Section 12.29 (015Y): Injectives and adjoint functors

12.29 Injectives and adjoint functors

Here are some lemmas on adjoint functors and their relationship with injectives. See also Lemma 12.7.4.

slogan

Lemma 12.29.1. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors with $u$ right adjoint to $v$ . Consider the following conditions:

$v$ transforms injective maps into injective maps,
$v$ is exact, and
$u$ transforms injectives into injectives.

Then (a) $\Leftrightarrow$ (b) $\Rightarrow$ (c). If $\mathcal{A}$ has enough injectives, then all three conditions are equivalent.

Proof. Observe that $v$ is right exact as a left adjoint (Categories, Lemma 4.24.6). Combined with Lemma 12.7.2 this explains why (a) $\Leftrightarrow$ (b).

Assume (a). Let $I$ be an injective object of $\mathcal{A}$ . Let $\varphi : N \to M$ be an injective map in $\mathcal{B}$ and let $\alpha : N \to uI$ be a morphism. By adjointness we get a morphism $\alpha : vN \to I$ and by assumption $v\varphi : vN \to vM$ is injective. Hence as $I$ is an injective object we get a morphism $\beta : vM \to I$ extending $\alpha$ . By adjointness again this corresponds to a morphism $\beta : M \to uI$ extending $\alpha$ . Hence (c) is true.

Assume $\mathcal{A}$ has enough injectives and (c) holds. Let $f : B \to B'$ be a monomorphism in $\mathcal{B}$ , and let $A = \mathop{\mathrm{Ker}}(v(f))$ . Choose a monomorphism $g : A \to I$ with $I$ injective (it exists by assumption). Then $g$ extends to $g' : v(B) \to I$ , whence by adjunction a morphism $B \to u(I)$ . Since $u(I)$ is injective, this morphism extends to $h : B' \to u(I)$ , hence by adjunction a morphism $k : v(B') \to I$ extending $g'$ . But then $k$ "extends” $g$ , which forces $A = 0$ since $g$ was a monomorphism. Thus (a) is true. $\square$

Remark 12.29.2. Let $R \to S$ be a ring map. Let $u : \text{Mod}_ S \to \text{Mod}_ R$ be $u(N) = N_ R$ and let $v : \text{Mod}_ R \to \text{Mod}_ S$ be $v(M) = M \otimes _ R S$ . Then $u$ is right adjoint to $v$ , and $u$ is exact and $v$ is right exact. But conditions (a), (b), (c) of Lemma 12.29.1 do not hold in general. For example, if $R = \mathbf{Z}$ and $S = \mathbf{Z}/p\mathbf{Z}$ , then the injective $S$ -module $\mathbf{Z}/p\mathbf{Z}$ is not an injective $\mathbf{Z}$ -module. In fact, the lemma shows all injective $S$ -modules are injective as $R$ -modules if and only if $R \to S$ is a flat ring map.

Lemma 12.29.3. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume

$u$ is right adjoint to $v$ ,
$v$ transforms injective maps into injective maps,
$\mathcal{A}$ has enough injectives, and
$vB = 0$ implies $B = 0$ for any $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ .

Then $\mathcal{B}$ has enough injectives.

Proof. Pick $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ . Pick an injection $vB \to I$ for $I$ an injective object of $\mathcal{A}$ . According to Lemma 12.29.1 and the assumptions the corresponding map $B \to uI$ is the injection of $B$ into an injective object. $\square$

Remark 12.29.4. Let $\mathcal{A}$ , $\mathcal{B}$ , $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be as In Lemma 12.29.3. In the presence of conditions (1) and (2) condition (4) is equivalent to $v$ being faithful. Moreover, condition (4) is needed. An example is to consider the case where the functors $u$ and $v$ are both the zero functor.

Lemma 12.29.5. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume

$u$ is right adjoint to $v$ ,
$v$ transforms injective maps into injective maps,
$\mathcal{A}$ has enough injectives,
$vB = 0$ implies $B = 0$ for any $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ , and
$\mathcal{A}$ has functorial injective embeddings.

Then $\mathcal{B}$ has functorial injective embeddings.

Proof. Let $A \mapsto (A \to J(A))$ be a functorial injective embedding on $\mathcal{A}$ . Then $B \mapsto (B \to uJ(vB))$ is a functorial injective embedding on $\mathcal{B}$ . Compare with the proof of Lemma 12.29.3. $\square$

Lemma 12.29.6. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ be a functor. If there exists a subset $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ such that

every object of $\mathcal{B}$ is a quotient of an element of $\mathcal{P}$ , and
for every $P \in \mathcal{P}$ there exists an object $Q$ of $\mathcal{A}$ such that $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(Q, A) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P, u(A))$ functorially in $A$ ,

then there exists a left adjoint $v$ of $u$ .

Proof. By the Yoneda lemma (Categories, Lemma 4.3.5) the object $Q$ of $\mathcal{A}$ corresponding to $P$ is defined up to unique isomorphism by the formula $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(Q, A) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P, u(A))$ . Let us write $Q = v(P)$ . Denote $i_ P : P \to u(v(P))$ the map corresponding to $\text{id}_{v(P)}$ in $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P), v(P))$ . Functoriality in (2) implies that the bijection is given by

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P), A) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P, u(A)),\quad \varphi \mapsto u(\varphi ) \circ i_ P$

For any pair of elements $P_1, P_2 \in \mathcal{P}$ there is a canonical map

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_2, P_1) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P_2), v(P_1)),\quad \varphi \mapsto v(\varphi )$

which is characterized by the rule $u(v(\varphi )) \circ i_{P_2} = i_{P_1} \circ \varphi$ in $\mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_2, u(v(P_1)))$ . Note that $\varphi \mapsto v(\varphi )$ is compatible with composition; this can be seen directly from the characterization. Hence $P \mapsto v(P)$ is a functor from the full subcategory of $\mathcal{B}$ whose objects are the elements of $\mathcal{P}$ .

Given an arbitrary object $B$ of $\mathcal{B}$ choose an exact sequence

$P_2 \to P_1 \to B \to 0$

which is possible by assumption (1). Define $v(B)$ to be the object of $\mathcal{A}$ fitting into the exact sequence

$v(P_2) \to v(P_1) \to v(B) \to 0$

Then

$\begin{align*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(B), A) & = \mathop{\mathrm{Ker}}(\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P_1), A) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P_2), A)) \\ & = \mathop{\mathrm{Ker}}(\mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_1, u(A)) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_2, u(A))) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(B, u(A)) \end{align*}$

Hence we see that we may take $\mathcal{P} = \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ , i.e., we see that $v$ is everywhere defined. $\square$

Comments (2)

Comment #8711 by Bruno Kahn on October 24, 2023 at 12:09

I think a slightly more general variant of Remark 03B8 is the following: given (1) in Lemma 015Z, the converse to its conclusion is true at least if $\mathcal{A}$ has enough injectives. Indeed, let $f:B\to B'$ be a monomorphism in $\mathcal{B}$ , and let $A=\mathrm{Ker}\ v(f)$ . Choose a monomorphism $g:A\to I$ with $I$ injective (it exists by assumption). Then $g$ extends to $g':v(B)\to I$ , whence by adjunction a morphism $B\to u(I)$ . If $u(I)$ is injective, this morphism extends to $h:B'\to u(I)$ , hence by adjunction a morphism $k:v(B')\to I$ extending $g'$ . But then $k$ "extends'' $g$ , which forces $A=0$ since $g$ was a monomorphism.

Maybe one can weaken this hypothesis on $\mathcal{A}$ ?

Comment #9362 by Stacks project on June 04, 2024 at 13:28

Thanks very much. I have simply improved the statement of the lemma with your remark. I don't know what happens if $\mathcal{A}$ does not have enough injectives. Changes are here.

The Stacks project

12.29 Injectives and adjoint functors

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