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The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful

Statement

Let $C$ be a locally small cocomplete abelian category and $P$ a small projective generator of $C$ . Put $E = {End}_{C} (P)$ , $A = E^{o p}$ , and $H = C (P, -) : C \to A -Mod$ , where $H (Y)$ carries the left $A$ -action $(e^{o p} \cdot h) = h \circ e$ . Then:

$H$ is an additive functor.
$H$ is exact: it preserves kernels, cokernels, and every finite limit and colimit that exists in $C$ .
$H$ preserves every set-indexed coproduct.
$H$ is faithful: for all $u \neq v : X \to Y$ there is $h : P \to X$ with $u h \neq v h$ .
If $H (Z) = 0$ then $Z = 0$ . No choice is used.

Facts & Assumptions

Given: A locally small cocomplete abelian category $C$ , a small projective generator $P$ of $C$ , $E = {End}_{C} (P)$ , $A = E^{o p}$ , and $H = C (P, -)$ with the left $A$ -action $(e^{o p} \cdot h) = h \circ e$ on $H (Y) = C (P, Y)$ .

[F1]

$P$ is projective, is a generator, and $C (P, -)$ preserves every set-indexed coproduct; $C$ is locally small, cocomplete and abelian (Small projective generators and progenerators, Abelian category, Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

$E = C (P, P)$ is a unital ring under addition and composition with identity $1_{P}$ , composition is bilinear, and $A = E^{o p}$ is a unital ring; left $A$ -modules and their homomorphisms form the category $A -Mod$ (Endomorphisms of an object of a preadditive category form a ring, Preadditive category, Left modules over a fixed ring and module homomorphisms form the large locally small category $R - M o d$ ).

[F3]

The hom-assignment $C (P, -)$ is a functor whose values are abelian groups and whose action on morphisms is postcomposition $u \mapsto u_{*}$ , where $u_{*} (h) = u \circ h$ , and postcomposition is additive (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category, The hom-bifunctor of a preadditive category takes values in abelian groups).

[F4]

Because $P$ is projective, every short exact sequence $0 \to K \to E^{'} \to M \to 0$ in $C$ induces an exact sequence $0 \to H (K) \to H (E^{'}) \to H (M) \to 0$ ; equivalently every epimorphism onto $P$ splits (Projective object characterisations).

[F5]

The covariant hom-functor $C (P, -)$ preserves every existing finite limit (Hom functors on a preadditive category are left exact).

[F6]

Because $P$ is a generator and $C$ is a locally small abelian category satisfying AB3, the functor $C (P, -)$ is faithful, and for every object $Z$ the canonical morphism $∐_{u \in C (P, Z)} P \to Z$ is an epimorphism (The cancellation and epimorphism descriptions of a generator agree, The axioms AB3 and AB3*, Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

[F7]

A one-object coproduct is canonically the object itself, and the empty coproduct is an initial object (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations, Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects).

[F8]

For every object $Z$ , the identity $1_{Z}$ is a cokernel of the zero morphism $0 \to Z$ out of the zero object (The cokernel of the zero map out of the zero object is the target, and dually for kernels).

[F9]

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

[F10]

A functor between additive categories is additive when its maps on hom-groups are group homomorphisms (Additive functor); a functor is exact when it is additive, left exact and right exact, and one-sided exactness means preservation of the corresponding finite limits or colimits (Exact functor between abelian categories, Left exact and right exact functors).

[F11]

An additive functor between abelian categories is exact exactly when it preserves kernels and cokernels (An additive functor is exact exactly when it preserves kernels and cokernels).

Proof

technique · direct

1.1F2F3F10given

( $H$ is an additive functor to $A -Mod$ .) By [F2] the ring $A = E^{o p}$ is unital with identity $1_{P}^{o p}$ and composition in $E$ is bilinear. For $Y \in C$ , $H (Y) = C (P, Y)$ is an abelian group by [F3], and the prescription $e^{o p} \cdot h : = h \circ e$ makes it a left $A$ -module: $((e + f)^{o p}) \cdot h = h \circ (e + f) = h \circ e + h \circ f$ and $(e^{o p} + f^{o p}) \cdot h$ agree, $(e^{o p} f^{o p}) \cdot h = ((f e)^{o p}) \cdot h = h \circ (f e) = (h \circ f) \circ e = e^{o p} \cdot (f^{o p} \cdot h)$ , and $1_{P}^{o p} \cdot h = h \circ 1_{P} = h$ . For $u : X \to Y$ , postcomposition $u_{*} = H (u)$ is additive by [F3] and $A$ -linear: $u_{*} (e^{o p} \cdot h) = u \circ h \circ e = e^{o p} \cdot u_{*} (h)$ . Identities and composites are preserved because postcomposition is associative and unital, so $H$ is a functor $C \to A -Mod$ whose hom-maps are group homomorphisms, that is, an additive functor by [F10].

1.2F6given

( $H$ is faithful.) By [F6] the functor $C (P, -)$ is faithful: for all $u \neq v : X \to Y$ there is $h : P \to X$ with $u \circ h \neq v \circ h$ . The underlying functions of $H (u)$ and $C (P, -) (u)$ coincide, so $H (u) \neq H (v)$ , and $H$ is faithful in the stated elementwise form.

1.3F6F7F8F9givenalgebra

(Nonzero objects are detected.) Suppose $H (Z) = 0$ , so $H (Z)$ is the zero group and every morphism $P \to Z$ is zero. By [F6] the canonical morphism $f : ∐_{u \in C (P, Z)} P \to Z$ is an epimorphism; its index set is the singleton ${0_{P, Z}}$ , so by [F7] the coproduct is $P$ and $f$ is the zero morphism $0 : P \to Z$ . The image of $0 : P \to Z$ is the zero subobject, the same image as that of the zero morphism $0 \to Z$ out of the zero object, so $coker f ≅ coker (0 \to Z) ≅ Z$ by [F8]. Since $f$ is epic, [F9] gives $coker f = 0$ ; hence $Z ≅ 0$ , that is, $Z = 0$ .

2.1F1step 1.1given

( $H$ preserves coproducts.) By clause (iii) of [F1], the functor $C (P, -)$ preserves every set-indexed coproduct, and the additional $A$ -module structure of step 1.1 only adds structure to the values: the canonical comparison maps for $H$ have the same underlying group homomorphisms as those of $C (P, -)$ , which are isomorphisms. Hence $H$ preserves every set-indexed coproduct.

2.2F4step 1.1given

( $H$ carries short exact sequences to short exact sequences.) Let $0 \to K \to E^{'} \to M \to 0$ be short exact in $C$ . By [F4] the induced sequence $0 \to H (K) \to H (E^{'}) \to H (M) \to 0$ is exact; the maps are $A$ -linear by step 1.1, and $H$ is additive by step 1.1.

3.1F5step 2.2given

( $H$ preserves kernels and cokernels.) Let $g : X \to Y$ be a morphism of $C$ . Applying step 2.2 to $0 \to \ker g \to X \to im g \to 0$ and using left exactness [F5] identifies $H (\ker g)$ with $\ker H (g)$ ; applying step 2.2 to $0 \to im g \to Y \to coker g \to 0$ and using that $H (X) \to H (im g)$ is surjective with kernel $H (\ker g)$ identifies $H (coker g)$ with $coker H (g)$ . Hence $H$ preserves kernels and cokernels.

4.1F5F10F11step 1.1step 3.1

( $H$ is exact.) By step 1.1, $H$ is additive, by [F5] it is left exact, and step 3.1 with [F11] makes it right exact as well: an additive functor preserving kernels and cokernels is exact. By the definitions of [F10] this means precisely that $H$ preserves every finite limit and every finite colimit that exists in $C$ , in addition to the kernels and cokernels just exhibited.

5.1step 1.1step 1.2step 1.3step 2.1step 2.2step 3.1step 4.1∎

Steps 1.1-1.3 establish that $H$ is an additive functor and is faithful, and show that $H (Z) = 0$ forces $Z = 0$ ; step 2.1 gives coproduct preservation, and steps 2.2, 3.1 and 4.1 give exactness with preservation of kernels, cokernels and finite limits and colimits. Every construction uses only the given category, the object $P$ and postcomposition, so no element is chosen and no choice principle is used.

Depends on

Used by

Dependency tree · two levels

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

Sources

P. Etingen, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, printed p.10 (projectivity gives exactness, the generator condition gives faithfulness) (standard reference, not scraped)
W. Crawley-Boevey, Noncommutative Algebra, §3.12 (finitely generated projective generator; faithfulness argument for Hom(P,-)) (standard reference, not scraped)