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Along with მამუკა ჯიბლაძე's example in the comments of sheaves on a complete boolean algebra, this makes it pretty clear that there are plenty of locally presentable abelian categories with enough projectives which are not additive presheaf categories.

But none of these categories are locally finitely presentable. In this case, I think it may be true that having enough projectives entails being a module category.

Here is a partial result in that direction, strengthening a classical result of Nastacescu in the Artinian case:

Proposition: Let $X$ be a Noetherian scheme and $\mathcal C = \mathrm{QCoh}(X)$ its category of quasicoherent sheaves. Then $\mathcal C$ has enough projectives (if and) only if $X$ is "noncommutatively affine", i.e iff there is a (possibly noncommutative) ring $R$ such that $\mathcal C = Mod_R$. If $X$ is defined over a field of characteristic 0, then $R$ may be chosen to be commutative, i.e. $X$ is affine.

The proof will use the following facts (which derive from $X$ being Noetherian):

• $\mathcal C$ is a locally finitely presentable category (with a compact generator)

• The compact objects of $\mathcal C$ are the coherent sheaves.

• Any quasicoherent subsheaf of a coherent sheaf is coherent.

So we in fact have the following more general result:

If $\mathcal C$ is a locally finitely presentable abelian category which is "Noetherian" in the sense that any subobject of a compact object is compact, then $\mathcal C$ has enough projectives iff it is an additive presheaf category; if in addition $\mathcal C$ has a compact generator, then $\mathcal C$ is a module category.

Note that this meaning of "Noetherian" implies the usual ascending chain condition for subobjects / descending chain condition for quotient objects of compact objects.

Proof:

If: $\checkmark$

Only if: Let $G_0$ be a compact generator of $\mathcal C$, and choose (the zeroth step of) a projective resolution $P \twoheadrightarrow G_0$. Take an epimorphism $G_0^{\oplus I} \twoheadrightarrow P$. Because $X$ is Noetherian and $G_0$ is coherent, there must be a finite subsum $G = G_0^n$ such that the composite $G \to P \to G_0$ is an epimorphism. Because $P$ is projective, we may choose a map $P \to G$ which commutes with the epimorphisms to $G_0$.

Composing, we obtain an endomorphism $G \overset e \to G$ lying over the epimorphism to $G_0$, which factors through $P \twoheadrightarrow G_0$.

Now let $\bar G$ be the cokernel of $G \overset {1-e} \to G$. Note that $e$ descends to the identity map on $\bar G$. I claim that this map factors through $P$. To see this, it suffices to show that the composite $G \overset {1-e} \to G \to P \to G \twoheadrightarrow \bar G$ is zero. But this is the same as the composite $G \overset e \to G \overset {1-e} \to G \twoheadrightarrow \bar G$, which is zero. So $\bar G$ is a retract of $P$, and hence projective.

Moreover, $\bar G$ is compact, and since $e$ restricts to the identity on $G_0$, we obtain an epimorphism $\bar G \twoheadrightarrow G_0$, so that $\bar G$ is a generator. That is, $\bar G$ is a compact projective generator, so $\mathcal C$ is a module category. If we start with a generating set of compact objects, we can find a compact projective cover of each of them in this manner, and so we get a generating set of compact projectives.

Affineness: Now we assume characteristic 0 and show that $X$ is affine. Let $G$ be a compact projective generator. Because $G$ is compact projective, it is locally free of finite rank. Moreover, if $E$ is locally free of finite rank, then $G \otimes E$ is projective -- to see this, note that if $A \twoheadrightarrow B$ is an epimorphism, then $\mathcal{Hom}(E,A) \to \mathcal{Hom}(E,B)$ is an epimorphism (clear on stalks), and use tensor-hom adjointness. In particular, $G \otimes G^\vee$ is projective, where $G^\vee$ is the dual of $G$. Then we have maps $\mathcal O_X \to G \otimes G^\vee \to \mathcal O_X$ which compose to the scalar $n$, where $n$ is the rank of $G$; in characteristic 0, this is invertible so that $\mathcal O_X$ is a retract of $G \otimes G^\vee$ and hence projective. It follows that the higher cohomology of any quasicoherent sheaf vanishes, a well-known criterion for affineness of $X$.

But ideally I would like a statement with looser hypotheses.

And incidentally, what's an example of a scheme (in the standard, "commutative" sense) which is "noncommutatively affine" but not actually affine, i.e. $\mathrm{QCoh}(X) = \mathrm{Mod}_R$ for $R$ which is not commutative (even up to Morita equivalence)? Since $\mathrm{QCoh}(X)$ has a nice symmetric monoidal structure, such rings $R$ must be very special.

For my own sanity, here is a proof of the bullet-pointed items. Let $X$ be a Noetherian scheme.

1. For a Noetherian affine scheme $U$, $\mathrm{QCoh}(U)$ has the stated properties.

2. If $i: U \to X$ is the inclusion of an affine open, then $i_\ast$ preserves filtered colimits, so $i^\ast$ preserves compact objects. Hence if $F \in \mathrm{QCoh}(X)$ is compact, then $i^\ast F$ is compact, hence coherent, and so $F$ is coherent because coherence is a local property. Moreover, because $i^\ast$ is exact and coherent sheaves on Noetherian affine schemes are closed under subobjects and quotients, any subobject or quotient object of a compact $F \in \mathrm{QCoh}(X)$ is coherent.

3. Conversely, if $F$ is coherent, then it embeds into a sum of pushforwards of its restrictions to a finite open affine cover, which is compact, so it is a subobject of a compact object. Hence coherent sheaves coincide with subobjects of compact sheaves. Because coherent sheaves are closed under quotients, every quotient of a compact sheaf is coherent, but we don't at this point have the converse.

4. Every coherent sheaf satisfies the ascending chain condition. This can be seen by passing to an affine open cover and applying the pigeonhole principle.

5. If $F$ satisfies the ascending chain condition, then for any filtered colimit $G_\bullet$, the map $\varinjlim Hom(F,G_\bullet) \to Hom(F,\varinjlim G_\bullet)$ is injective.

6. Let $0 \to E \to F \to G \to 0$ be a short exact sequence with $F$ compact. Since $E,F,G$ are coherent, the previous item and a 5-lemma argument imply that $E$ and $F$ are compact. Thus all coherent sheaves are compact.

7. Every $F \in \mathrm{QCoh}(X)$ is a colimit of coherent sheaves. For, embedding $F$ into the sum of its restrictions $F_i$ to a finite affine open cover, the pullback of any product of finitely-generated submodules $G_1,\dots,G_n$ of $F_1,\dots,F_n$ is a coherent subsheaf of $F$, and as we let $G_\bullet$ range over all such finitely-generated submodules, we obtain a system of coherent sheaves whose colimit is $F$.

So $\mathrm{Coh}(X)$ forms a compact set of generators of $\mathrm{QCoh}(X)$. Since $\mathrm{QCoh}(X)$ is cocomplete, it is locally finitely presentable. Since $\mathrm{Coh}(X)$ is closed under colimits, it coincides with the finitely presentable objects.

Along with მამუკა ჯიბლაძე's example in the comments of sheaves on a complete boolean algebra, this makes it pretty clear that there are plenty of locally presentable abelian categories with enough projectives which are not additive presheaf categories. I'm still not sure about categories of quasicoherent sheaves, though.

Along with მამუკა ჯიბლაძე's example in the comments of sheaves on a complete boolean algebra, this makes it pretty clear that there are plenty of locally presentable abelian categories with enough projectives which are not additive presheaf categories. I'm still not sure about categories of quasicoherent sheaves, though.

Along with მამუკა ჯიბლაძე's example in the comments of sheaves on a complete boolean algebra, this makes it pretty clear that there are plenty of locally presentable abelian categories with enough projectives which are not additive presheaf categories.

But none of these categories are locally finitely presentable. In this case, I think it may be true that having enough projectives entails being a module category.

Here is a partial result in that direction, strengthening a classical result of Nastacescu in the Artinian case:

Proposition: Let $X$ be a Noetherian scheme and $\mathcal C = \mathrm{QCoh}(X)$ its category of quasicoherent sheaves. Then $\mathcal C$ has enough projectives (if and) only if $X$ is "noncommutatively affine", i.e iff there is a (possibly noncommutative) ring $R$ such that $\mathcal C = Mod_R$. If $X$ is defined over a field of characteristic 0, then $R$ may be chosen to be commutative, i.e. $X$ is affine.

The proof will use the following facts (which derive from $X$ being Noetherian):

• $\mathcal C$ is a locally finitely presentable category (with a compact generator)

• The compact objects of $\mathcal C$ are the coherent sheaves.

• Any quasicoherent subsheaf of a coherent sheaf is coherent.

So we in fact have the following more general result:

If $\mathcal C$ is a locally finitely presentable abelian category which is "Noetherian" in the sense that any subobject of a compact object is compact, then $\mathcal C$ has enough projectives iff it is an additive presheaf category; if in addition $\mathcal C$ has a compact generator, then $\mathcal C$ is a module category.

Note that this meaning of "Noetherian" implies the usual ascending chain condition for subobjects / descending chain condition for quotient objects of compact objects.

Proof:

If: $\checkmark$

Only if: Let $G_0$ be a compact generator of $\mathcal C$, and choose (the zeroth step of) a projective resolution $P \twoheadrightarrow G_0$. Take an epimorphism $G_0^{\oplus I} \twoheadrightarrow P$. Because $X$ is Noetherian and $G_0$ is coherent, there must be a finite subsum $G = G_0^n$ such that the composite $G \to P \to G_0$ is an epimorphism. Because $P$ is projective, we may choose a map $P \to G$ which commutes with the epimorphisms to $G_0$.

Composing, we obtain an endomorphism $G \overset e \to G$ lying over the epimorphism to $G_0$, which factors through $P \twoheadrightarrow G_0$.

Now let $\bar G$ be the cokernel of $G \overset {1-e} \to G$. Note that $e$ descends to the identity map on $\bar G$. I claim that this map factors through $P$. To see this, it suffices to show that the composite $G \overset {1-e} \to G \to P \to G \twoheadrightarrow \bar G$ is zero. But this is the same as the composite $G \overset e \to G \overset {1-e} \to G \twoheadrightarrow \bar G$, which is zero. So $\bar G$ is a retract of $P$, and hence projective.

Moreover, $\bar G$ is compact, and since $e$ restricts to the identity on $G_0$, we obtain an epimorphism $\bar G \twoheadrightarrow G_0$, so that $\bar G$ is a generator. That is, $\bar G$ is a compact projective generator, so $\mathcal C$ is a module category. If we start with a generating set of compact objects, we can find a compact projective cover of each of them in this manner, and so we get a generating set of compact projectives.

Affineness: Now we assume characteristic 0 and show that $X$ is affine. Let $G$ be a compact projective generator. Because $G$ is compact projective, it is locally free of finite rank. Moreover, if $E$ is locally free of finite rank, then $G \otimes E$ is projective -- to see this, note that if $A \twoheadrightarrow B$ is an epimorphism, then $\mathcal{Hom}(E,A) \to \mathcal{Hom}(E,B)$ is an epimorphism (clear on stalks), and use tensor-hom adjointness. In particular, $G \otimes G^\vee$ is projective, where $G^\vee$ is the dual of $G$. Then we have maps $\mathcal O_X \to G \otimes G^\vee \to \mathcal O_X$ which compose to the scalar $n$, where $n$ is the rank of $G$; in characteristic 0, this is invertible so that $\mathcal O_X$ is a retract of $G \otimes G^\vee$ and hence projective. It follows that the higher cohomology of any quasicoherent sheaf vanishes, a well-known criterion for affineness of $X$.

But ideally I would like a statement with looser hypotheses.

And incidentally, what's an example of a scheme (in the standard, "commutative" sense) which is "noncommutatively affine" but not actually affine, i.e. $\mathrm{QCoh}(X) = \mathrm{Mod}_R$ for $R$ which is not commutative (even up to Morita equivalence)? Since $\mathrm{QCoh}(X)$ has a nice symmetric monoidal structure, such rings $R$ must be very special.

For my own sanity, here is a proof of the bullet-pointed items. Let $X$ be a Noetherian scheme.

1. For a Noetherian affine scheme $U$, $\mathrm{QCoh}(U)$ has the stated properties.

2. If $i: U \to X$ is the inclusion of an affine open, then $i_\ast$ preserves filtered colimits, so $i^\ast$ preserves compact objects. Hence if $F \in \mathrm{QCoh}(X)$ is compact, then $i^\ast F$ is compact, hence coherent, and so $F$ is coherent because coherence is a local property. Moreover, because $i^\ast$ is exact and coherent sheaves on Noetherian affine schemes are closed under subobjects and quotients, any subobject or quotient object of a compact $F \in \mathrm{QCoh}(X)$ is coherent.

3. Conversely, if $F$ is coherent, then it embeds into a sum of pushforwards of its restrictions to a finite open affine cover, which is compact, so it is a subobject of a compact object. Hence coherent sheaves coincide with subobjects of compact sheaves. Because coherent sheaves are closed under quotients, every quotient of a compact sheaf is coherent, but we don't at this point have the converse.

4. Every coherent sheaf satisfies the ascending chain condition. This can be seen by passing to an affine open cover and applying the pigeonhole principle.

5. If $F$ satisfies the ascending chain condition, then for any filtered colimit $G_\bullet$, the map $\varinjlim Hom(F,G_\bullet) \to Hom(F,\varinjlim G_\bullet)$ is injective.

6. Let $0 \to E \to F \to G \to 0$ be a short exact sequence with $F$ compact. Since $E,F,G$ are coherent, the previous item and a 5-lemma argument imply that $E$ and $F$ are compact. Thus all coherent sheaves are compact.

7. Every $F \in \mathrm{QCoh}(X)$ is a colimit of coherent sheaves. For, embedding $F$ into the sum of its restrictions $F_i$ to a finite affine open cover, the pullback of any product of finitely-generated submodules $G_1,\dots,G_n$ of $F_1,\dots,F_n$ is a coherent subsheaf of $F$, and as we let $G_\bullet$ range over all such finitely-generated submodules, we obtain a system of coherent sheaves whose colimit is $F$.

So $\mathrm{Coh}(X)$ forms a compact set of generators of $\mathrm{QCoh}(X)$. Since $\mathrm{QCoh}(X)$ is cocomplete, it is locally finitely presentable. Since $\mathrm{Coh}(X)$ is closed under colimits, it coincides with the finitely presentable objects.

Here is an example of the kind of result I have in mind:

Proposition: Let $X$ be a Noetherian scheme and $\mathcal C = \mathrm{QCoh}(X)$ its category of quasicoherent sheaves. Then $\mathcal C$ has enough projectives (if and) only if $X$ is "noncommutatively affine", i.e iff there is a (possibly noncommutative) ring $R$ such that $\mathcal C = Mod_R$. If $X$ is defined over a field of characteristic 0, then $R$ may be chosen to be commutative, i.e. $X$ is affine.

The proof will use the following facts (which derive from $X$ being Noetherian):

• $\mathcal C$ is a locally finitely presentable category (with a compact generator)

• The compact objects of $\mathcal C$ are the coherent sheaves.

• Any quasicoherent subsheaf of a coherent sheaf is coherent.

So we in fact have the following more general result:

If $\mathcal C$ is a locally finitely presentable abelian category which is "Noetherian" in the sense that any subobject of a compact object is compact, then $\mathcal C$ has enough projectives iff it is an additive presheaf category; if in addition $\mathcal C$ has a compact generator, then $\mathcal C$ is a module category.

Note that this meaning of "Noetherian" implies the usual ascending chain condition for subobjects / descending chain condition for quotient objects of compact objects.

Proof:

If: $\checkmark$

Only if: Let $\mathcal G$ be a set of compact objects. We construct a new set of compact objects $\mathcal G'$, all of which are quotients of objects of $\mathcal G$, as follows. For $G \in \mathcal G$, pick a projective cover $P \twoheadrightarrow G$. Choose an epimorphism $\oplus_i G_i \to P$ where $G_i \in \mathcal G$. Because $G$ is compact, we may choose a minimal finite subsum such that $G_1 \oplus \dots \oplus G_n \to P \to G$ is an epimorphism. Let $\mathcal G'$ be the collection of the images $G_1',\dots,G_n'$ of $G_1,\dots,G_n$ under the resulting maps $G_i \to G$. The key point is that the maximal subcategory of $\mathcal C$ in which $\mathcal G$ is a generator (in the sense that $\prod_{G \in \mathcal G} Hom(G,-)$ is faithful) is the same as the maximal subcategory in which $\mathcal G'$ is a generator.

If $\mathcal G$ is finite, then so is $\mathcal G'$. Moreover if $\mathcal G$ is finite, then by Noetherianness (and Konig's lemma), if we iterate the passage $\mathcal G \mapsto \mathcal G'$, the process eventually stabilizes at some $\mathcal G^\ast$. Now, $\mathcal G^\ast$ has the property that for every $G \in \mathcal G^\ast$, there is a $G' \in \mathcal G^\ast$ and maps $G' \to P \to G$ such that $P$ is projective and $G' \to G$ is an epimorphism. Since $\mathcal G^\ast$ is finite, it breaks into cycles under the iteration of $G \mapsto G'$, and so we may always take $G' = G$. But by Noetherianness, any epimorphism $G \twoheadrightarrow G$ is an isomorphism. So every $G \in \mathcal G^\ast$ is a retract of a projective and hence projective.

So if we start with a compact generator $G$, then $\oplus_{H \in \{G\}^\ast} H$ is a compact projective generator. If we start with a set of compact generators $\mathcal G$, then $\cup_{G \in \mathcal G} \{G\}^\ast$ is is a set of compact projective generators.

Affineness: Now we assume characteristic 0 and show that $X$ is affine. Let $G$ be a compact projective generator. Because $G$ is compact projective, it is locally free of finite rank. Moreover, if $E$ is locally free of finite rank, then $G \otimes E$ is projective -- to see this, note that if $A \twoheadrightarrow B$ is an epimorphism, then $\mathcal{Hom}(E,A) \to \mathcal{Hom}(E,B)$ is an epimorphism (clear on stalks), and use tensor-hom adjointness. In particular, $G \otimes G^\vee$ is projective, where $G^\vee$ is the dual of $G$. Then we have maps $\mathcal O_X \to G \otimes G^\vee \to \mathcal O_X$ which compose to the scalar $n$, where $n$ is the rank of $G$; in characteristic 0, this is invertible so that $\mathcal O_X$ is a retract of $G \otimes G^\vee$ and hence projective. It follows that the higher cohomology of any quasicoherent sheaf vanishes, a well-known criterion for affineness of $X$.

  But ideally I would like a statement with looser hypotheses.

And incidentally, what's an example of a scheme (in the standard, "commutative" sense) which is "noncommutatively affine" but not actually affine, i.e. $\mathrm{QCoh}(X) = \mathrm{Mod}_R$ for $R$ which is not commutative (even up to Morita equivalence)? Since $\mathrm{QCoh}(X)$ has a nice symmetric monoidal structure, such rings $R$ must be very special.

For my own sanity, here is a proof of the bullet-pointed items. Let $X$ be a Noetherian scheme.

1. For a Noetherian affine scheme $U$, $\mathrm{QCoh}(U)$ has the stated properties.

2. If $i: U \to X$ is the inclusion of an affine open, then $i_\ast$ preserves filtered colimits, so $i^\ast$ preserves compact objects. Hence if $F \in \mathrm{QCoh}(X)$ is compact, then $i^\ast F$ is compact, hence coherent, and so $F$ is coherent because coherence is a local property. Moreover, because $i^\ast$ is exact and coherent sheaves on Noetherian affine schemes are closed under subobjects and quotients, any subobject or quotient object of a compact $F \in \mathrm{QCoh}(X)$ is coherent.

3. Conversely, if $F$ is coherent, then it embeds into a sum of pushforwards of its restrictions to a finite open affine cover, which is compact, so it is a subobject of a compact object. Hence coherent sheaves coincide with subobjects of compact sheaves. Because coherent sheaves are closed under quotients, every quotient of a compact sheaf is coherent, but we don't at this point have the converse.

4. Every coherent sheaf satisfies the ascending chain condition. This can be seen by passing to an affine open cover and applying the pigeonhole principle.

5. If $F$ satisfies the ascending chain condition, then for any filtered colimit $G_\bullet$, the map $\varinjlim Hom(F,G_\bullet) \to Hom(F,\varinjlim G_\bullet)$ is injective.

6. Let $0 \to E \to F \to G \to 0$ be a short exact sequence with $F$ compact. Since $E,F,G$ are coherent, the previous item and a 5-lemma argument imply that $E$ and $F$ are compact. Thus all coherent sheaves are compact.

7. Every $F \in \mathrm{QCoh}(X)$ is a colimit of coherent sheaves. For, embedding $F$ into the sum of its restrictions $F_i$ to a finite affine open cover, the pullback of any product of finitely-generated submodules $G_1,\dots,G_n$ of $F_1,\dots,F_n$ is a coherent subsheaf of $F$, and as we let $G_\bullet$ range over all such finitely-generated submodules, we obtain a system of coherent sheaves whose colimit is $F$.

So $\mathrm{Coh}(X)$ forms a compact set of generators of $\mathrm{QCoh}(X)$. Since $\mathrm{QCoh}(X)$ is cocomplete, it is locally finitely presentable. Since $\mathrm{Coh}(X)$ is closed under colimits, it coincides with the finitely presentable objects.

Let $\mathcal C_0$ be a small additive category with countable coproducts, and let $\mathcal C$ be the category of additive functors $\mathcal C^{op}_0 \to \mathsf{Ab}$ preserving countable products. Then by Lemma 1.3 and Prop 2.5 here, $\mathcal C$ is a cocomplete abelian category which is easily seen to be locally $\aleph_1$-presentable. Moreover, cokernels are computed "levelwise", so that the representables are a generating set of projective objects; in particular, $\mathcal C$ has enough projectives. But $\mathcal C$ is pretty far from being an additive presheaf category.

Along with მამუკა ჯიბლაძე's example in the comments of sheaves on a complete boolean algebra, this makes it pretty clear that there are plenty of locally presentable abelian categories with enough projectives which are not additive presheaf categories. I'm still not sure about categories of quasicoherent sheaves, though.