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COHOMOLOGY of SCHEMES 01X6 Contents 1. Introduction 1 2. Čech COHOMOLOGY OF SCHEMES 01X6 Contents 1. Introduction 1 2. Čech cohomology of quasi-coherent sheaves 2 3. Vanishing of cohomology 4 4. Quasi-coherence of higher direct images 6 5. Cohomology and base change, I 9 6. Colimits and higher direct images 12 7. Cohomology and base change, II 12 8. Cohomology of projective space 14 9. Coherent sheaves on locally Noetherian schemes 20 10. Coherent sheaves on Noetherian schemes 23 11. Depth 25 12. Devissage of coherent sheaves 26 13. Finite morphisms and affines 31 14. Coherent sheaves on Proj, I 33 15. Coherent sheaves on Proj, II 37 16. Higher direct images along projective morphisms 40 17. Ample invertible sheaves and cohomology 42 18. Chow’s Lemma 45 19. Higher direct images of coherent sheaves 47 20. The theorem on formal functions 49 21. Applications of the theorem on formal functions 55 22. Cohomology and base change, III 57 23. Coherent formal modules 58 24. Grothendieck’s existence theorem, I 63 25. Grothendieck’s existence theorem, II 64 26. Being proper over a base 67 27. Grothendieck’s existence theorem, III 71 28. Grothendieck’s algebraization theorem 73 29. Other chapters 76 References 78 1. Introduction 01X7 In this chapter we first prove a number of results on the cohomology of quasi- coherent sheaves. A fundamental reference is [DG67]. Having done this we will elaborate on cohomology of coherent sheaves in the Noetherian setting. See [Ser55]. This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 COHOMOLOGY OF SCHEMES 2 2. Čech cohomology of quasi-coherent sheaves 01X8 Let X be a scheme. Let U ⊂ X be an affine open. Recall that a standard open Sn covering of U is a covering of the form U : U = i=1 D(fi) where f1, . , fn ∈ Γ(U, OX ) generate the unit ideal, see Schemes, Definition 5.2. 01X9 Lemma 2.1. Let X be a scheme. Let F be a quasi-coherent OX -module. Let Sn U : U = i=1 D(fi) be a standard open covering of an affine open of X. Then Hˇ p(U, F) = 0 for all p > 0. Proof. Write U = Spec(A) for some ring A. In other words, f1, . , fn are elements of A which generate the unit ideal of A. Write F|U = Mf for some A-module M. Clearly the Čech complex Cˇ•(U, F) is identified with the complex Y Y Y M → M → M → ... fi0 fi0 fi1 fi0 fi1 fi2 i0 i0i1 i0i1i2 We are asked to show that the extended complex Y Y Y (2.1.1)01XA 0 → M → M → M → M → ... fi0 fi0 fi1 fi0 fi1 fi2 i0 i0i1 i0i1i2 (whose truncation we have studied in Algebra, Lemma 24.1) is exact. It suffices to show that (2.1.1) is exact after localizing at a prime p, see Algebra, Lemma 23.1. In fact we will show that the extended complex localized at p is homotopic to zero. There exists an index i such that fi 6∈ p. Choose and fix such an element ifix. Note that M = M . Similarly for a localization at a product f . f and p we fifix ,p p i0 ip can drop any fij for which ij = ifix. Let us define a homotopy Y Y h : M −→ M fi0 ...fip+1 ,p fi0 ...fip ,p i0...ip+1 i0...ip by the rule h(s)i0...ip = sifixi0...ip (This is “dual” to the homotopy in the proof of Cohomology, Lemma 10.4.) In other words, h : Q M → M is projection onto the factor M = M and in i0 fi0 ,p p fifix ,p p general the map h equal projection onto the factors M = M . fifix fi1 ...fip+1 ,p fi1 ...fip+1 ,p We compute p X j (dh + hd)(s)i ...i = (−1) h(s) ˆ + d(s)i i ...i 0 p j=0 i0...ij ...ip fix 0 p p p X j X j+1 = (−1) s ˆ + si ...i + (−1) s ˆ j=0 ifixi0...ij ...ip 0 p j=0 ifixi0...ij ...ip = si0...ip This proves the identity map is homotopic to zero as desired. The following lemma says in particular that for any affine scheme X and any quasi- coherent sheaf F on X we have Hp(X, F) = 0 for all p > 0. 01XB Lemma 2.2. Let X be a scheme. Let F be a quasi-coherent OX -module. For any affine open U ⊂ X we have Hp(U, F) = 0 for all p > 0. COHOMOLOGY OF SCHEMES 3 Proof. We are going to apply Cohomology, Lemma 11.9. As our basis B for the topology of X we are going to use the affine opens of X. As our set Cov of open coverings we are going to use the standard open coverings of affine opens of X. Next we check that conditions (1), (2) and (3) of Cohomology, Lemma 11.9 hold. Note that the intersection of standard opens in an affine is another standard open. Hence property (1) holds. The coverings form a cofinal system of open coverings of any element of B, see Schemes, Lemma 5.1. Hence (2) holds. Finally, condition (3) of the lemma follows from Lemma 2.1. Here is a relative version of the vanishing of cohomology of quasi-coherent sheaves on affines. 01XC Lemma 2.3. Let f : X → S be a morphism of schemes. Let F be a quasi-coherent i OX -module. If f is affine then R f∗F = 0 for all i > 0. i Proof. According to Cohomology, Lemma 7.3 the sheaf R f∗F is the sheaf asso- i −1 ciated to the presheaf V 7→ H (f (V ), F|f −1(V )). By assumption, whenever V is affine we have that f −1(V ) is affine, see Morphisms, Definition 11.1. By Lemma i −1 2.2 we conclude that H (f (V ), F|f −1(V )) = 0 whenever V is affine. Since S has a basis consisting of affine opens we win. 089W Lemma 2.4. Let f : X → S be an affine morphism of schemes. Let F be a i i quasi-coherent OX -module. Then H (X, F) = H (S, f∗F) for all i ≥ 0. Proof. Follows from Lemma 2.3 and the Leray spectral sequence. See Cohomology, Lemma 13.6. The following two lemmas explain when Čech cohomology can be used to compute cohomology of quasi-coherent modules. 0BDX Lemma 2.5. Let X be a scheme. The following are equivalent (1) X has affine diagonal ∆ : X → X × X, (2) for U, V ⊂ X affine open, the intersection U ∩ V is affine, and S (3) there exists an open covering U : X = i∈I Ui such that Ui0...ip is affine open for all p ≥ 0 and all i0, . , ip ∈ I. In particular this holds if X is separated. Proof. Assume X has affine diagonal. Let U, V ⊂ X be affine opens. Then U ∩ V = ∆−1(U × V ) is affine. Thus (2) holds. It is immediate that (2) implies S (3). Conversely, if there is a covering of X as in (3), then X × X = Ui × Ui0 is an −1 affine open covering, and we see that ∆ (Ui × Ui0 ) = Ui ∩ Ui0 is affine. Then ∆ is an affine morphism by Morphisms, Lemma 11.3. The final assertion follows from Schemes, Lemma 21.7. S 01XD Lemma 2.6. Let X be a scheme. Let U : X = i∈I Ui be an open covering such that Ui0...ip is affine open for all p ≥ 0 and all i0, . , ip ∈ I. In this case for any quasi-coherent sheaf F we have Hˇ p(U, F) = Hp(X, F) as Γ(X, OX )-modules for all p. Proof. In view of Lemma 2.2 this is a special case of Cohomology, Lemma 11.6. COHOMOLOGY OF SCHEMES 4 3. Vanishing of cohomology 01XE We have seen that on an affine scheme the higher cohomology groups of any quasi- coherent sheaf vanish (Lemma 2.2). It turns out that this also characterizes affine schemes. We give two versions. 01XF Lemma 3.1. Let X be a scheme. Assume that [Ser57], [DG67, II, (1) X is quasi-compact, Theorem 5.2.1 (d’) 1 (2) for every quasi-coherent sheaf of ideals I ⊂ OX we have H (X, I) = 0. and IV (1.7.17)] Then X is affine. Proof. Let x ∈ X be a closed point. Let U ⊂ X be an affine open neighbourhood of x. Write U = Spec(A) and let m ⊂ A be the maximal ideal corresponding to x. Set Z = X \U and Z0 = Z ∪{x}. By Schemes, Lemma 12.4 there are quasi-coherent sheaves of ideals I, resp. I0 cutting out the reduced closed subschemes Z, resp. Z0. Consider the short exact sequence 0 → I0 → I → I/I0 → 0. Since x is a closed point of X and x 6∈ Z we see that I/I0 is supported at x. In fact, the restriction of I/I0 to U corresponds to the A-module A/m. Hence we see that Γ(X, I/I0) = A/m. Since by assumption H1(X, I0) = 0 we see there exists a global section f ∈ Γ(X, I) which maps to the element 1 ∈ A/m as a section of 0 I/I . Clearly we have x ∈ Xf ⊂ U. This implies that Xf = D(fA) where fA is the image of f in A = Γ(U, OX ). In particular Xf is affine.
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