Quasi-coherent envelope of a module - MathOverflow

Quasi-coherent envelope of a module

2010-09-30T08:31:07Z

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?

Answer (after reading the article Leo Alonso has cited)

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).

Answer by Leo Alonso for Quasi-coherent envelope of a module

2010-09-30T16:26:08Z

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.

Answer by Bugs Bunny for Quasi-coherent envelope of a module

2010-10-01T14:24:11Z

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!