There are two practical ways to complete an abelian group A at a prime p. One could complete A with respect to the neighborhood base of zero given by the subgroups pA. This completion, called p-completion, is limA/pA. Or one could complete A with respect to the neighborhood base of zero given by subgroups B ⊆ A of finite p-power index. This completion, called p-pro-finite completion, is limAα w...

Let Φ be a root system and let Φ(Zp) be the standard Chevalley Zp-Lie algebra associated to Φ. For any integer t > 1, let G be the uniform pro-p group corresponding to the powerful Lie algebra pΦ(Zp) and suppose that p > 5. Then the Iwasawa algebra ΩG has no nontrivial reflexive ideals. This was previously proved by authors for the root system A1.

0965 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...

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 1...