PRO-ÉTALE COHOMOLOGY Contents

Contents 1. Introduction 1 2. Some topology 2 3. Local isomorphisms 4 4. Ind-Zariski algebra 6 5. Constructing w-local affine schemes 6 6. Identifying local rings versus ind-Zariski 10 7. Ind-´ etale algebra 14 8. Constructing ind-´ etale algebras 15 9. Weaklyétale versus pro-´ etale 18 10. Constructing w-contractible covers 18 11. The pro-´ etale site 21 12. Points of the pro-´ etale site 29 13. Compact generation 31 14. Generalities on derived completion 31 15. Application to theorem on formal functions 39 16. Derived completion in the constant Noetherian case 40 17. Derived completion on the pro-´ etale site 41 18. Comparison with thé etale site 43 19. Cohomology of a point 47 20. Weakly contractible hypercoverings 48 21. Functoriality of the pro-´ etale site 50 22. Finite morphisms and pro-´ etale sites 51 23. Closed immersions and pro-´ etale sites 51 24. Extension by zero 54 25. Constructible sheaves on the pro-´ etale site 56 26. Constructible adic sheaves 57 27. A suitable derived category 61 28. Proper base change 65 29. Other chapters 66 References 67