X$ ; when $M$ is a $(B, A)$ -bimodule and $X$ a left $A$ -module there is a unique left $B$ -module structure with $b (m \otimes x) = (b m) \otimes x$ , and for a left $A$ -linear $u : X \to Y$ the map $1_{M} \otimes u$ is $B$ -linear, functorial, additive and $k$ -homogeneous in $u$ (Universal property of the tensor product for balanced maps into abelian groups, A commuting outer scalar action descends to a tensor product, Module homomorphisms induce tensor-product homomorphisms functorially).

[F3]

For an additive functor $F : A - M o d \to B - M o d$ the module $F (A)$ carries a $(B, A)$ -bimodule structure with $m a = F (r_{a}) (m)$ for the right multiplications $r_{a}$ , commuting with the left $B$ -action, using only functoriality on the maps $r_{a} : A \to A$ ( $F (A)$ is a $(B, A)$ -bimodule for every additive functor $F$ , $(S, R)$ -bimodules and commuting left and right scalar actions).

[F4]

For an additive $F$ and $M = F (A)$ as in [F3], the pairing $β_{X} (m, x) = F (ℓ_{x}) (m)$ is balanced in $m$ and $B$ -linear, so it induces a $B$ -linear map $τ_{X} : M \otimes_{A} X \to F (X)$ that is natural in $X$ ; the computation uses only additivity and functoriality of $F$ on the maps $ℓ_{x} : A \to X$ and the universal property of the tensor product (The canonical comparison to the tensor functor of $F (A)$ is balanced and natural, Universal property of the tensor product for balanced maps into abelian groups).

[F5]

An additive functor between additive categories preserves finite biproducts (An additive functor preserves finite biproducts, Additive functor).

[F6]

For every left $A$ -module $X$ the unit map $ρ_{M^{'}} : M^{'} \otimes_{A} A \to M^{'}$ , $m \otimes a \mapsto m a$ , is an isomorphism natural in the right module $M^{'}$ (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[F7]

Every finite-dimensional left $A$ -module $X$ admits a finite presentation $A^{s} \overset{α}{\to} A^{r} \overset{β}{\to} X \to 0$ : finitely many generators give the surjection $β$ , and the kernel is a submodule of the finite-dimensional space $A^{r}$ , hence finite-dimensional, so finitely many of its generators give $α$ ; right exact functors preserve the exactness of this sequence (Every module is a quotient of a free module, Generated submodule, cyclic and finitely generated modules, module basis and free module, Exact sequences and short exact sequences of modules, Left exact and right exact functors, Module homomorphism and isomorphism, kernel, image and cokernel).

[F8]

Every natural transformation $η : T_{M} \Rightarrow T_{M^{'}}$ between tensor functors of $(B, A)$ -bimodules is determined by its component at $A$ : with $f = ρ_{M^{'}} \circ η_{A} \circ ρ_{M}^{- 1}$ one has $f (m a) = f (m) a$ , so $f$ is a $(B, A)$ -bimodule map, and $η_{X} = f \otimes 1_{X}$ for every left $A$ -module $X$ ; conversely every bimodule map $f$ gives such a natural transformation, and the two assignments are inverse bijections compatible with addition, identities and vertical composition, using only the maps $ℓ_{x}$ and the unit isomorphisms (Natural transformations between tensor functors are bimodule maps).

[F9]

The general comparison theorem proves that $τ$ is a natural isomorphism for every additive, right exact, coproduct-preserving functor on all modules, by the cokernel-universality argument on the canonical free presentation of an arbitrary module; its hypotheses are stronger than those available on $A - m o d$ , where $F$ is defined only on finite modules and only finite presentations occur (Canonical free presentations force the comparison to be an isomorphism).

Proof

technique · direct

1.1F2given

(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 $b (m \otimes x) = (b m) \otimes x$ ; 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 $A - m o d \to B - m o d$ .

1.2F3F4givenalgebra

(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 $ℓ_{x} : A \to X$ , $ℓ_{x} (a) = a x$ , is left $A$ -linear between finite-dimensional modules, so $F (ℓ_{x})$ 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 $τ_{X} : F (A) \otimes_{A} X \to F (X)$ , $τ_{X} (m \otimes x) = F (ℓ_{x}) (m)$ , 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.

2.1F1F2step 1.1

(i, right exactness.) Let $X \overset{u}{\to} Y \overset{v}{\to} Z \to 0$ be exact in $A - m o d$ . Applying [F1] gives the exact sequence $M \otimes_{A} X \overset{1 \otimes u}{\to} M \otimes_{A} Y \overset{1 \otimes v}{\to} M \otimes_{A} Z \to 0$ : 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.

2.2F5F6step 1.2

(ii, the comparison is an isomorphism on free modules.) For $X = A$ the map $τ_{A} : F (A) \otimes_{A} A \to F (A)$ sends $m \otimes a$ to $F (ℓ_{a}) (m) = F (r_{a}) (m) = m a$ by [F3], so it is the unit isomorphism $ρ_{F (A)}$ of [F6], an isomorphism. Both $F$ and $T_{F (A)}$ are additive and therefore preserve finite biproducts by [F5], and $τ$ is natural; hence for every $r \geq 0$ the map $τ_{A^{r}}$ is the direct sum of $r$ copies of $τ_{A}$ and is an isomorphism.

2.3F6F8step 1.1

(iii) Let $M, M^{'}$ be finite-dimensional $(B, A)$ -bimodules. Every natural transformation $η : T_{M} \Rightarrow T_{M^{'}}$ has, by [F8], the form $η_{X} = f \otimes 1_{X}$ for the bimodule map $f = ρ_{M^{'}} η_{A} ρ_{M}^{- 1} : M \to M^{'}$ , 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 $ℓ_{x}$ , and the unit isomorphisms) lies in the finite module categories, the classification restricts verbatim from all modules to $A - m o d$ .

3.1F7F9step 2.1step 1.2step 2.2algebra

(ii, isomorphism for all finite-dimensional $X$ .) Let $X \in A - m o d$ and choose a finite presentation $A^{s} \overset{α}{\to} A^{r} \overset{β}{\to} X \to 0$ by [F7]. By naturality of $τ$ and right exactness of $F$ and of $T_{F (A)}$ (step 2.1) there is a commutative diagram with exact rows comparing $τ$ 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 $q^{'}$ are the cokernel maps of $T (α)$ and $F (α)$ , then $τ_{X}$ is characterized by $τ_{X} q = q^{'} τ_{r}$ , and the map $s$ defined by $s q^{'} = q τ_{r}^{- 1}$ satisfies $s τ_{X} = 1$ and $τ_{X} s = 1$ after composing with the epimorphisms $q, q^{'}$ . 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 $A - m o d$ , so [F9] is not applied as a statement, but its argument is reproduced here with finite presentations. Hence $τ_{X}$ is an isomorphism, so $F ≅ T_{F (A)}$ naturally.

4.1F6step 3.1step 2.3

(iv) Define $Φ$ on finite-dimensional $(B, A)$ -bimodules by $Φ (M) = T_{M}$ 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 $T_{F (A)} ≅ F$ , 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: $F (A) \otimes_{A} A ≅ F (A)$ by [F6] and $T_{F (A)} ≅ F$ by step 3.1, so $Φ$ 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 $η : F \Rightarrow G$ , naturality at $ℓ_{x}$ gives $η_{X} τ_{X}^{F} (m \otimes x) = G (ℓ_{x}) (η_{A} (m)) = τ_{X}^{G} (η_{A} (m) \otimes x)$ . The tensor-unit isomorphisms are natural in $M$ by [F6].

5.1step 1.1step 2.1step 1.2step 3.1step 2.3step 4.1given∎

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.

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Finite left exact functors are Hom functors with dual bimodule kernels

Statement

Let $A, B$ be finite-dimensional unital algebras over a field $k$ . Let $_{A} A_{A}$ be the regular bimodule and let $A^{*} = {Hom}_{k} (A, k)$ be its $k$ -dual with the commuting actions $(a \cdot λ) (x) = λ (x a)$ and $(λ \cdot a) (x) = λ (a x)$ , regarded as a left $A$ -module. Let $F : A - m o d \to B - m o d$ be a $k$ -linear left exact functor and put $M = F (A^{*})$ with the right $A$ -action $m \cdot a = F (λ \mapsto λ \cdot a) (m)$ ; then $M$ is a finite-dimensional $(B, A)$ -bimodule. There is a natural isomorphism of left $B$ -modules

$F (X) ≅ {Hom}_{A} (M^{*}, X)$

for every finite-dimensional left $A$ -module $X$ , where $M^{*}$ is the $(A, B)$ -bimodule dual to $M$ and the left $B$ -action on the Hom is $(b φ) (u) = φ (u \cdot b)$ ; the isomorphism is natural in $X$ . Consequently $M \mapsto {Hom}_{A} (M^{*}, -)$ is an equivalence of categories between finite-dimensional $(B, A)$ -bimodules with bimodule maps and $k$ -linear left exact functors $A - m o d \to B - m o d$ with all natural transformations, with quasi-inverse $F \mapsto F (A^{*})$ . No commutativity of $A$ or $B$ and no choice are used.

Facts & Assumptions

Given: The scalar and size conventions above, a field $k$ , finite-dimensional unital $k$ -algebras $A$ and $B$ , the $k$ -dual $A^{*} = {Hom}_{k} (A, k)$ of the regular bimodule with the commuting actions displayed in the statement, and a $k$ -linear left exact functor $F : A - m o d \to B - m o d$ on finite-dimensional left modules.

[F1]

Duality $(-)^{*} = {Hom}_{k} (-, k)$ is a contravariant $k$ -linear functor on finite-dimensional modules, exact, with $X ≅ X^{* *}$ naturally; it is a contravariant equivalence between finite-dimensional left $A$ -modules and finite-dimensional left $A^{o p}$ -modules, and between finite-dimensional $(A, B)$ -bimodules and finite-dimensional $(B, A)$ -bimodules, and it carries the left/right module structures into one another (Finite module duality is exact with commuting bimodule actions, The opposite ring $R^{o p}$ , Unital left and right modules over a ring; unqualified module means left module).

[F2]

Every $k$ -linear right exact functor between finite-dimensional module categories over finite-dimensional algebras is naturally isomorphic to $T_{K} = K \otimes_{A^{'}} -$ for the bimodule kernel $K = G (A^{'})$ , and the assignment is an equivalence with quasi-inverse $G \mapsto G (A^{'})$ (Finite Eilenberg–Watts for right exact linear functors).

[F3]

If $Y$ is an $(A, A)$ -bimodule with compatible $k$ -actions, each right multiplication $t_{a} (y) = y a$ is left $A$ -linear. For a $k$ -linear $F : A - m o d \to B - m o d$ , the formulas $m a = F (t_{a}) (m)$ give a right $A$ -action on $F (Y)$ commuting with its left $B$ -action. Indeed $t_{a a^{'}} = t_{a^{'}} t_{a}$ , $t_{1} = 1$ , and $t_{a + a^{'}} = t_{a} + t_{a^{'}}$ ; the agreeing $k$ -actions follow from $F (t_{c 1_{A}}) = c 1_{F (Y)}$ ( $F (A)$ is a $(B, A)$ -bimodule for every additive functor $F$ , $(S, R)$ -bimodules and commuting left and right scalar actions).

[F4]

Tensor-hom adjunction: for a $(k, A^{o p})$ -bimodule $K$ , a left $A^{o p}$ -module $Y$ and a $k$ -vector space $V$ , a $k$ -linear map $K \otimes_{A^{o p}} Y \to V$ corresponds naturally to an $A^{o p}$ -linear map $Y \to {Hom}_{k} (K, V)$ ; equivalently, a balanced $k$ -bilinear pairing out of $K \times Y$ induces a unique map out of the tensor product (Tensor-Hom adjunction for bimodules over arbitrary unital rings, Universal property of the tensor product for balanced maps into abelian groups).

[F5]

A functor is left exact when it preserves every finite limit, in particular kernels, and right exact when it preserves finite colimits, in particular cokernels (Left exact and right exact functors).

[F6]

The Yoneda lemma identifies natural transformations ${Hom}_{A} (U, -) \Rightarrow G$ with elements of $G (U)$ ; concretely, a natural transformation between represented functors ${Hom}_{A} (U, -) \Rightarrow {Hom}_{A} (V, -)$ is determined by, and determined as precomposition with, a map $V \to U$ (The Yoneda bijection $Nat (C (a, -), F) ≅ F (a)$ is natural in both $a$ and $F$ ).

[F7]

For a finite-dimensional $(A, B)$ -bimodule $U$ , tensor-Hom adjunction gives $U \otimes_{B} - ⊣ {Hom}_{A} (U, -)$ , restricting to finite-dimensional modules because both tensor products and Hom-spaces remain finite-dimensional. Hence ${Hom}_{A} (U, -)$ preserves finite limits and is left exact; it is $k$ -linear by postcomposition and the agreeing scalar actions (Tensor-Hom adjunction for bimodules over arbitrary unital rings, Right adjoints preserve every limit that exists, k-linear categories and k-linear functors).

Proof

technique · direct

1.1F1F3given

The $k$ -dual $A^{*}$ with the actions $(a \cdot λ) (x) = λ (x a)$ and $(λ \cdot a) (x) = λ (a x)$ is a finite-dimensional $(A, A)$ -bimodule (the actions commute by associativity of multiplication in $A$ ), and $λ \mapsto λ \cdot a$ is left $A$ -linear for each $a$ , since $((b \cdot λ) \cdot a) (x) = λ (a x b) = (b \cdot (λ \cdot a)) (x)$ . Hence $F$ is defined on $A^{*}$ and, by [F3] applied to the bimodule $A^{*}$ in place of the regular module, $M = F (A^{*})$ is a finite-dimensional $(B, A)$ -bimodule, the right action being $m \cdot a = F (λ \mapsto λ \cdot a) (m)$ .

1.2F1F5given

Define $F^{d} (Y) = F (Y^{*})^{*}$ for finite-dimensional left $A^{o p}$ -modules $Y$ (equivalently right $A$ -modules): $Y^{*}$ is a finite-dimensional left $A$ -module by [F1], so $F (Y^{*})$ is in $B - m o d$ and its dual is a left $B^{o p}$ -module, so $F^{d}$ is a functor $A^{o p} - m o d \to B^{o p} - m o d$ , $k$ -linear by [F1]. It is right exact: if $Y_{1} \to Y_{2} \to Y_{3} \to 0$ is exact, then dualizing gives the exact sequence $0 \to Y_{3}^{*} \to Y_{2}^{*} \to Y_{1}^{*}$ by exactness of the duality [F1], left exactness of $F$ gives $0 \to F (Y_{3}^{*}) \to F (Y_{2}^{*}) \to F (Y_{1}^{*})$ , and dualizing again gives the exact sequence $F^{d} (Y_{1}) \to F^{d} (Y_{2}) \to F^{d} (Y_{3}) \to 0$ .

2.1F1F2step 1.1step 1.2

By [F2] applied to the finite-dimensional algebras $A^{o p}$ and $B^{o p}$ , the right exact $k$ -linear functor $F^{d}$ is naturally isomorphic to $K \otimes_{A^{o p}} -$ with kernel $K = F^{d} (A^{o p})$ , a finite-dimensional $(B^{o p}, A^{o p})$ -bimodule. The dual of the regular left $A^{o p}$ -module is $A^{*}$ with the left $A$ -action $(a \cdot λ) (x) = λ (x a)$ of step 1.1, so $K = F (A^{*})^{*} = M^{*}$ under the identifications of [F1]; thus $K$ is the $(A, B)$ -bimodule dual of $M$ .

3.1F1step 1.2step 2.1

For $X \in A - m o d$ double duality of [F1] gives $F (X) ≅ F (X^{* *}) ≅ F^{d} (X^{*})^{*}$ , and step 2.1 gives $F^{d} (X^{*}) ≅ K \otimes_{A^{o p}} X^{*}$ ; hence $F (X) ≅ (K \otimes_{A^{o p}} X^{*})^{*}$ , naturally in $X$ .

3.2F1F4step 1.1step 2.1algebra

There is a natural left $B$ -module isomorphism $(K \otimes_{A^{o p}} X^{*})^{*} ≅ {Hom}_{A} (M^{*}, X)$ . Here $K = M^{*}$ is a left $A$ -module and right $B$ -module, and is regarded as a right $A^{o p}$ -module by $u \cdot a^{o p} = a u$ ; $X^{*}$ is a left $A^{o p}$ -module by $a^{o p} λ = λ \cdot a$ . For $A$ -linear $φ : M^{*} \to X$ , the pairing $u \otimes λ \mapsto λ (φ (u))$ is balanced since $λ (φ (a u)) = λ (a φ (u)) = (λ \cdot a) (φ (u))$ . Conversely a functional $ω$ defines $u \mapsto [λ \mapsto ω (u \otimes λ)]$ into $X^{* *} ≅ X$ , and balancing makes this map $A$ -linear. These constructions are inverse, $k$ -linear and natural in $X$ . The tensor product carries a right $B$ -action $(u \otimes λ) b = (u b) \otimes λ$ , so its dual has left action $(b ω) (u \otimes λ) = ω (u b \otimes λ)$ . This corresponds to $(b φ) (u) = φ (u b)$ , proving $B$ -linearity.

4.1step 1.1step 3.1step 3.2

Combining steps 3.1 and 3.2 gives a natural isomorphism $F (X) ≅ {Hom}_{A} (M^{*}, X)$ of left $B$ -modules for every finite-dimensional left $A$ -module $X$ , with $M = F (A^{*})$ a finite-dimensional $(B, A)$ -bimodule by step 1.1.

5.1F1F6F7step 4.1

(Equivalence on hom-categories.) The assignment lands in $k$ -linear left exact functors by [F7]. For finite-dimensional $(B, A)$ -bimodules $M, N$ , a natural transformation $η : {Hom}_{A} (M^{*}, -) \Rightarrow {Hom}_{A} (N^{*}, -)$ with $B$ -linear components corresponds by the Yoneda computation [F6] to the map $f = η_{M^{*}} (1_{M^{*}}) : N^{*} \to M^{*}$ , which is $A$ -linear by construction and right $B$ -linear: naturality at $r_{b} : M^{*} \to M^{*}$ gives $r_{b}^{M^{*}} f = η_{M^{*}} (r_{b}^{M^{*}})$ , while $B$ -linearity of $η_{M^{*}}$ gives $η_{M^{*}} (r_{b}^{M^{*}}) = b f = f r_{b}^{N^{*}}$ ; conversely every $(A, B)$ -bimodule map $N^{*} \to M^{*}$ gives such a natural transformation by precomposition. By the bimodule duality [F1] these correspond bijectively to $(B, A)$ -bimodule maps $M \to N$ ; so the assignment $M \mapsto {Hom}_{A} (M^{*}, -)$ is full and faithful. It is essentially surjective by step 4.1: every $k$ -linear left exact $F$ is naturally isomorphic to ${Hom}_{A} (F (A^{*})^{*}, -)$ with $F (A^{*})$ a $(B, A)$ -bimodule. Hence the assignment is an equivalence of categories, and it has quasi-inverse $F \mapsto F (A^{*})$ : on objects this returns $F$ up to the isomorphism of step 4.1, and on morphisms it sends a natural transformation to its component at $A^{*}$ , a $(B, A)$ -bimodule map by naturality against the right-action maps of $A^{*}$ . The other composite is naturally isomorphic to $M$ : ${Hom}_{A} (M^{*}, A^{*}) ≅ M$ sends $φ$ to the unique $m$ with $u (m) = φ (u) (1)$ for every $u \in M^{*}$ , using double duality. Its inverse sends $m$ to $u \mapsto [a \mapsto u (m a)]$ ; these formulas respect both actions and are natural in $M$ .

6.1step 1.1step 4.1step 5.1given∎

Steps 1.1, 4.1 and 5.1 prove the statement: $M = F (A^{*})$ is a finite-dimensional $(B, A)$ -bimodule, $F (X) ≅ {Hom}_{A} (M^{*}, X)$ naturally in $X$ , and $M \mapsto {Hom}_{A} (M^{*}, -)$ is an equivalence with quasi-inverse $F \mapsto F (A^{*})$ . No commutativity of $A$ or $B$ was used, and all dualities, tensor products and presentations involved are finite-dimensional, so no choice is used.

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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.

CorollaryStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Exact finite tensor functors have projective right-module kernels

Statement

Let $A, B$ be finite-dimensional unital algebras over a field $k$ and let $M$ be a finite-dimensional $(B, A)$ -bimodule with associated tensor functor $T_{M} = M \otimes_{A} - : A - m o d \to B - m o d$ . Then the following are equivalent: (1) $T_{M}$ is exact on finite-dimensional left $A$ -modules, equivalently $M$ is flat as a right $A$ -module; (2) $M$ is a projective right $A$ -module. Since $M$ is finite-dimensional, (2) is also equivalent to $M$ being a direct summand of a finite free right $A$ -module and to $M$ being finitely generated and projective. No choice is used.

Facts & Assumptions

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 $T_{M} = M \otimes_{A} - : A - m o d \to B - m o d$ .

[F1]

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).

[F2]

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).

[F3]

For a $(B, A)$ -bimodule $N$ and finite-dimensional left $A$ -modules $X$ , duality and the tensor-hom adjunction give a natural isomorphism $(N \otimes_{A} X)^{*} ≅ {Hom}_{A^{o p}} (N, X^{*})$ of left $B^{o p}$ -modules: a functional $ω$ corresponds to $u \mapsto (x \mapsto ω (u \otimes x))$ , and the balancing relation for $ω$ is exactly $A^{o p}$ -linearity of that map, because $u \cdot a$ denotes the right $A$ -action on $N$ and $λ \cdot a$ the right $A$ -action on $X^{*}$ (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).

[F4]

Duality $X \mapsto X^{*}$ is an exact contravariant equivalence between finite-dimensional left $A$ -modules and finite-dimensional left $A^{o p}$ -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).

[F5]

The category $A - m o d$ is abelian; a finite-dimensional left $A$ -module is finitely generated, and a finite-dimensional right $A$ -module has a finite free cover $A^{n} ↠ N$ 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).

[F6]

If $N$ is a direct summand of the free right module $A^{n}$ with inclusion $h : N \to A^{n}$ and retraction $p : A^{n} ↠ N$ , then $N$ is projective: given a surjection $e : E ↠ N^{'}$ and $f : N \to N^{'}$ , lift $f p : A^{n} \to N^{'}$ 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).

Proof

technique · direct

1.1F1F2

(2) $\Rightarrow$ (1). If $M$ is a projective right $A$ -module, then $M$ is flat by [F1], that is, $M \otimes_{A} -$ is exact on left $A$ -modules; restricting to finite-dimensional left modules, $T_{M}$ is exact on $A - m o d$ . Equivalently, $M$ is a direct summand of a free right module by [F2], tensoring with a free module is a direct sum of copies of the identity, and a direct summand of an exact functor is exact.

1.2F3F4

((1), transport of exactness.) Suppose $T_{M}$ is exact on finite-dimensional left $A$ -modules. By [F3] there are natural isomorphisms $(M \otimes_{A} X)^{*} ≅ {Hom}_{A^{o p}} (M, X^{*})$ for finite-dimensional left $A$ -modules $X$ ; since $X \mapsto X^{*}$ is an exact contravariant equivalence between $A - m o d$ and $A^{o p} - m o d$ by [F4], exactness of $M \otimes_{A} -$ on the finite left modules is equivalent to exactness of ${Hom}_{A^{o p}} (M, -)$ on the finite right $A$ -modules.

2.1F5F6step 1.2chooseconstruct

((1) $\Rightarrow$ (2), the cover splits.) Keep the hypothesis of step 1.2. Since $M$ is finite-dimensional it is finitely generated as a right $A$ -module, so a finite $k$ -basis of $M$ induces a surjection $p : A^{n} ↠ M$ of right $A$ -modules by [F5]. Exactness of ${Hom}_{A^{o p}} (M, -)$ at this surjection gives that ${Hom}_{A^{o p}} (M, A^{n}) \to {Hom}_{A^{o p}} (M, M)$ is surjective, so the identity of $M$ lifts to $h : M \to A^{n}$ with $p h = 1_{M}$ ; hence $M$ is a direct summand of the finite free right $A$ -module $A^{n}$ , and is projective by [F6].

3.1F1F2F5step 1.1step 2.1

The finite-generation clause: a finite-dimensional module is finitely generated by [F5]; conversely a finitely generated projective right module is a direct summand of a finite free module by [F2], giving the stated equivalences. Steps 1.1 and 2.1 prove (2) $\Rightarrow$ (1) and (1) $\Rightarrow$ (2), so the exactness of $T_{M}$ , the flatness of $M$ , the projectivity of $M$ , and the finite-summand and finite-generation conditions are all equivalent.

4.1step 1.1step 1.2step 2.1step 3.1given∎

All covers, bases and summands used above are finite (they come from finite $k$ -bases of finite-dimensional modules), and no commutativity of $A$ or $B$ was used, so no choice is used.

CorollaryStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Finite Eilenberg–Watts is a biequivalence

Statement

Let $A, B$ range over finite-dimensional unital algebras over a field $k$ , and consider the assignment $Φ (A) = A - m o d$ , $Φ (M) = T_{M} = M \otimes_{A} -$ , $Φ (f) = f \otimes 1$ on finite-dimensional bimodules and bimodule maps. (i) For all $A, B$ the assignment $M \mapsto T_{M}$ is an equivalence of categories between finite-dimensional $(B, A)$ -bimodules with bimodule maps and $k$ -linear right exact functors $A - m o d \to B - m o d$ with all natural transformations, so it is full, faithful and essentially surjective. (ii) The composition and unit comparisons of the pseudofunctor Tensoring defines a schematic pseudofunctor with interchange restrict to finite-dimensional algebras and finite-dimensional modules: the associativity isomorphism $N \otimes_{B} (M \otimes_{A} -) ≅ (N \otimes_{B} M) \otimes_{A} -$ with components $N \otimes_{B} (M \otimes_{A} X) ≅ (N \otimes_{B} M) \otimes_{A} X$ and the inverse unit isomorphism $X ≅ A \otimes_{A} X$ are natural isomorphisms of finite-dimensional modules, and the pseudofunctor coherence equations remain satisfied. Consequently these data define a biequivalence from the bicategory of finite-dimensional algebras, finite-dimensional bimodules and bimodule maps to the 2-category of finite-dimensional module categories, $k$ -linear right exact functors and natural transformations. No commutativity and no choice are used.

Facts & Assumptions

Given: A field $k$ and the assignment $Φ$ on finite-dimensional unital $k$ -algebras, finite-dimensional bimodules and bimodule maps described in the statement.

[F1]

For finite-dimensional unital $k$ -algebras $A, B$ , the assignment $M \mapsto T_{M}$ is an equivalence of categories between finite-dimensional $(B, A)$ -bimodules with bimodule maps and $k$ -linear right exact functors $A - m o d \to B - m o d$ with all natural transformations; in particular it is full and faithful with $Nat (T_{M}, T_{M^{'}}) ≅ {Hom}_{B - A} (M, M^{'})$ and essentially surjective (Finite Eilenberg–Watts for right exact linear functors, Equivalence, quasi-inverse, and adjoint equivalence of categories, Natural transformation and its components).

[F2]

The explicit tensor comparison maps in Tensoring defines a schematic pseudofunctor with interchange have composition comparison $T_{N} \circ T_{M} ≅ T_{N \otimes_{B} M}$ built from the associativity isomorphism $N \otimes_{B} (M \otimes_{A} X) ≅ (N \otimes_{B} M) \otimes_{A} X$ , identity comparison the inverse unit isomorphism $X ≅ A \otimes_{A} X$ , and coherence given by the pentagon and triangle diagrams; horizontal composition corresponds to tensoring bimodule maps (Tensoring defines a schematic pseudofunctor with interchange, The Morita bicategory of rings and bimodules, Bicategories, pseudofunctors, and biequivalences).

[F3]

The finite-dimensional module categories $A - m o d$ are well formed and the right exact $k$ -linear functors between them with all natural transformations form a strict 2-category, with composition of functors and identity transformations as structure (Finite-dimensional module categories satisfy the intrinsic finiteness conditions, Strict 2-category, Functor category $[C, D]$ , Natural transformation and its components, Left exact and right exact functors); the restriction of the Morita bicategory to finite-dimensional algebras and finite-dimensional bimodules is a bicategory, since tensor products of finite-dimensional bimodules over finite-dimensional algebras are again finite-dimensional and the associators and unitors are the same isomorphisms (The Morita bicategory of rings and bimodules, $(S, R)$ -bimodules and commuting left and right scalar actions).

[F4]

A pseudofunctor is a biequivalence when each local functor on hom-categories is an equivalence of categories and every object of the target is equivalent to $F Y$ for some object $Y$ of the source (Bicategories, pseudofunctors, and biequivalences).

Proof

technique · direct

1.1F1

(i) By [F1] the assignment $M \mapsto T_{M}$ on finite-dimensional $(B, A)$ -bimodules is full and faithful and essentially surjective onto the $k$ -linear right exact functors $A - m o d \to B - m o d$ , so it is an equivalence of categories for the given $A, B$ .

1.2F2F3

(ii, restriction of the pseudofunctor.) Let $A, B, C$ be finite-dimensional unital $k$ -algebras and let $M$ be a finite-dimensional $(B, A)$ -bimodule, $N$ a finite-dimensional $(C, B)$ -bimodule. Use the explicit maps of [F2]: the comparison 2-cells of $Φ$ are the associativity isomorphism $N \otimes_{B} (M \otimes_{A} X) ≅ (N \otimes_{B} M) \otimes_{A} X$ and the inverse unit isomorphism $X ≅ A \otimes_{A} X$ ; for finite-dimensional $X$ all objects occurring are quotients of finite tensor products of finite-dimensional spaces, hence finite-dimensional, and tensoring finite-dimensional bimodules over finite-dimensional algebras gives a finite-dimensional bimodule, so the source and target data together with these comparisons lie in the smaller bicategory and 2-category of [F3]. The coherence equations hold directly: on an elementary tensor every associativity path sends each nested tensor to the same reassociation of its factors, and every unit path multiplies the same adjacent unit factor. Elementary tensors generate the iterated tensor products, so equality there proves the required equations. Horizontal composition corresponds to tensoring bimodule maps under these comparisons, since both send $n \otimes m \otimes x$ to $g (n) \otimes f (m) \otimes x$ . The set-sized construction here uses the supplier's explicit maps and componentwise equations.

2.1F3F4step 1.1step 1.2

(ii, biequivalence.) The restricted assignment is a pseudofunctor by step 1.2. Its local functor at $(A, B)$ is the equivalence of step 1.1, so every local functor is an equivalence of categories; and every object $A - m o d$ of the target is $Φ (A)$ for the finite-dimensional algebra $A$ itself, so essential surjectivity holds trivially. Hence by [F4] the restriction is a biequivalence between the bicategory of finite-dimensional algebras, finite-dimensional bimodules and bimodule maps and the 2-category of finite-dimensional module categories, $k$ -linear right exact functors and natural transformations.

3.1step 1.1step 1.2step 2.1given∎

Steps 1.1 and 2.1 prove (i) and (ii). All algebras, bimodules, modules and comparison isomorphisms occurring above are finite-dimensional, no commutativity of any ring was used, and no choice is used.

5 · Examples, counterexamples and false statements

None yet.

Finite Abelian Categories and Eilenberg–Watts

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Superfluous subobjects and projective covers in an abelian category

Definition

Finite module duality is exact with commuting bimodule actions

Statement

Facts & Assumptions

Proof

Finite-support families of finite-dimensional vector spaces are locally finite but not finite

Statement

Facts & Assumptions

Proof

Projective epimorphisms onto the simples generate every finite-length object

Statement

Facts & Assumptions

Proof

Finite-dimensional module categories satisfy the intrinsic finiteness conditions

Statement

Facts & Assumptions

Proof

Intrinsic finite category hypotheses give a finite projective generator

Statement

Facts & Assumptions

Proof

Finite abelian categories admit finite-dimensional module models

Statement

Facts & Assumptions

Proof

Finite Eilenberg–Watts for right exact linear functors

Statement

Facts & Assumptions

Proof

Finite left exact functors are Hom functors with dual bimodule kernels

Statement

Facts & Assumptions

Proof

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

Statement

Facts & Assumptions

Proof

Exact finite tensor functors have projective right-module kernels

Statement

Facts & Assumptions

Proof

Finite Eilenberg–Watts is a biequivalence

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources