Lemma 19.14.1 (0F5S)—The Stacks project

Lemma 19.14.1. The functor $G$ above has a left adjoint $F : \text{Mod}_ R \to \mathcal{A}$.

Proof. We will give two proofs of this lemma.

The first proof will use the adjoint functor theorem, see Categories, Theorem 4.25.3. Observe that $G : \mathcal{A} \to \text{Mod}_ R$ is left exact and sends products to products. Hence $G$ commutes with limits. To check the set theoretical condition in the theorem, suppose that $M$ is an object of $\text{Mod}_ R$. Choose a suitably large cardinal $\kappa $ and denote $E$ a set of objects of $\mathcal{A}$ such that every object $A$ with $|A| \leq \kappa $ is isomorphic to an element of $E$. This is possible by Lemma 19.11.4. Set $I = \coprod _{A \in E} \mathop{\mathrm{Hom}}\nolimits _ R(M, G(A))$. We think of an element $i \in I$ as a pair $(A_ i, f_ i)$. Finally, let $A$ be an arbitrary object of $\mathcal{A}$ and $f : M \to G(A)$ arbitrary. We are going to think of elements of $\mathop{\mathrm{Im}}(f) \subset G(A) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(U, A)$ as maps $u : U \to A$. Set

\[ A' = \mathop{\mathrm{Im}}(\bigoplus \nolimits _{u \in \mathop{\mathrm{Im}}(f)} U \xrightarrow {u} A) \]

Since $G$ is left exact, we see that $G(A') \subset G(A)$ contains $\mathop{\mathrm{Im}}(f)$ and we get $f' : M \to G(A')$ factoring $f$. On the other hand, the object $A'$ is the quotient of a direct sum of at most $|M|$ copies of $U$. Hence if $\kappa = |\bigoplus _{|M|} U|$, then we see that $(A', f')$ is isomorphic to an element $(A_ i, f_ i)$ of $E$ and we conclude that $f$ factors as $M \xrightarrow {f_ i} G(A_ i) \to G(A)$ as desired.

The second proof will give a construction of $F$ which will show that “$F(M) = M \otimes _ R U$” in some sense. Namely, for any $R$-module $M$ we can choose a resolution

\[ \bigoplus \nolimits _{j \in J} R \to \bigoplus \nolimits _{i \in I} R \to M \to 0 \]

Then we define $F(M)$ by the corresponding exact sequence

\[ \bigoplus \nolimits _{j \in J} U \to \bigoplus \nolimits _{i \in I} U \to F(M) \to 0 \]

This construction is independent of the choice of the resolution and is functorial; we omit the details. For any $A$ in $\mathcal{A}$ we obtain an exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(F(M), A) \to \prod \nolimits _{i \in I} G(A) \to \prod \nolimits _{j \in J} G(A) \]

which is isomorphic to the sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M, G(A)) \to \mathop{\mathrm{Hom}}\nolimits _ R(\bigoplus \nolimits _{i \in I} R, G(A)) \to \mathop{\mathrm{Hom}}\nolimits _ R(\bigoplus \nolimits _{j \in J} R, G(A)) \]

which shows that $F$ is the left adjoint to $G$. $\square$

Comments (1)

Comment #11008 by thesnakefromthelemma on November 26, 2025 at 01:00

Adding on to Comment #11001, a few remarks on the second proof given of Lemma 0F5S:

Is it worth inserting a sentence or so clarifying that/how the application of $F$ to the map $\bigoplus _ { j \in J } R \overset { \varphi } { \to } \bigoplus _ { i \in I } R$ (of which $M$ is presented as the cokernel) is well-defined in terms of the canonical coproduct inclusions of the domain and codomain by virtue of $F$ 's commuting with coproducts? This is a separate question from the functoriality/presentation-independence of $F$ , and as far as I can tell not quite trivial—any argument I can think of implicitly uses that $R$ is a compact object of $R\text{-mod}$ , or at least its consequence that for any map $R \to \bigoplus _ { i \in I } R$ there exists a finite subset $I' \subseteq I$ such that the former factors through the evident inclusion $\bigoplus _ { i \in I' } R \to \bigoplus _ { i \in I } R$ . (This subtlety is common to various instances of the Eilenberg-Watts argument.)

Relatedly, is it worth specifying how $F$ acts on morphisms of $R\text{-mod}$ ? Imho this is the actual meat of the argument (specifically in that the equivalence of two presentations of the morphism is witnessed by a "chain homotopy", which $F$ then carries into $\mathcal{A}$ ).

In the case as hand, we don't actually need to quantify/choose over generic presentations $\bigoplus _ { j \in J } R \overset { \varphi } { \to } \bigoplus _ { i \in I } R$ and then verify well-definition: Instead we can for each $M$ construct the specific presentation where (I) $I$ is the set of elements of $M$ , (II) $J$ is the set of elements of the kernel of the evident map $\bigoplus _ { i \in I } R \to M$ , (III) $\varphi$ is the evident insertion of kernel elements, and (IV) it's evident how each morphism should be presented. It's easy to see that this construction yields a functor $R\text{-mod} \to \text{Ar} (\mathcal{A})$ (here $\text{Ar}$ denotes the arrow category), which can in turn be composed with the cokernel functor $\text{Ar} (\mathcal{A}) \to \mathcal{A}$ to give a functor $R\text{-mod} \to \mathcal{A}$ , no further verification of well-definition required.

If any of these ideas are worth implementing at this time I'm happy to try to help...

There are also:

4 comment(s) on Section 19.14: The Gabriel-Popescu theorem

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