# Quasi-coherent envelope of a module

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?

We have the following description of the quasi-coherator: Let $X$ be a concentrated scheme, i.e. quasi-compact and quasi-separated. If $X$ is separated, say $X = \cup U_i$ with finitely many affines $U_i$ such that the intersections $U_i \cap U_j$ are affine, then the quasi-coherator of a module $M$ on $X$ is the kernel of the obvious map

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

where $u_i : U_i \to X$ and $u_{ij} : U_i \cap U_j \to X$ are the inclusions. If $X$ is 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 affine part, the quasi-coherator consists of sections of all other affine parts over it, which are compatible in the obvious sense.

If we have no finiteness conditions, the description is basically also valid, but you have to take the quasi-coherators of the products or the direct images, since they don't have to be quasi-coherent. In general there is no nice description. Also in nice special cases, b) and c) have no nice answers (and infinite products are not exact, even in the category of quasi-coherent modules on nice schemes).

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Hi Martin. I don't think that the "coherator", i.e. the right adjoint to '$\text{QCoh}(X)\to\text{Mod}(X)$, can be described in a nice way, because taking the coherator doesn't even commute with restriction to open subsets. For example, the coherator of the $\text{Mod}(X)$-product of a family of quasi-coherent sheaves is a $\text{QCoh}(X)$`-product, but there are examples where taking the product of quasi-coherent sheaves is not exact, and in particular not local. See the beautiful paper [Krause, The stable derived category of a Noetherian scheme, Example 4.9]. –  Hanno Becker Sep 30 '10 at 9:02
a) What is wrong with taking an affine open cover, section $M(U_i)$ and associating the quasicoherent module to these sections? It appears that it is independent of the choice of affine open cover... –  Bugs Bunny Sep 30 '10 at 11:03
@Hanno: Thank you. Nevertheless, I think there are some clear constructions which are not local. @Bugs: Hm, are you sure that we have isomorphisms on the intersections? –  Martin Brandenburg Sep 30 '10 at 12:45
My vague recollection is that Thomason-Trobaugh has some discussion of the coherator and quasi-coherent cohomology (possibly in an appendix), starting with roughly the terms Bugs describes - have you looked at this paper? –  Tyler Lawson Sep 30 '10 at 12:56
@ Martin B. It seems to follow from $M$ being a sheaf... –  Bugs Bunny Sep 30 '10 at 14:05

A very nice reference for the coherator functor together with a nice description of this functor is written down in Thomason and Trobaugh "Higher algebraic $K$-theory of schemes and of derived categories" in The Grothendieck Festschrift, Vol. III, 247--435, Progr. Math., 88, Birkhäuser, Boston, 1990. (MR11069118). Look for appendix B.

The original reference goes back to SGA6 (exposé II 3.2, by Illusie). It contains an appendix with counterexamples due to Verdier showing that:

• An affine scheme $\mathrm{Spec}(A)$ together with an injective $A$-module $I$ such that $\widetilde{I}$ is not injective as a quasi-coherent sheaf.
• A morphism $f$ between concentrated schemes such that the right derived functors of $f_*$ are different when considered from all modules or from quasi-coherent modules
• A concentrated scheme $S$ together with a quasi-coherent sheaf that it is not acyclic for the quasi-coherator.

The word concentrated is a shorthand for quasi-compact and quasi-separated. Under separation (or just semi-separation) hypothesis the last two pathologies do not show up.

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Let us do the case of an affine scheme $X$ first. This is easy. If $M$ is an $O_X$-module, we define $\tilde{M}$ as the quasicoherent $O_X$-module defined by the global sections $M(X)$. Notice that the restriction gives the canonical map $\tilde{M}->M$.

As step 2, we extend to the case of quasiaffine $X$. Quasiaffinity means that $X-> Spec O_X(X)$ is an open embedding. Equivalently, all its quasicoherent modules are generated by global section. Hence, the coherator functor $M |-> \tilde{M}$ is defined in the same way. BTW, it is clear how the coherator works on maps too.

Finally, we have everything ready as a general scheme admits an affine open cover $X=\cup_i U_i$. The quasiaffine case is useful as double and triple intersection $U_{i,j}$, $U_{i,j,k}$ are all quasiaffine. Given an $O_X$-module $M$, we use its open pieces $M_i=M|_{U_i}$ and gluing maps $\phi_{i,j}$ from ${M_i}|_{U(i,j)}$ to ${M_j}|_{U(i,j)}$,

where $U(i,j)=U_{i,j}$ (because tex translator is finding it difficult to comprehend it too without each formula starting in a new line), that satisfy the cocycle conditions on triple intersection.

Now the coherator is glued from open pieces $\tilde{M}_i$

using isomorphisms $\tilde{\phi}_{i,j}$ which inherit the cocycle conditions. And That's All Folks!

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In EGA quasi-affine schemes are assumed to be quasi-compact. It then follows that open subschemes of affine schemes have the property which you have stated. I don't understand why $U_i \cap U_j$ has the property. Also I don't see why your construction works in the quasi-affine case. –  Martin Brandenburg Oct 1 '10 at 21:05
You got me worried because I don't want to require quasicompactnes. I define quasiaffine as an open subset in affine, i.e. in a spectrum of a ring. All I need is for this to imply that every quasicoherent sheaf is generated by global sections. My feeling is that the implication works without quasicompactness... I will think a bit more... –  Bugs Bunny Oct 3 '10 at 20:55