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EDIT: With the background of

Answer (after reading the paper "Relative homological algebra in article Leo Alonso has cited)

We have the category following description of quasi-coherent shaves" by Enochs, Estrada (which was mentioned by Thomas Nevins in the other question) , it seems to quasi-coherator: Let $X$ be clear that a ) has nothing to do with schemesconcentrated scheme, but rather with quiversi.e. quasi-compact and quasi-separated. If $X$ is separated, for example say $\bullet \to X = \bullet$. So we fix a ring homomorphism cup U_i$ with finitely many affines $R U_i$ such that the intersections $U_i \to S$cap U_j$ are affine, and consider then the categories $Mod$, consisting quasi-coherator of $R$-modules a module $M$ and $S$-modules $N$ together with a on $S$-linear X$ is the kernel of the obvious map

$M \prod_i (u_i)_* \otimes_R S tilde{M(U_i)} \to N$\prod_{i,j} (u_{i,j})_* \tilde{M(U_i \cap U_j)}$,

where $u_i : U_i \to X$ and $u_{ij} : U_i \cap U_j \to X$ are the full subcategory inclusions. If $Qcoh$, where this map X$ is assumed to just quasi-separated, there is a similar description using the separated case.

The idea is quite simple and can be generalized to every flat ring representation of a finite partial order, which has suprema (for example the dual of the affine subsets of a quasi-compact separated scheme). On an isomorphismaffine part, the quasi-coherator consists of sections of all other affine parts over it, which are compatible in the obvious sense.How can

If we describe the right adjoint to $Qcoh \subseteq Mod$?EDIT2: I have found a no finiteness conditions, the description for this eample if $R \to S$ is a flat epimorphism basically also valid, but you have to take the quasi-coherators of ringsthe products or the direct images, since they don't have to be quasi-coherent. Note that this In general there is satisfied no nice description. Also in nice special cases, b) and c) have no nice answers (and infinite products are not exact, even in the scheme examplecategory of quasi-coherent modules on nice schemes).
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Let $X$ be a scheme. It is known that $Qcoh(X)$ is cocomplete, co-wellpowered and has a generating set. The special adjoint functor theorem tells us that then every(!) cocontinuous functor $Qcoh(X) \to A$ has a right-adjoint. Here $A$ is an arbitrary category (which I always assume to be locally small).

a) Is there a nice description of the right-adjoint to the forgetful functor $Qcoh(X) \to Mod(X)$? Here, you may impose finiteness conditions on $X$. This functor may be called a quasi-coherator.

b) Let $f : X \to Y$ be a morphism of schemes. Then $f^* : Qcoh(Y) \to Qcoh(X)$ is cocontinuous, since $f^* : Mod(Y) \to Mod(X)$ is cocontinuous and the forgetful functor preserves and reflects colimits. In particular, there is a right-adjoint $f_+ : Qcoh(X) \to Qcoh(Y)$. If $f$ is quasi-separated ans quasi-compact, then this is the direct image functor $f_*$. Is there a nice description in general? Note that $f_+$ is the composition $Qcoh(X) \to Mod(X) \to Mod(Y) \to Qcoh(Y)$, where the latter is the quasi-coherator. This is only nice if we have answered a).

c) Since $Mod(X)$ is complete and $Qcoh(X) \to Mod(X)$ has a right adjoint, $Qcoh(X)$ is also complete. Is there a nice description for the products? They are given by taking the quasi-coherator of the product, can we simplify this? I mean, perhaps they turn out to be exact although the products in $Mod(X)$ are not exact?

EDIT: With the background of the paper "Relative homological algebra in the category of quasi-coherent shaves" by Enochs, Estrada (which was mentioned by Thomas Nevins in the other question) , it seems to be clear that a) has nothing to do with schemes, but rather with quivers, for example $\bullet \to \bullet$. So we fix a ring homomorphism $R \to S$, and consider the categories $Mod$, consisting of $R$-modules $M$ and $S$-modules $N$ together with a $S$-linear map $M \otimes_R S \to N$, and the full subcategory $Qcoh$, where this map is assumed to be an isomorphism. How can we describe the right adjoint to $Qcoh \subseteq Mod$? EDIT2: I have found a description for this eample if $R \to S$ is a flat epimorphism of rings. Note that this is satisfied in the scheme example.
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Let $X$ be a scheme. It is known that $Qcoh(X)$ is cocomplete, co-wellpowered and has a generating set. The special adjoint functor theorem tells us that then every(!) cocontinuous functor $Qcoh(X) \to A$ has a right-adjoint. Here $A$ is an arbitrary category (which I always assume to be locally small).

a) Is there a nice description of the right-adjoint to the forgetful functor $Qcoh(X) \to Mod(X)$? Here, you may impose finiteness conditions on $X$. This functor may be called a quasi-coherator.

b) Let $f : X \to Y$ be a morphism of schemes. Then $f^* : Qcoh(Y) \to Qcoh(X)$ is cocontinuous, since $f^* : Mod(Y) \to Mod(X)$ is cocontinuous and the forgetful functor preserves and reflects colimits. In particular, there is a right-adjoint $f_+ : Qcoh(X) \to Qcoh(Y)$. If $f$ is quasi-separated ans quasi-compact, then this is the direct image functor $f_*$. Is there a nice description in general? Note that $f_+$ is the composition $Qcoh(X) \to Mod(X) \to Mod(Y) \to Qcoh(Y)$, where the latter is the right adjoint to $Qcoh(Y) \to Mod(Y)$quasi-coherator. This is only nice if we have answered a).

c) Since $Mod(X)$ is complete and $Qcoh(X) \to Mod(X)$ has a right adjoint, $Qcoh(X)$ is also complete. Is there a nice description for the products? They are given by taking the quasi-coherator of the product, can we simplify this? I mean, perhaps they turn out to be exact although the products in $Mod(X)$ are not exact?

EDIT: With the background of the paper "Relative homological algebra in the category of quasi-coherent shaves" by Enochs, Estrada (which was mentioned by Thomas Nevins in the other question) , it seems to be clear that a) has nothing to do with schemes, but rather with quivers, for example $\bullet \to \bullet$. So we fix a ring homomorphism $R \to S$, and consider the categories $Mod$, consisting of $R$-modules $M$ and $S$-modules $N$ together with a $S$-linear map $M \otimes_R S \to N$, and the full subcategory $Qcoh$, where this map is assumed to be an isomorphism. How can we describe the right adjoint to $Qcoh \subseteq Mod$?
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