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Basics of Sheaves

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Basics of Sheaves Basics of Sheaves Eric Auld Revised October 8, 2016 Contents 1 Kernel, Cokernel, and Image in the Category of Presheaves 1 2 Stalk and Kernel, Cokernel, and Image 2 3 Sheaves 4 4 Kernel, Cokernel, and Image in the Category of Sheaves 6 5 Sheaves and the Definition of a Manifold 10 6 The Pullback Sheaf 10 7 Relation With Covering Spaces and Lifting 12 1 Kernel, Cokernel, and Image in the Category of Presheaves Let F ÝÑΦ G be a morphism of presheaves of abelian groups. For each U Ă X, associate the abelian group kerpΦU q ¤ FpUq (resp ImpΦU q ¤ GpUq, resp cokpΦU q). Claim. U ÞÑ kerpΦU q (resp U ÞÑ ImpΦU q, resp U ÞÑ cokpΦU q) naturally has the structure of a presheaf. These are the categorical kernel, cokernel, image, in the category of presheaves of abelian groups. Proof. Look at the following diagrams, and convince yourself that it works. In each case note that the map on sections should exist uniquely, and then just confirm the maps comprise a map of presheaves. 1 ΨU iU ΦU HpUq kerpΦU q FpUq GpUq r r r r iV ΦV HpV q kerpΦV q FpV q GpV q ΨV Above, the restriction map on ker is just that of F. So it's not surprising that the map to the kernel is a map of presheaves. ΨU ΦU πU FpUq GpUq cokpΦU q HpUq r r r r ΦV πV FpV q GpV q cokpΦV q HpV q ΨV In the above case, the restriction map should work if the restriction map respects representatives, i.e., if 1 1 g ´ g P ImpΦU q ùñ g|V ´ g |V P ImpΦV q. This is true, and in fact it's why the restriction map works below. ΨU iU πU HpUq ImpΦU q GpUq pcok ΦqpUq r r r r iV πV HpV q ImpΦV q GpV q pcok ΦqpV q ΨV 2 Stalk and Kernel, Cokernel, and Image Let F ÝÑΦ G be a map of presheaves of abelian groups. Claim. The stalk of the presheaf kernel (resp. cokernel, resp. image) is isomorphic to the kernel (resp. cokernel, resp. image) of the stalk map, i.e. • pKer Φqx – KerpΦxq • pIm Φqx – ImpΦxq • pCok Φqx – CokpΦxq 2 Remark. Clearly a map of presheaves is injective (surjective) on all sections iff its presheaf kernel (cokernel) is zero. Since the stalk of the kernel is the kernel of the stalk, we might think that this implies that a map of presheaves is injective on all sections iff it is injective on all stalks. But when we try to make it work, we see Φ is monic (epic) in presheaves Φ injective (surjec- kerpre Φ “ 0 pcokpre Φ “ 0q tive) on all sections Φ injective (surjec- kerpΦ q “ 0 pcokpΦ q “ 0q, all x tive) on all stalks x x Stalk of kerpre Φ (cokpre Φ) is zero everywhere and the dashed arrow does not work, because it is very possible for a (unseparated) presheaf to have zero stalks but not be the zero presheaf. It just has to be \locally" zero, like the presheaf cokernel of the exponential map. On the other hand, it holds for the kernel if F is separated. If we work in the category of sheaves (once we find out what the kernel and cokernel are), we will find that the diagram has a different failure: Φ is monic (epic) in sheaves Φ injective (surjective) on all sections ker Φ “ 0 pcok Φ “ 0q Φ injective (surjective) on all stalks kerpΦxq “ 0 pcokpΦxq “ 0q, all x and we find that the dashed arrow holds only in the injective case, when the presheaf kernel is the sheaf kernel. Claim. Presheaves of abelian groups form an abelian category. pre Let Γp‚;Xq be the global section functor AbX Ñ Ab. Claim.Γ p‚;Xq is exact. Proof. This is clear, since Γ commutes with the taking of kernels and images. (Note that this works in presheaves though it fails in sheaves because it is harder for a sequence to be exact in the category of presheaves. Some wily exact sequences sneak in, in the category of sheaves, whose global sections are not exact.) 3 3 Sheaves Example. Let ' : X Ñ Y be continuous. Then for each V Ă Y , let ΓpV q be the continuous sections of '. This is a sheaf. (Note glueability might fail if we just talked about general subsets (incoherent cover).) Example. In fact, we need not even take continuous sections. Just any old sections as set functions form a sheaf (even sections of a discontinuous function). Claim that the "any old sections" sheaf is flabby if ' is surjective, and so any sheaf is a subsheaf of a flabby sheaf (the discontinuous sections of the projection map from the etale space). Given a presheaf F on X, we may form its sheafification in the "compatible germs" way. Another method is the \etale space" way. Here we form the etale space EF of the presheaf, which as a set is Fx, and has topology the strongest (?) such that f ÞÑ EF is a homeomorphism for all f. xPX ² Let U Ă X, and let µ P Fx. xPU Claim. ± (a) µ represents a compatible choice of germs ðñ µ is a continuous section U Ñ EF . ´1 (b) Stalks of ΓpEF Ñ Xq are isomorphic to Fx “ p rxs. 1 Proof. (a) is omitted. For (b), for any sections σ; σ : U Ñ EF , take pf; V q compatible around x, and 1 1 σpxq “ σ pxq σx “ σx D nbd where their compatible nbds are the same D nbd where σ “ σ1 ´1 1´1 with the LHS vertical arrow holding because take σ pV q X σ pV q where V is a nbd of x in EF . Claim. A presheaf F is separated ðñ Every section is determined by its germs. Proof. ùñ If two distinct sections f, f 1 give rise to the same germs, then for each point x in their domain we may choose a neighborhood Ux where they agree. Therefore F can't be separated. ðù If distinct f, f 1 agree on an open cover, then they surely give rise to the same germs. Remark. Note that this implies that if F Ñ G is a map of presheaves, and G is separated, then Φ is determined by its actions on the stalks. (Though note that germs don't determine a section in F, so Φ could for example be injective on stalks but not on sections, if F has some sections which are locally zero.) Claim. A presheaf F is separated ðñ pF ÝÑshf F shf q (the map sending each section to its collection of germs) is injective on each section, i.e., iff it is a monomorphism of presheaves. Proof. This follows from the previous claim. 4 In general, the glueability axiom does not play as nicely with the "compatible collection of germs" notion. We'd like to say Every collection pf ;U q agreeing Glueable i i on sections glues in the presheaf Every collection Sheafification pfi;Uiq agreeing surjective on stalks glues in the presheaf Unfortunately, the dashed arrow doesn't necessarily hold: if F is not separated, then germs do not determine sections, so there may be more collections agreeing on germs than there are collections agreeing on sections. However, we can see from this that Claim. A presheaf F is glueable ðù pF ÝÑshf F shf q is surjective on each section. One way to get the dashed arrow to work (even in the absence of F being separated) is to require that we're working on a paracompact hausdorff space (that is, to assume paracompactness of the open set you're concerned with. Or if you want it to work for all sections, assume hereditary paracompactness.) If we're given an element i in the sheafification (a compatible collection of germs represented by a cover f ;Ui agreeing on germs), we seek to find another cover with the same germs, which actually agrees as sections, and then use that F is assumed glueable. Our strategy for doing this will be to take our cover Ui, and to pick for each x a neighborhood Mx, and a subset Ix of the index set I, such that (a) Ix is finite for all x (b) Mx meets My ùñ Ix meets Iy. (c) i P Ix ùñ Mx Ă Ui. x i Then, if we can do this, we can just choose f to be a representative of tf : i P Ixu. (We may have to shrink x Mx in order to say that f is defined on Mx, but note that shrinking the Mx does not affect any of the above x y conditions.) Condition (b) will then guarantee that if Mx meets My, then f and f are both restrictions of x y some common Ui for i P Ix X Iy, and so f agrees with f as sections. Note also that if we have conditions (a) and (b), we can get condition (c) easily, since we can just shrink Mx to be inside the open set XiPIx Ui, remembering that shrinking Mx does not affect our conditions. So condi- tions (a) and (b) suffice. Given an open cover Ui, we might call such a choice x ÞÑ pIx;Mxq an indexed refinement of Ui. Lemma. Any locally finite open cover on a normal topological space has an indexed refinement. 5 df Proof. Take a shrinking of the cover Ui to a locally finite closed cover Zi, and take Ix “ ti P I : x P Ziu.
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