Descent on the étale site Wouter Zomervrucht, October 14, 2014


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1 Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasicoherent sheaves. All will also be true on the larger pp and pqc sites. 1. Descent o morphisms Let S be a scheme. Recall: the étale site (Sch/S) et is the category Sch/S together with the étale coverings; a shea on (Sch/S) et is a unctor (Sch/S) op Set that satisies the shea property or all étale coverings. Theorem 1.1 (Grothendieck). Let be a scheme over S. Then : (Sch/S) op Set, T (T) := Hom S (T, ) is a shea on (Sch/S) et. Corollary 1.2. Morphisms can be deined étalelocally on the source. Corollary 1.3. Morphisms can be deined étalelocally on S. The irst corollary is actually a reormulation o the theorem. It generalizes the rather trivial operation o gluing morphisms along opens. Proo. Reduction to the aine case. Let V U be an étale covering. Since morphisms glue along opens, we may assume U is aine. Since U is quasicompact and étale morphisms are open, there is a inite collection o aine opens W 1,..., W n V that jointly surject to U. Then also W = n i=1 W i is aine and W U is an étale covering. Take y (V) with equal pullbacks to (V U V). Assuming that has the shea property or W U, there is a unique x (U) with x W = y W. It is the sole candidate to satisy x V = y. We claim that this indeed is true. The identity x V = y may be veriied on an open covering. It holds certainly on each W i. Now just observe that x does not depend on the choice o W; compare choices W, W via W W. Homological algebra. We have reduced to étale coverings W U with U, W aine. Write U = Spec A and W = Spec B. Then A B is aithully lat. Deine r : B B A B by r(b) = 1 b b 1. We claim that the complex r 0 A B B A B is an exact sequence o Amodules. Since A B is aithully lat, it suices to prove that r 0 B B B A B B A B A B is an exact sequence o Bmodules. Here the middle map is given by b 1 b and r B by b c 1 b c b 1 c. But now the maps B A B B, B A B A B B A B, b c bc, b c d c bd show that the identity on this complex is nullhomotopic. So the complex is exact. 1
2 Conclusion o the proo. We have to show that any commutative diagram W U W W W U g can be extended uniquely by a map g : U. Reasoning as beore, we may assume is aine. Write = Spec R. Since A B is the equalizer o the coprojections B B A B in AMod, there is a unique Amodule homomorphism g : R A compatible with. It is automatically a ring map. Example 1.4. The ollowing unctors Sch op Ab are étale sheaves. G a (T) = O(T) is representable by Spec Z[t]. G m (T) = O(T) is representable by Spec Z[t, t 1 ]. µ n (T) = O(T) [n] is representable by Spec Z[t]/(t n 1). Example 1.5. Let A be an abelian group. The constant preshea (Sch/S) op Ab, T A is not a shea on (Sch/S) zar. Its Zariski sheaiication is A : (Sch/S) op Ab, T {continuous unctions T A}. But A is representable by a A S, so is also the étale sheaiication o the constant preshea. Many properties o morphisms are preserved under descent. Theorem 1.6. Let : Y be a morphism o schemes over S. Let S S be an étale surjection, and let : Y be the base change o. I is an isomorphism aine universally open inite universally closed an open immersion quasicompact a closed immersion separated lat (locally) o inite type smooth (locally) o inite presentation unramiied proper étale then so is. Warning: (quasi)projectivity is not preserved in general! Proo. We may assume Y = S. Each o the properties is Zariski local on the target, so we may assume S is aine. There is an aine scheme S, étale over S, that surjects to S. Each o the properties is stable under base change, so : S has the property as well. Thus we may assume S is aine. Summarizing, let a cartesian diagram S q p S 2
3 o schemes be given, in which p is an étale surjection o aine schemes. For each o the properties we have to show that i has the property, then so does. Isomorphism. Suppose is an isomorphism. Let g : S be its inverse. Its two pullbacks to S are inverse to, so identical, hence by theorem 1.1 g descends to a morphism g : S. Étalelocally both g and g are the identity. By the shea property, g and g are the identity globally, hence g is an inverse to and is an isomorphism. Universally open. Suppose is universally open. Let T S be any morphism, and let U T be open. Then p 1 T ( T(U)) = T (q 1 T (U)) is open in S T. Since étale surjections are topological quotients, this implies T (U) is open in S T. Universally closed. Analogous. Quasicompact. A morphism to an aine scheme is quasicompact i and only i the source is quasicompact. Since q is surjective, i is quasicompact then so is. Separated. A morphism is separated i and only i the diagonal is universally closed. The diagram S q S is an étale base change, so we are done. (Locally) o inite type. By descent o quasicompactness it suices to consider locally o inite type morphisms. Being locally o inite type is Zariski local on the source, so we may assume that is aine. Write S = Spec R, S = Spec R, = Spec A, and A = A R R. Suppose R A is o inite type, i.e. there exist b 1,..., b n A that generate A as R algebra. Write b i = j a ij r ij. Then the map R[x ij : i, j] A, x ij a ij is surjective ater the base change R R. Since R R is aithully lat, the map itsel is also surjective. Hence R A is o inite type. (Locally) o inite presentation. Use notation as beore. Suppose R A is o inite presentation. Certainly R A is o inite type. Take a surjection R[x 1,..., x n ] A with kernel I. Since R R is lat, the base change R [x 1,..., x n ] A has kernel I R R. It is initely generated, say by h 1,..., h m. Write h i = j g ij r ij. Arguing as beore, I is generated by {g ij : i, j}. Proper. Proper means separated, o inite type, and universally closed. Aine. Suppose is aine, i.e. is aine. Write S = Spec R, S = Spec R, and = Spec A. Also write A = O (). Since is quasicompact and separated, so is. Now lat base change says p O = O, in other words A = A R R. Consider the canonical map Spec A. The base change Spec A is an isomorphism, so Spec A is an isomorphism. Finite. Finite is equivalent to aine and proper. Open immersion. Suppose is an open immersion. Then it is universally open, so is universally open. Replace S by (); we need to prove is an isomorphism. But now and are surjective, so is an isomorphism. Closed immersion. Suppose is a closed immersion. Then it is aine, so is aine. In particular and are aine. Use notation as beore. Since is a closed immersion, R A is surjective. As R R is aithully lat, also R A is surjective, and is a closed immersion. Flat. Being lat is Zariski local on the source, so we may assume that is aine. Let R, R, A, and A be as beore. Suppose R A is lat. Let M N be an injection o Rmodules. Since R R A is lat, M R A N R A is injective. Since A A is aithully lat, M R A N R A is also injective. Hence R A is lat. 3
4 Smooth. A morphism is smooth i and only i it is locally o inite presentation, lat, and has smooth ibers. The irst two properties descend, so we are reduced to the case S = Spec k, S = Spec k or some inite separable ield extension k /k. But or /k locally o inite type, /k is smooth i and only i is geometrically regular over k. Unramiied. A morphism is unramiied i and only i it is locally o inite type and the diagonal is an open immersion. Étale. A morphism is étale i and only i it is locally o inite presentation, lat, and unramiied. 2. Descent o quasicoherent sheaves Deinition 2.1. Let U = {U i S} i I be an étale covering. A descent datum o quasicoherent sheaves or U consists o a quasicoherent shea F i QCoh U i or all i I, and an isomorphism ϕ ji : F i Uij F j Uij in QCoh U ij or all i, j I, such that the cocycle condition ϕ ki = ϕ kj ϕ ji holds on U ijk or all i, j, k I. Deinition 2.2. A morphism (F i, ϕ ji ) (G i, ψ ji ) o descent data or U consists o a morphism α i : F i G i in QCoh U i or all i I, such that ψ ji α i = α j ϕ ji holds on U ij or all i, j I. The category o descent data o quasicoherent sheaves or U is denoted QCoh U. Let F be a quasicoherent shea on S. Then (F Ui, id ji ) is a descent datum. This construction is unctorial. Theorem 2.3 (Grothendieck). The unctor QCoh S QCoh U is an equivalence. In other words, quasicoherent sheaves and their morphisms descend uniquely along étale coverings. This is not true or arbitrary sheaves o modules. The proo is very similar to that o theorem 1.1. Indeed, descent o quasicoherent sheaves is a shea condition o sorts; more precisely, the theorem states that QCoh is a 2shea or stack over (Sch/S) et. Remark 2.4. Corollary 1.3 states that Hom S (, Y) is a shea on (Sch/S) et. This ollows directly rom theorem 2.3. Indeed, we may reduce to étale coverings V U with U, V aine. We may also assume that S, and Y are aine. But an aine scheme over Y is a quasicoherent algebra on Y. Descent o quasicoherent algebras ollows rom that o quasicoherent sheaves. The ollowing proposition is an important application. Proposition 2.5. Let F be a quasicoherent shea on S. Then F : (Sch/S) op Ab, T F(T) := F T (T) is a shea on (Sch/S) et. Proo. A section t F(T) is a morphism O T F T. For instance, we can speak o the structure shea on (Sch/S) et. Any quasicoherent shea on S has associated étale cohomology. 4
5 Reerences Brochard, Topologies de Grothendieck, descente, quotients. This lecture and a bit more is covered in chapter 2. de Jong, Stacks Project. Descent is the topic o chapter More in particular, theorem 1.1 is in section 023P, theorem 1.6 in section 02YJ, and theorem 2.3 in section 023R. 5
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