Quasi-coherent envelope of a module - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:58:09Z http://mathoverflow.net/feeds/question/40587 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40587/quasi-coherent-envelope-of-a-module Quasi-coherent envelope of a module Martin Brandenburg 2010-09-30T08:31:07Z 2010-10-04T12:09:00Z <p>Let $X$ be a scheme. It is known that $Qcoh(X)$ is cocomplete, co-wellpowered and has a <a href="http://mathoverflow.net/questions/39941/does-qcohx-admit-a-generating-set" rel="nofollow">generating set</a>. 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).</p> <p>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.</p> <p>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).</p> <p>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?</p> <p><strong>Answer</strong> (after reading the article Leo Alonso has cited)</p> <p>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</p> <p>$\prod_i (u_i)_* \tilde{M(U_i)} \to \prod_{i,j} (u_{i,j})_* \tilde{M(U_i \cap U_j)}$,</p> <p>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.</p> <p>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.</p> <p>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).</p> http://mathoverflow.net/questions/40587/quasi-coherent-envelope-of-a-module/40641#40641 Answer by Leo Alonso for Quasi-coherent envelope of a module Leo Alonso 2010-09-30T16:26:08Z 2010-09-30T16:26:08Z <p>A very nice reference for the <em>coherator</em> 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. (<a href="http://ams.rice.edu/mathscinet/search/publdoc.html?pg1=IID&amp;s1=172225&amp;vfpref=html&amp;r=12&amp;mx-pid=1106918" rel="nofollow">MR11069118</a>). Look for appendix B.</p> <p>The original reference goes back to SGA6 (exposé II 3.2, by Illusie). It contains an appendix with counterexamples due to Verdier showing that:</p> <ul> <li>An affine scheme $\mathrm{Spec}(A)$ together with an injective $A$-module $I$ such that $\widetilde{I}$ is <strong>not</strong> injective as a quasi-coherent sheaf.</li> <li>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</li> <li>A concentrated scheme $S$ together with a quasi-coherent sheaf that it is not acyclic for the quasi-coherator.</li> </ul> <p>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.</p> http://mathoverflow.net/questions/40587/quasi-coherent-envelope-of-a-module/40750#40750 Answer by Bugs Bunny for Quasi-coherent envelope of a module Bugs Bunny 2010-10-01T14:24:11Z 2010-10-01T14:52:12Z <p>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 $\tilde{M}->M$.</p> <p>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.</p> <p>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 ${M_j}|_{U(i,j)}$, </p> <p>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.</p> <p>Now the coherator is glued from open pieces $\tilde{M}_i$ </p> <p>using isomorphisms $\tilde{\phi}_{i,j}$ which inherit the cocycle conditions. And That's All Folks!</p>