foundations for almost ring theory november 4, 2016

FOUNDATIONS FOR ALMOST RING THEORY

OFER GABBER AND LORENZO RAMERO

January 23, 2018

Release 7

Ofer GabberI.H.E.S.Le Bois-Marie35, route de ChartresF-91440 Bures-sur-Yvettee-mail address: [email protected]

Lorenzo RameroUniversite de Lille ILaboratoire de MathematiquesF-59655 Villeneuve dAscq Cedexe-mail address: [email protected] page: http://math.univ-lille1.fr/ramero

Acknowledgements We thank Niels Borne, Michel Emsalem, Pierre-Yves Gaillard, Lutz Geissler, W.-P. Heidorn,Fabrice Orgogozo, Olaf Schnurer and Peter Scholze for pointing out some mistakes in earlier drafts, and for usefulsuggestions.

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mailto:[email protected]:[email protected]://math.univ-lille1.fr/~ramero

2 OFER GABBER AND LORENZO RAMERO

CONTENTS

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. Basic category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1. Categories, functors and natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2. Presheaves and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3. Adjunctions and Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.4. Special properties of the categories of presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.5. Final and cofinal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.6. Localizations of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612. 2-Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.1. 2-Categories and pseudo-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.2. Pseudo-natural transformations and their modifications . . . . . . . . . . . . . . . . . . . . . . . . 822.3. The formalism of base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.4. Adjunctions in 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.5. 2-Limits and 2-colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282.6. 2-Categorical Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383. Special categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543.1. Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543.2. 2-Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793.3. Fibrations in groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1953.4. Sieves and descent theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1983.5. Profinite groups and Galois categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2133.6. Tensor categories and abelian categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214. Sites and topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.1. Topologies and sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.2. Continuous and cocontinuous functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2484.3. Morphisms of sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2564.4. Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2704.5. Fibred sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.6. Fibred topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2894.7. Localization and points of a topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3044.8. Algebra on a topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3204.9. Torsors on a topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3365. Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3435.1. Prestacks and stacks on a site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3435.2. Covering morphisms of prestacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3565.3. Local calculus of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3655.4. Functorial properties of the categories of stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3705.5. Sheaves of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3875.6. Stacks in groupoids and ind-finite stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.7. Stacks on fibred sites and fibred topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4166. Monoids and polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4346.1. Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4356.2. Integral monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3. Polyhedral cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4. Fine and saturated monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4756.5. Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4896.6. Special subdivisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5047. Homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

FOUNDATIONS FOR ALMOST RING THEORY 3

7.1. Complexes in an additive category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.2. Filtered complexes and spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5337.3. Derived categories and derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5417.4. Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5617.5. Simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5877.6. Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6207.7. Differential graded algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6307.8. Koszul algebras and regular sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6357.9. Filtered rings and Rees algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6497.10. Some homotopical algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6557.11. Injective modules, flat modules and indecomposable modules . . . . . . . . . . . . . . . . . 6698. Complements of topology and topological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6808.1. Spectral spaces and constructible subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6808.2. Topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7028.3. Topological rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7128.4. Topologically local and topologically henselian rings . . . . . . . . . . . . . . . . . . . . . . . . . . 7298.5. Graded structures on topological rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7358.6. Homological algebra for topological modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7419. Complements of commutative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7549.1. Valuation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7549.2. Hubers theory of the valuation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7699.3. Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7879.4. Fontaine rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.5. Divided power modules and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8219.6. Regular rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8319.7. Excellent rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84610. Cohomology and local cohomology of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85710.1. Cohomology of topoi and topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85710.2. Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87010.3. Quasi-coherent modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88310.4. Depth and cohomology with supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89310.5. Depth and associated primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90610.6. Cohomology of projective schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91911. Duality theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93711.1. Duality for quasi-coherent modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93711.2. Cousin complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94411.3. Duality over coherent schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95511.4. Schemes over a valuation ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97611.5. Local duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99111.6. Hochsters theorem and Stanleys theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100712. Logarithmic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102012.1. Log topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102012.2. Log schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103112.3. Logarithmic differentials and smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .104612.4. Logarithmic blow up of a coherent ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106012.5. Regular log schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107612.6. Resolution of singularities of regular log schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . .109412.7. Local properties of the fibres of a smooth morphism . . . . . . . . . . . . . . . . . . . . . . . . . .111313. Etale coverings of schemes and log schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112013.1. Acyclic morphisms of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1120


13.2. Local asphericity of smooth morphisms of schemes . . . . . . . . . . . . . . . . . . . . . . . . . .113513.3. Etale coverings of log schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114713.4. Local acyclicity of smooth morphisms of log schemes . . . . . . . . . . . . . . . . . . . . . . . .116714. The almost purity toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118014.1. Non-flat almost structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118014.2. Inverse systems of almost modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120414.3. Almost pure pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122414.4. Normalized lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124214.5. Finite group actions on almost algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126714.6. Almost Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127514.7. Complements : locally measurable algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128615. Continuous valuations and adic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130315.1. Formal schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130315.2. Analytically noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131915.3. Continuous valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133215.4. Affinoid rings and affinoid schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134215.5. Adic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135615.6. Special loci of quasi-affinoid schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137815.7. Etale coverings of quasi-affinoid schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138316. Perfectoid rings and perfectoid spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139616.1. Distinguished elements and transversal pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139616.2. P-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140116.3. Perfectoid rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141016.4. Homological theory of perfectoid rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143916.5. Perfectoid quasi-affinoid rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145616.6. Graded perfectoid rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148416.7. Perfectoid spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150416.8. Almost purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152216.9. Perfectoid Tate rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154117. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154917.1. Model algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154917.2. Almost purity : the log regular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1564References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1577

It is not incumbent upon you to complete the work,but neither are you at liberty to desist from it.

(Avot 2:21)

0. INTRODUCTION

Both the focus of this monograph and its subject matter have evolved considerably in thelast few years. On the one hand, the insistence on making the text self-contained (aside from areduced canon of basic references, which should ideally contain only EGA and some parts ofBourbakis Elements), has resulted in a rather weighty mass of material of independent interest,that is applied to, but is completely separate from almost ring theory, and whose relationship top-adic Hodge theory is thus even more indirect. Rather than stemming from a well-thought-outplan, this part is the outcome of a haphazard process, lumbering between alternating phases ofaccretion and consolidation, with new topics piled up as dictated by need, or occasionally bywhim, when we just branched out from the main flow to pursue a certain line of thought to its


logical destination. Nevertheless, a few themes have spontaneously emerged, around which theoriginally amorphous magma has been able to settle, to the point where by now a distinct shapeis finally discernible, and it is perhaps time to pause and take stock of its broad outlines.

Now then, we may distinguish : First of all, a rather thorough exposition of the foundations of logarithmic algebraic geom-

etry, comprising chapters 6, 12 and 13. Inevitably, our treatment owes a lot to the works of Katoand his school : our contribution is foremost that of gathering and tidying up the subject, whichuntil now was scattered in a disparate number of research articles, many of them still unpub-lished and even unfinished. A closer scrutiny would also reveal a few technical innovations thatwe hope will become standard issue of the working algebraic log-geometer : we may mentionthe systematic use of pointed monoids and pointed modules, the projective fan associated witha graded monoid, or a definition of -flatness for log structures which refines and generalizesan older notion of toric flatness. Furthermore, we took the occasion to repair a few small (andnot so small) mistakes and inaccuracies that we detected in the literature. Two other chapters are dedicated to local cohomology and Grothendiecks duality theory.

Early on, the emphasis here was on generalizations : especially, we were interested in removingfrom the theory the pervasive noetherian assumptions, to pave the way for our recasting ofFaltingss almost purity in the framework of valuation theory. Applications of local cohomologyto non-archimedean analytic geometry furnished another influential motivation, though one thathas remained, so far, hidden from view. More recently, the noetherian aspects have also becomerelevant to our project, and this latest release contains a detailed account of the most importantproperties of noetherian rings endowed with a dualizing complex. The latter, in turn, could bedealt with satisfactorily only after a thorough revisitation of the general theory of the dualizingcomplex, so that our chapters 10 and 11 can also be regarded as complementary to Conradsbook [31] (dedicated to the trace morphism and the deeper aspects of duality) : totaling ourrespective efforts, it should eventually become possible to bypass entirely Hartshornes notes[59] which, as is well known, are wanting in many ways. Chapters 7 and 9 present (for the time being, anyway) a looser structure : a miscellanea

of self-contained units devoted to more or less independent topics. However, there is at leastone thread running through several sections, and whose stretch can be traced all the way backto the earliest beginnings of almost ring theory; it connects sections 7.4 and 7.5 on simplicialhomotopy theory to a section 7.10 dedicated to homotopical algebra, then on to sections 9.6and 9.7, which make extensive use of the cotangent complex to derive important characteriza-tions of regular and excellent rings, including an up-to-date presentation of classical results dueto Andre, extracted from his monograph [2], and from his paper [3] on localization of formalsmoothness. This homotopical algebraic thread resurfaces again in section 14.1, but there weare already squarely into almost ring theory proper.

On the other hand, two recent notable developments are compelling a revision of our un-derstanding of almost ring theory itself, and of its situation within commutative algebra andalgebraic geometry at large : The first is Scholzes PhD thesis [97] on perfectoid spaces, that contains both a maximal

generalization of the almost purity theorem, and a major simplification of its proof, based on histilting technique (and completely different from Faltingss). However, the range of Scholzestheory transcends the domain of p-adic Hodge theory (which was not even his original moti-vation) : to drive the point home, his thesis concludes with a clever application to the longstanding weight monodromy conjecture, thus affording the unusual spectacle of a tool whichwas fashioned out of purely p-adic concerns, and ends up playing a crucial role in the solutionof a purely `-adic problem.


The second spectacular development is Yves Andres proof of the direct summand con-jecture ([4]); the latter is a deceptively simple assertion that has been a central problem incommutative algebra for the last thirty years : it asserts that every finite injective ring homo-morphism f : B A from a regular local ring B, admits a B-linear splitting. The relevanceof almost purity to this question was first surmised by Paul Roberts in 2001 (after a talk by thesecond author at the University of Utah), and has been widely advertised by him ever since.Andres solution uses perfectoid techniques, and builds on earlier work by Bhargav Bhatt, whoin [12] proved the conjecture in the case where B is essentially smooth over a mixed character-istic discrete valuation ring and f Z Q is etale outside a relative normal crossings divisor ofSpecB. Moreover, Bhatt has subsequently simplified some of Andres arguments and shownhow the same method yields a more general derived version of the conjecture, for properschemes over any regular ring : see [13].

We see then, that almost ring theory has emancipated itself from its former ancillary rolein the exclusive service of p-adic Hodge theory, and is now elbowing out a niche in the widerecosystem of algebraic geometry.

The present release completes the project announced in the introduction of the 6th release : First we introduce a class of topological rings that generalize the perfectoid rings of [97]; it

is very easy to say what a (generalized) perfectoid Fp-algebra is : namely, it is just a perfect andcomplete topological Fp-algebra whose topology is linear, defined by an ideal of finite type. Thegeneral definition is somewhat more involved, but we prove the following characterization. Forany perfectoid Fp-algebra E, we consider the ring of Witt vectors W (E), and we endow it witha natural topology, induced from that of E; then every perfectoid ring is a topological quotientof the form A := W (E)/aW (E), for such a suitable E, and where a W (E) is what we call adistinguished element : see definition 16.1.6. Moreover, just as in Scholzes work, the perfectoidFp-algebra E can be recovered from A via a tilting functor that establishes, more precisely, anequivalence between the category of all perfectoid rings and that of pairs (E,I ) consisting of aperfectoid Fp-algebra E and a principal ideal I W (E) generated by a distinguished element(as it is well known, this construction is rooted in Fontaine and Winterbergers theory of thefield of norms). The distinguished ideal I represents an extra parameter that remains hiddenin Scholzes original approach : the reason is that he fixes from the start a base perfectoid fieldK, thereby implicitly fixing as well a distinguished element a in the ring of Witt vectors of thetilt of K, and then every perfectoid ring in his work is supposed to be a K-algebra, which from our viewpoint amounts to restricting to perfectoid rings whose associated distinguishedideal is generated by a. Having thus removed the parameter I , he can then also do away withWitt vectors altogether, and the inverse to the tilting construction is obtained in [97] via a moreabstract deformation theoretic argument. This route is precluded to us, so we rely instead ondirect and rather concrete Witt vectors calculations. A similar strategy has been proposed in[75], and our viewpoint can indeed be described fairly as an interpolation of those of Scholzeand Kedlaya-Liu, though we only studied [97] in detail. The first three sections of chapter 16 are devoted to exploring this new class of perfectoid

rings and its manifold remarkable features. The rest of the chapter then merges the theory ofperfectoid rings with Hubers adic spaces, to forge the perfectoid spaces that are the main toolfor our proof of almost purity, whose most general form is given by theorem 16.8.39 and appliesto formal perfectoid rings, i.e. to topological rings whose completion is perfectoid. The proofproceeds via several preliminary reductions : first, to the case of a perfectoid quasi-affinoidring, covered by theorem 16.8.28, then by exploiting the local geometry of adic spaces tothe case of a perfectoid valuation ring, which was treated already in our monograph [48]. Whatenables here this localization argument is a basic feature of the etale topology of arbitrary adicspaces : the fibred category of finite etale coverings of the affinoid subsets of an adic space is astack. The latter result is in turn a special case of our theorem 15.7.6.


We also include a detailed treatment of the foundations of the theory of adic spaces, thatessentially follows [65], but contains some modest improvement : notably, the systematic useof analytically noetherian rings (borrowed from [45]) allows us to unify the two classes oftopological rings that Huber dealt with separately in his work (the strongly noetherian rings andthe f-adic rings with a noetherian ring of definition). We also point out a henselian variant ofthe structure sheaf that is available on the adic spectrum of any f-adic ring, with no restrictionwhatsoever. The last chapter proposes for now a couple of applications : in section 17.1 we introduce a

class of model algebras over any rank one valuation ring K+ of mixed characteristic (0, p), andwe show that when K+ is deeply ramified, such algebras are formal perfectoid rings for theirp-adic topology; hence the theory of chapter 16 immediately yields an almost purity theorem formodel algebras. Likewise, section 17.2 proves an almost purity theorem for certain very ram-ified towers of log-regular rings; again, after some preliminary reductions, the proof amountsto the observation that the inductive limits of such towers are formal perfectoid for their p-adictopology. These instances of almost purity were already contained in a previous draft of ourwork (Release 6), where they were proven by an extension of Faltingss method, that relied ondeep results from logarithmic algebraic geometry, and also entailed the construction of certainnormalized lengths for torsion modules over model algebras, and respectively over the ringsoccurring in section 17.2. Neither of these two ingredients intervenes any longer in the newproofs; however, we have found worthwhile to explain how model algebras arise from suitablevery ramified towers of log-smooth K+-algebras, and we have also retained the construction ofnormalized lengths for model algebras and for limits of towers log-regular rings, since they aresufficiently interesting in their own right, and might be useful for other applications (normalizedlengths for torsion K+-modules are exploited in [98]).

In the next release of this treatise, we shall complete chapter 17 with our account of Andreswork on the direct summand conjecture, and with some further applications of our theory ofgeneralized perfectoid rings.

1. BASIC CATEGORY THEORY

The purpose of this chapter is to fix some notation that shall stand throughout this work, andto collect, for ease of reference, a few well known generalities on categories and functors thatare frequently used. Our main reference on general nonsense is the treatise [15], and anothergood reference is the more recent [72].

Sooner or later, any honest discussion of categories and topoi gets tangled up with somefoundational issues revolving around the manipulation of large sets. For this reason, to beable to move on solid ground, it is essential to select from the outset a definite set-theoreticalframework (among the several currently available), and stick to it unwaveringly.

Thus, throughout this work we will accept the so-called Zermelo-Fraenkel system of axiomsfor set theory. (In this version of set theory, everything is a set, and there is no primitive notionof class, in contrast to other axiomatisations.)

Additionally, following [5, Exp.I, 0], we shall assume that, for every set S, there exists auniverse V such that S V. (For the notion of universe, the reader may also see [15, 1.1].)

Throughout this chapter, we fix some universe U such that N U (where N is the set ofnatural numbers; the latter condition is required, in order to be able to perform some standardset-theoretical operations without leaving U). A set S is U-small (resp. essentially U-small), ifS U (resp. if S has the cardinality of a U-small set). If the context is not ambiguous, we shalljust write small, instead of U-small.


1.1. Categories, functors and natural transformations. A category C is the datum of a setOb(C ) of objects and, for every A,B Ob(C ), a set of morphisms from A to B, denoted :

HomC (A,B)

and as usual we write f : A B to signify f HomC (A,B). Furthermore, we setMorph(C ) := {(A,B, f) | A,B Ob(C ), f HomC (A,B)}.

For any f := (A,B, f) Morph(C ), the object A is called the source of f , and B is the targetof f . We also often use the notation

EndC (A) := HomC (A,A)

and the elements of EndC (A) are called the endomorphisms of A in C . We say that a pair ofelements (f, g) of Morph(A ) is composable if the target of f equals the source of g. Moreover,for every A,B,C Ob(C ) we have a composition law

HomC (A,B) HomC (B,C) HomC (A,C) : (f, g) 7 g ffulfilling the following two standard axioms :

For every A Ob(C ) there exists an identity endomorphism 1A of A, such that1A f = f g 1A = g for every B,C Ob(C ) and every f : B A and g : A C.

The composition law is associative, i.e. we have(h g) f = h (g f)

for every A,B,C,D Ob(C ) and every f : A B, g : B C and h : C D.Clearly, it follows that (EndC (A), ,1A) is a monoid, and we get a group :

AutC (A) EndC (A)of invertible endomorphisms, i.e. the automorphisms of the object A.

1.1.1. We say that the category C is U-small (or just small), if both Ob(C ) and Morph(C ) aresmall sets. We say that C has small Hom-sets if HomC (A,B) U for every A,B Ob(C ).

A subcategory of C is a category B with Ob(B) Ob(C ) and Morph(B) Morph(C ).The opposite category C o is the category with Ob(C o) = Ob(C ), and such that :

HomC o(A,B) := HomC (B,A) for every A,B Ob(C )(with composition law induced by that of C , in the obvious way). Given A Ob(C ), some-times we denote by Ao the same object, viewed as an element of Ob(C o); likewise, given amorphism f : A B in C , we write f o for the corresponding morphism Bo Ao in C o.

1.1.2. A morphism f : A B in C is said to be a monomorphism if the induced mapHomC (X, f) : HomC (X,A) HomC (X,B) g 7 f g

is injective, for everyX Ob(C ). Dually, we say that f is an epimorphism if f o is a monomor-phism in C o. Also, f is an isomorphism if there exists a morphism g : B A such thatg f = 1A and f g = 1B. Obviously, an isomorphism is both a monomorphism and anepimorphism. The converse does not necessarily hold, in an arbitrary category.

Two monomorphisms f : A B and f : A B are equivalent, if there exists anisomorphism h : A A such that f = f h. A subobject of B is defined as an equivalenceclass of monomorphisms A B. Dually, a quotient of B is a subobject of Bo in C o.

One says that C is well-powered if, for every A Ob(C ), the set :Sub(A)

of all subobjects of A is essentially small. Dually, C is co-well-powered, if C o is well-powered.


1.1.3. Let A and B be any two categories; a functor F : A B is a pair of mapsOb(A ) Ob(B) Morph(A ) Morph(B)

both denoted also by F , such that F assigns to any morphism f : A A in A , a morphism Ff : FA FA in B F1A = 1FA for every A Ob(A ) F (g f) = Fg Ff for every A,A, A Ob(A ) and every pair of morphismsf : A A, g : A A in A .

If F : A B and G : B C are any two functors, we get a compositionG F : A C

which is the functor whose maps on objects and morphisms are the compositions of the respec-tive maps for F and G. We denote by

Fun(A ,B)

the set of all functors A B. Moreover, any such F induces a functor F o : A o Bo withF oAo := (FA)o and F of o := (Ff)o for every A Ob(A ) and every f Morph(A ).

Definition 1.1.4. Let F : A B be a functor.(i) We say that F is faithful (resp. full, resp. fully faithful), if for every A,A Ob(A ) it

induces injective (resp. surjective, resp. bijective) maps :

HomA (A,A) HomB(FA, FA) : f 7 Ff.

(ii) We say that F reflects monomorphisms (resp. reflects epimorphisms, resp. is conserva-tive) if the following holds. For every morphism f : A A in A , if the morphism Ff of Bis a monomorphism (resp. epimorphism, resp. isomorphism), then the same holds for f .

(iii) If A is a subcategory of B, and F is the natural inclusion functor, then F is obviouslyfaithful, and we say that A is a full subcategory of B, if F is fully faithful.

(iv) The essential image of F is the full subcategory of B whose objects are the objects ofB that are isomorphic to an object of the form FA, for some A Ob(A ). We say that F isessentially surjective if its essential image is B.

(v) We say that F is an equivalence, if it is fully faithful and essentially surjective.

Remark 1.1.5. For later use, it is convenient to introduce the notion of n-faithful functor, for allintegers n 2. Namely : if n < 0, every functor is n-faithful; a functor F : A B (betweenany two categories A and B) is 0-faithful, if it is faithful; F is 1-faithful, if it is fully faithful;finally, we say that F is 2-faithful, if it is an equivalence.

Example 1.1.6. (i) The collection of all small categories, together with the functors betweenthem, forms a category

U-Cat.Unless we have to deal with more than one universe, we shall usually omit the prefix U, andwrite just Cat. It is easily seen that Cat is a category with small Hom-sets.

(ii) The category of all small sets shall be denoted U-Set or just Set, if there is no need toemphasize the chosen universe. There is a natural fully faithful embedding :

Set Cat.Indeed, to any set S one may assign its discrete category also denoted S, i.e. the unique categorysuch that Ob(S) = S and Morph(S) = {(s, s,1s) | s S}. If S and S are two discretecategories, the datum of a functor S S is clearly the same as a map of sets Ob(S) Ob(S ).

Notice also the natural functor

Ob : Cat Set C 7 Ob(C )


that assigns to each functor F : C D the underlying map Ob(C ) Ob(D) : C 7 FC.(iii) Recall that a preordered set is a pair (I,) consisting of a set I and a binary relation

on I which is reflexive and transitive. In this case, we also say that is a preordering on I . Wesay that (I,) is a partially ordered set, if is also antisymmetric, i.e. if we have

(x y and y x) x = y for every x, y I.

We say that (I,) is a totally ordered set, if it is partially ordered and any two elements arecomparable, i.e. for every x, y I we have either x y or y x. An order-preserving mapf : (I,) (J,) between preordered sets is a mapping f : I J such that

x y f(x) f(y) for every x, y I.

We denote by Preorder (resp. POSet) the category of small preordered (resp. partiallyordered) sets, with morphisms given by the order-preserving maps. To any preordered set (I,)one assigns a category whose set of objects is I , and whose morphisms are given as follows.For every i, j I , the set of morphisms i j contains exactly one element when i j, and isempty otherwise. Clearly, this rule defines a fully faithful functor

Preorder Cat.

Notice that if a category C lies in the essential image of this functor, then the same holds for Co.Indeed, if C corresponds to the preordered set (I,), then Co corresponds to the preorderedset (Io,) with Io := I and x y in Io if and only if y x in I , for every x, y I . Clearly(I,) is a partially ordered set if and only if the same holds for (Io,).

1.1.7. Let A , B be two categories, F,G : A B two functors. A natural transformation

(1.1.8) : F G

from F to G is a family of morphisms (A : FA GA | A Ob(A )) of B such that, forevery morphism f : A B in A , the diagram :

(1.1.9)FA

A //

Ff

GA

Gf

FBB // GB

commutes. If A is an isomorphism for every A Ob(A ), we say that is a natural iso-morphism of functors. For instance, the rule that assigns to any object A the identity morphism1FA : FA FA, defines a natural isomorphism 1F : F F . A natural transformation(1.1.8) is also indicated by a diagram of the type :

AF ))

G

55 B.

1.1.10. The natural transformations between functors A B can be composed; namely, if : F G and : G H are two such transformations, we obtain a natural transformation

: F H by the rule : A 7 A A for every A Ob(A ).

With this composition, Fun(A ,B) is the set of objects of a category which we shall denote

Fun(A ,B).


There is also a second composition law for natural transformations : if C is another category,and we have a diagram of functors and natural transformations

AF ((

G

66 B

F ''

G77

C

we get a natural transformation

: F F G G : A 7 GA F (A) = G(A)FA for every A Ob(A )

called the Godement product of and ([15, Prop.1.3.4]). Especially, if H : B C (resp.H : C A ) is any functor, we write H (resp. H) instead of 1H (resp. 1H).

Both composition laws are associative, i.e., if we have additional natural transformations

AH ((

K

66 B C

F ''

G77

D

then we get the identities

( ) = ( ) ( ) = ( ) .

Moreover, the composition laws are related as follows. Suppose that A , B and C are threecategories, and we have a diagram of six functors and four natural transformations :

A

F1

1

??

H1

1G1 // B

F2

2

??

H2

2G2 // C .

Then we have the identity :

(1.1.11) (2 1) (2 1) = (2 2) (1 1).

The proofs are left as exercises for the reader (see [15, Prop.1.3.5]).

Remark 1.1.12. (i) In the situation of (1.1.10), if A and B are small categories, the sameholds for Fun(A ,B).

(ii) Also, if A is small, B has small Hom-sets and Ob(B) U, then Fun(A ,B) has smallHom-sets, and Ob(Fun(A ,B)) U.

(iii) Assertion (ii) depends on our choices on how to encode arbitrary maps of sets : accord-ing to our (implicit) convention, a map of sets f : S S is the graph (f) S S . Thisdoes not agree, e.g. with the definition found in Bourbakis treatise [21], where such a mapf is the triple (S, S ,(f)). With Bourbakis convention, assertion (ii) fails. Other referencesare not so explicit about their choices for encoding maps, but for instance the SGA4 treatise([5], [6], [7]) appears to follow Bourbakis conventions, in view of [5, Exp.I, Rem.1.1.2], whichstates that Fun(A ,B) is not necessarily a subset of U, and Fun(A ,B) does not necessarilyhave small Hom-sets, under the assumptions of (ii). On the other hand, under the same assump-tions, it is stated in [49, Ch.II, Prop.1] that Fun(A ,B) has small Hom-sets, so the set-theoreticconventions of the latter are not compatible with those of SGA4.


1.1.13. Adjoint pairs of functors. Let A and B be two categories, F : A B and G : B A two functors. We say that G is left adjoint to F if there exist bijections

A,B : HomA (GB,A) HomB(B,FA) for every A Ob(A ) and B Ob(B)

and these bijections are natural in both A and B, i.e.

AB(gf Gh)=FgAB(f)h for all morphisms GBfA gA in A and B hB in B.

Then one also says that F is right adjoint to G, that (G,F ) is an adjoint pair of functors, andthat is an adjunction for the pair (G,F ).

Especially, to any object B of B (resp. A of A ), the adjunction assigns a morphismGB,B(1GB) : B FGB (resp. 1A,FA(1FA) : GFA A), whence a natural transformation

(1.1.14) : 1B F G (resp. : G F 1A )

called the unit (resp. counit) of the adjunction. The naturality of follows from the calculation:

FG(f) B = FG(f) GB,B(1GB) = GB,B(Gf 1GB) = GB,B(1GB Gf) = B f

for every morphism f : B B in B. A similar computation shows the naturality of . Thenaturality of implies that we have commutative diagrams

BB //

A,B(f) ""FFF

FFFF

F FGB

Ff

GBGg //

1A,B(g) ##HHH

HHHH

HHH GFA

A

FA A

for every morphism f : GB A in A and g : B FA in B. Taking f = A and g = B, wesee that the unit and counit are related by the so-called triangular identities expressed by thecommutative diagrams :

FF +3

1F FFFF

FFFF

F

FFFF

FFFF

F FGF

F

GG +3

1G FFFF

FFFF

F

FFFF

FFFF

F GFG

G

F G.

Conversely, we have ([15, Th.3.1.5] or [72, Prop.1.5.4]) :

Proposition 1.1.15. Let F : A B and G : B A be two functors.(i) If , are natural transformations as in (1.1.14), fulfilling the triangular identities of

(1.1.13), then there is a unique adjunction for the pair (G,F ), with unit and counit .(ii) Suppose that (G,F ) is an adjoint pair, and is the unit (resp. is the counit) of an

adjunction for (G,F ), then there exists a unique natural transformation (resp. ) asin (1.1.14), fulfilling the triangular identities of (1.1.13).

Proof. (i): Let A Ob(A ) and B Ob(B) be any two objects; in light of the discussion of(1.1.13), we see that the sought natural bijection A,B must be given by the rule :

f 7 Ff B for every f HomA (GB,A)

and its inverse must be the mapping

g 7 A Gg for every g HomB(B,FA).

So we come down to checking that the triangular identities imply that these rules do inducemutually inverse bijections on the respective Hom-sets. We leave the verification to the reader.

(ii) is clear from the explicit construction of in (i).


Example 1.1.16. To every preordered set (F ,) we may attach its partially ordered quotient(F/,)

where denotes the equivalence relation such that x y if and only if x y and y x, forevery x, y F . The ordering on F/ is the unique one such that the quotient map F F/defines a morphism

qF : (F ,) (F/,)of preordered sets, and it is easily seen that the rule (F ,) 7 (F/,) defines a left adjointto the inclusion functor

POSet Preorder.Moreover, the rule (F ,) 7 qF is a unit for this adjunction.

The following observations are borrowed from, and are further developed in [53, I.6].

Remark 1.1.17. (i) Consider two adjoint pairs (G1, F1) and (G2, F2) :

AF1 // B

F2 //

G1

oo CG2

oo

and suppose that for i = 1, 2 we are given adjunctions i, for the pair (Gi, Fi). Then clearly(G1 G2, F2 F1) is an adjoint pair, and we get an induced adjunction for this pair, by thecomposition :

HomA (G1G2C,A)1,A,G2C HomB(G2C,F1A)

2,F1A,C HomB(C,F2F1A)for every A Ob(A ) and C Ob(C ). We denote this adjunction by

(2 1)and we call it the composition of the adjunctions 1 and 2. If (i, i) are the units and counitsof i, (for i = 1, 2), then the unit and counit of (2 1) are respectively :

(F2 1 G2) 2 and 1 (G1 2 F1).

Moreover, suppose that CF3 // DG3

oo is another adjoint pair of functors, and 3, an adjunction

for this pair; with this notation, it is also then clear that

(3 (2 1)) = (3 (2 1)).(ii) Suppose that we have two pairs of adjoint functors and two natural transformations

AF // BGoo A

F // BGoo : F F : G G

and let us fix units and counits (, ) (resp. (, )) for the adjoint pair (G,F ) (resp. (G, F )).Then we obtain adjoint transformations

: G G : F F

given by the compositions :

GBG(B) GFGB G

(GB) GF GBGB GB for every B Ob(B)

FAFA F GFA F

(FA) F GFA F(A) F A for every A Ob(A ).

We claim that ( ) = and () = . Indeed, let (resp. ) be the adjunctions corre-sponding to (, ) (resp. to (, )), and notice that

B = 1GB,FGB(GB) G

(B) A = F

(A) GFA,FA(FA)


so we may compute :

( )A =F(A) GFA,FA(1GFA,FGFA(GFA) G

(FA))

=A,FA(A 1GFA,FGFA(GFA) G(FA))

=A,FA(1A,FGFA(F

(A) GFA) G(FA))=A,FA(

1A,FGFA(A F (A)) G

(FA))

=A,FA(1A,FA(A) G

F (A) G(FA))=A,FA(

1A,FA(A))

= A

where the second, third and fifth identities follow from the naturality of , the fourth from thenaturality of , and the sixth from the triangular identities of (1.1.13). We leave to the readerthe similar calculation which gives the second identity. Hence the rule

7 := (, , )

establishes a natural bijection from the set of natural transformations F F , to the set ofnatural transformations G G. Notice that this correspondence depends not only on (G,F )and (G, F ), but also on (, ) and (, ). Sometimes we denote this adjoint transformationalso by (, , ).

(iii) Moreover, using the triangular identities of (1.1.13), it is easily seen that the diagrams :

G F G +3

F

G F

1B +3

F G

G

G F +3 1A F GF +3 F G

commute. Also, (, , ) is characterized as the unique natural transformation G G suchthat the following diagram commutes for every A Ob(A ) and B Ob(B) :

(1.1.18)

HomA (GB,A)A,B //

HomA (B ,A)

HomB(B,FA)

HomB(B,A)

HomA (G

B,A)A,B // HomB(B,F

A)

Indeed, letting A := GB and recalling that GB,B(1GB) = B, we see easily that the commuta-tivity of (1.1.18) determines uniquely (details left to the reader). Conversely, if is definedas in (i), we may compute, for every morphism f : GB A in A :

A,B(f B) =

A,B(f 1GB,FGB(GB) G

(B))

=A,FGB(f 1GB,FGB(GB)) B=A,FGB(

1A,FGB(F

f GB)) B=F f GB B= A Ff B= A A,B(f).

(iv) Furthermore, suppose we have a third pair of adjoint functors

AF // BGoo and a natural transformation : F F


and let us fix an adjunction for the pair (G, F ). Then we have :

( , , ) = (, , ) (, , ).Indeed, this identity follows easily from the characterization of , and ( ) given in (iii)(details left to the reader).

(v) Lastly, in the situation of (i), suppose moreover that we have two other adjoint pairs

AF 1 // BG1

ooF 2 // CG2

oo and natural transformations 1 : F1 F 1 2 : F2 F 2

and for i = 1, 2, let us fix an adjunction i, for (Gi, F

i ). Then we get as in (ii) the natural

transformations 1 : G1 G1 and

2 : G

2 G2, and we have the identity

(2 1, 2 1, 2 1) = (1, 1, 1) (2, 2, 2).Indeed, taking into account (iii) we get the commutative diagram

HomA (G1G2C,A)1,A,G2C //

HomA (1,G2C

,A)

HomB(G2C,F1A)2,F1A,C //

HomB(G2C,1,C)

HomC (C,F2F1A)

HomC (C,F2(1,C))

HomA (G

1G2C,A)

1,A,G2C //

HomA (G1(1,C),A)

HomB(G2C,F1A)

2,F 1A,C //

HomB(2,C ,F

1A)

HomC (C,F2F1A)

HomC (C,2,F 1C)

HomA (G

1G2C,A)

1,A,G2C // HomB(G

2C,F

1A)

2,F 1A,C // HomC (C,F

2F1A)

for every A Ob(A ) and C Ob(C ). The sought identity follows after invoking again (iii).(vi) Especially, taking into account the triangular identities of (1.1.13), it is easily seen that :

(1F2 , 2, 2) = 1G2

(F2 1, 2 1, 2 1) = (1, 1, 1) G2(2 F1, 2 1, 2 1) = G1 (2, 2, 2).

Remark 1.1.19. (i) For any two categories C ,D we have a natural isomorphism of categories

Fun(C ,D)o Fun(C o,Do)

that assigns to any functor H : C D the opposite functor Ho : C o Do, and to any naturaltransformation : H K the opposite transformation o : Ko Ho such that oCo := (C)ofor every C Ob(C ). Also, in the situation of (1.1.10), notice the identities

( )o = o o and ( )o = o o.(ii) Let B and E be two other categories, f : B C and g : D E two functors; we get

an induced functor

Fun(f, g) : Fun(C ,D) Fun(B,E ) H 7 g H f ( : H K) 7 g f.Likewise, if f : B C and g : D E are any two other functors, every pair of naturaltransformations : f f and : g g induces a transformation

Fun(, ) : Fun(f, g) Fun(f , g) H 7 H .We shall usually write Fun(f,D) (resp. Fun(C , g)) instead of Fun(f,1D) (resp. of Fun(1C , g)).Furthermore, in the situation of (1.1.10), notice the identities

Fun(,D) Fun(,D) = Fun( ,D)Fun(,D) Fun(,D) = Fun( ,D).


(iii) In the situation of (ii), suppose that the functor f admits a left adjoint g : C B.Then f := Fun(f,D) is left adjoint to g := Fun(g,D). More precisely, let be a unit and a counit for the adjoint pair (g, f). From the triangular identities (1.1.13) for (g, f), and takinginto account (ii), we deduce commutative diagrams :

gFun(,D)g

+3

SSSSSSSS

SSSSSSSS

SSSSS

SSSSSSSS

SSSSSSSS

SSSSS gf g

gFun(,D)

f fFun(,D)

+3

TTTTTTTT

TTTTTTTT

TTTTT

TTTTTTTT

TTTTTTTT

TTTTT f gf

Fun(,D)f

g f

which, in light of proposition 1.1.15(i), says that Fun(,D) is a unit and Fun(,D) a counit forthe adjoint pair (f , g).

(iv) Let F : C D be a functor, and G : D C a left adjoint to F . Then F o is left adjointto Go. More precisely, let be a unit and a counit for the adjoint pair (G,F ); then it followseasily from (i) and proposition 1.1.15(i) that o is a unit and o is a counit for the adjoint pair(F o, Go) : details left to the reader.

Proposition 1.1.20. Let F : A B be a functor.(i) The following conditions are equivalent :

(a) F is fully faithful and has a fully faithful left adjoint.(b) There exist a functor G : B A and isomorphisms of functors

G F 1A 1B F G.

(c) F is an equivalence.(ii) Suppose that F admits a left adjoint G : B A , and let : 1B F G and

: G F 1A be a unit and respectively a counit for the adjoint pair (G,F ). ThenF (resp. G) is faithful if and only if X is an epimorphism for every X Ob(A ) (resp.Y is a monomorphism for every Y Ob(B)).

(iii) In the situation of (ii), the following conditions are equivalent :(a) F (resp. G) is fully faithful.(b) The counit (resp. the unit ) is an isomorphism of functors.(c) There exists an isomorphism of functors : GF 1A (resp. : 1B

F G).Moreover, if (c) holds, there exists a unique adjunction for the pair (G,F ) whosecounit is (resp. whose unit is ).

(iv) Suppose that F admits both a left adjoint G : B A and a right adjoint H : B A . Then G is fully faithful if and only if H is fully faithful.

Proof. (ii): In view of remark 1.1.19(iv), it suffices to consider the assertion relative to . Thus,suppose that Ff1 = Ff2 for two morphisms f1, f2 : X X . The naturality of yields theidentity : X GFfi = fi X for i = 1, 2, and if X is an epimorphism, we deduce f1 = f2.Conversely, suppose F is faithful and f1 X = f2 X ; from the triangular identities we get :

Ffi = F (X) FGF (fi) FX = F (fi X) FXso Ff1 = Ff2 and therefore f1 = f2 which shows that X is an epimorphism.

(iii): Again, remark 1.1.19(iv) reduces to considering the assertion for . We check first that(iii.a)(iii.b) : indeed, if F is fully faithful, for every X Ob(A ), there exists a morphismX : X GFX such that FX = FX : FX FGFX . From the triangular identities of(1.1.13) we deduce that F (X X) = F (X) FX = 1FX , whence X X = 1X , since Fis faithful. Next, let be the unique adjunction for the pair (G,F ) whose unit and counit are and ; by inspection of the explicit description of in the proof of proposition 1.1.15(i) we get

GFX,FX(X X) = F (X X) FX = FX F (X) FX = FX = GFX,FX(1GFX)


whence X X = 1GFX . Obviously (iii.b)(iii.c). Lastly, suppose that (iii.c) holds; for everyX, Y Ob(A ) we deduce a bijection

X,Y : HomA (X, Y )X,Y HomA (GFX, Y )

GX,Y HomB(FX,FY )where X,Y (f) := f X for every morphism f : X Y in A . It is easily seen that X,Y isnatural in both X and Y ; especially, for every f HomA (X, Y ) we have :

Y,Y (1Y ) Ff = X,Y (1Y f) = X,Y (f 1X) = Ff X,X(1X).Letting X = Y , and taking f : X X with X,X(f) = 1FX , we deduce that X,X(1X)is an isomorphism in B, for every X Ob(A ). Since X,Y (f) = Ff X,X(1X) for ev-ery f HomA (X, Y ), we conclude that the rule f 7 Ff is a bijection HomA (X, Y )

HomB(FX,FY ), whence (iii.a). Lastly, let us check the existence and uniqueness of the ad-junction whose counit is . To this aim, let : 1A

1A be the automorphism such that = , and set := (F 1 G) . We compute :

(F ) ( F ) = (F ) (F ) (F 1 GF ) ( F )= (F ) (F 1) (F ) ( F ) = 1F .

Likewise we check that ( G) (G ) = 1G, whence the contention, by proposition 1.1.15.(i): The equivalence (i.a)(i.b) follows immediately from (iii). Moreover, if (i.b) holds, then

for every Y Ob(B) we have an isomorphism Y FGY , so F is essentially surjective. Sincewe have just seen that (i.b) implies that F is fully faithful, we deduce as well that (i.b)(i.c).

(i.c)(i.a): We construct a left adjoint G : B A to F as follows. Since F is essentiallysurjective, for every B Ob(B) we may find an object A Ob(A ) with an isomorphismB : B

FA, and we set GB := A. Next, since F is fully faithful, for every morphismg : B B in B there exists a unique morphism f : GB GB in A which makes commutethe diagram

Bg //

B

B

B

FGBFf // FGB

and we let Gg := f . It is easily seen that these rules yield a well defined functor G as sought,and is then a natural isomorphism 1B FG. Moreover from the full faithfulness of F wededuce that G is fully faithful as well (details left to the reader). To conclude we remark, moregenerally :

Claim 1.1.21. Let F : A B and G : B A be two functors, : 1B FG an isomor-phism of functors, and suppose that F is fully faithful. Then there exists a unique adjunction for the pair (G,F ) whose unit is .

Proof of the claim. From the proof of proposition 1.1.15 we know that determines , bythe rule : A,B(f) := Ff B for every A Ob(A ), B Ob(B) and every morphismf : GB A in A . Conversely, our assumptions easily imply that this rule does yield anatural bijection HomA (GB,A)

HomB(B,FA), whence the contention (details left to thereader).

(iv): Let : 1B FG (resp. : 1A HF ) be the unit and : GF 1A (resp. : FH 1B) the counit of a given adjunction for the adjoint pair (G,F ) (resp. (F,H)). Byremark 1.1.19(iv) we may assume that H is fully faithful, and we show that the same holds forG. By (iii), this is the same as assuming that is an isomorphism, and we need to check thatthe same holds for . To this aim, denote : FG 1B the composition

FG(FG)1 FGFH FH FH

1B.


We show that is inverse to . Indeed we have

= (F H) ( FH) 1 = 1 = 1Bwhere the first identity holds by the naturality of , and the second follows from the triangularidentites of (1.1.13). Likewise, we have

= ( FG) (FH ) (F H) (FG 1)= ( FG) (F HFG) (FGFH ) (FG 1)= ( FG) (F HFG) (FG 1 FG) (FG )= ( FG) (F HFG) (FGF G) (FG )= ( FG) (F G) (F G) (FG )= (1F G) (F 1G) = 1FG

where the first and third identities follow from the naturality of , the second and fifth from thatof , the fourth and sixth from the triangular identities for the pairs (, ) and (, ).

Definition 1.1.22. Let F : A B be any equivalence of categories. A quasi-inverse for F isthe datum of a functor G : B A and an adjunction for the pair (G,F ). Then we also saythat F is a quasi-inverse for G.

Remark 1.1.23. It follows easily from proposition 1.1.20(i,iii) and claim 1.1.21, that a quasi-inverse for an equivalence F : A B is the same as the datum of a functor G : B Aand an isomorphism of functors 1B

FG, and moreover any such G is also an equivalence.Taking into account remark 1.1.19(iv), we see that a quasi-inverse for F is also the same as thedatum of such a G and an isomorphism of functors GF 1A .

1.1.24. Slice categories. A standard construction attaches to any X Ob(C ) a category :

C /X

as follows. The objects of C /X are all the pairs (A, f) where A Ob(C ) and f : A X isany morphism of C . For any two such objects (A, f), (B, g), the set HomC /X((A, f), (B, g))consists of all the commutative diagrams of morphisms of C :

Ah //

f @@@

@@@@

@ B

g~~~~~~

~~~~

X

with composition of morphisms induced by the composition law of C . We denote sometimessuch a morphism of C /X by

h/X : (A, f) (B, g).An object (resp. a morphism) of C /X is also called an X-object (resp. an X-morphism) of C .Dually, one defines

X/C := (C o/Xo)o

i.e. the objects of X/C are the pairs (A, f) with A Ob(C ) and f : X A any morphism ofC . We have an obvious faithful source functor

sX : C /X C (A, f) 7 A ((A, f)h/X (B, g)) 7 (A h B)

and likewise one obtains a target functor

tX := soXo : X/C C .


Moreover, any morphism f : X Y in C induces functors :

(1.1.25)f : C /X C /Y : (A, g : A X) 7 (A, fg := f g : A Y )f : Y/C X/C : (B, h : Y B) 7 (B, f h := h f : X B).

Furthermore, given a functor F : C B, any X Ob(C ) induces functors :

(1.1.26)F|X : C/X B/FX : (A, g) 7 (FA, Fg)X|F : X/C FX/B : (B, h) 7 (FB,Fh).

1.1.27. The categories C /X and X/C are special cases of the following more general con-struction. Let F : A B be any functor. For any B Ob(B), we define

FA /B

as the category whose objects are all the pairs (A, f), where A Ob(A ) and f : FA B isa morphism in B. The morphisms g : (A, f) (A, f ) are the morphisms g : A A in Asuch that f Fg = f . There are well-defined functors :F/B : FA /B B/B : (A, f) 7 (FA, f) and sB : FA /B A : (A, f) 7 A.Dually, we define :

B/FA := (F oA o/Bo)o

and likewise one has natural functors :

B/F : B/FA B/B and tB : B/FA A .Any morphism g : B B induces functors :

g/FA : B/FA B/FA : (A, f) 7 (A, f g)FA /g : FA /B FA /B : (A, f) 7 (A, g f).

Obviously, the category C/X (resp. X/C ) is the same as 1C C /X (resp. X/1C C ).

1.1.28. In the situation of (1.1.27), the categories of the form FA /B can be faithfully em-bedded in a single category FA /B. The latter is the category whose objects are all the triples(A,B, f), where A Ob(A ), B Ob(B) are any two objects, and f : FA B is anymorphism of B. If f : FA B and f : FA B are any two objects, the set

HomFA /B((A,B, f), (A, B, f ))

consists of all pairs (g, g) where g is a morphism in A and g a morphism in B, that makecommute the diagram :

(1.1.29)FA

f //

Fg

B

g

FA

f // B

with composition of morphisms induced by the composition laws of A and B, in the obviousway. There are two natural source and target functors :

As FA /B t B

such that s(FA B) := A, t(FA B) := B for any object FA B of FA /B, ands(g, g) = g, t(g, g) = g. Dually, we let

B/FA := (F oA o/Bo)o

and the corresponding source and target functors are switched :

At B/FA s B.


1.1.30. For the special case of the identity endofunctor 1C of any category C , we obtain thecategory of arrows of C

Morph(C ) := 1C C /C .

So, the set of objects of Morph(C ) is Morph(C ) (notation of (1.1)) and the morphisms arethe commutative square diagrams in C . The functor sX of (1.1.24) is the restriction of s :Morph(C ) C to the subcategory C/X , and likewise for tX . Likewise, in the situation of(1.1.28), the target functor on FA /B and the source functor on B/FA factor through functors

FA /BT Morph(B) S B/FA

where T(A,B, f) := (FA,B, f) for every object (A,B, f) of FA /B, and T assigns to anymorphism (g, g) : (A,B, f) (A, B, f ) the commutative square (1.1.29), regarded as amorphism (FA,B, f) (FA, B, f ) in Morph(B). Likewise one describes the functor S.

Notice also the natural transformation

Morph(C )

s

((

t

66 m C

where m(A,B, f) := f for every (A,B, f) Morph(C ). Furthermore, every functor F :B C induces a functor

Morph(F ) : Morph(B) Morph(C )

which maps (A,B, f) Morph(B) to (FA, FB, Ff) Morph(C ) and which sends everycommutative square diagram D in B to the commutative square diagram FD in C .

Notice that a natural transformation as in (1.1.8) is equivalent to the datum of a functor

: A Morph(B) such that s = F and t = G.

Namely, one defines by the rule : A 7 (FA,GA, A) for every A Ob(A ), and for everymorphism f : A B in A , one lets (f) be the commutative square diagram (1.1.9).

1.1.31. Let A ,B be two categories, F : A B a functor, G : B A a left adjoint for F ,and an adjunction for the pair (G,F ). Then for every A Ob(A ) the functor F|A : A /AB/FA of (1.1.26) admits a left adjoint that we denote

G|A : B/FA A /A.

Namely, to every (f : B FA) Ob(B/FA) we assign the object (1AB(f) : GB A) Ob(A /A), and to every morphism h/FA : (f : B FA) (f : B FA) in B/FA weassign the morphismGh/A : 1AB(f)

1AB(f

) in A /A. Indeed, induces an adjunction |Afor the pair (G|A, F|A) : to every (f : B FA) Ob(B/FA) and (g : A A) Ob(A /A)we assign the bijection

(|A)g,f : HomA /A(1AB(f), g)

HomB/FA(f, Fg) h/A 7 (h)/FA.

Dually, since F o is left adjoint to Go (remark 1.1.19(iv)), we see that for every B Ob(B) thefunctor B|G : B/B GB/A of (1.1.26) admits the right adjoint

B|F : GB/A B/B (GBf A) 7 (B AB(f) FA) GB/h 7 B/Fh.

The detailed verifications shall be left to the reader. See example 1.2.27 for a related result.


1.1.32. In the situation of (1.1.31) we get furthermore for every A Ob(A ) an isomorphismof categories :

A : GB/A B/FA (f : GB A) 7 (AB(f) : B FA)

that assigns to every morphism g : (f : GB A) (f : GB A) of GB/A the morphismg : (AB(f) : B FA) (AB(f ) : B FA) of B/FA. Also, for every morphismh : A A of A , we get a commutative diagram of categories :

GB/AA //

GB/h

B/FA

(Fh)

GB/AA// B/FA.

Clearly, the isomorphisms A are restrictions of a single isomorphism of categories :

GB/A B/FA .

The detailed verifications shall be again left to the reader.

1.2. Presheaves and limits. A very important construction associated with every category Cis the category

C U := Fun(Co,U-Set)

whose objects are called the U-presheaves on C (notation of (1.1.10)). We usually drop thesubscript U, unless we have to deal with more than one universe. If U is another universe withU U, the natural inclusion of categories :(1.2.1) C U C Uis fully faithful (verification left to the reader). For every functor F : B C and every naturaltransformation : F G we set(1.2.2) FU := Fun(F

o,Set) : C U BU U := Fun(o,Set) : GU FU(notation of remark 1.1.19(i,ii)). Again, we shall drop the subscript and write simply F and, unless the omission of U may be a source of ambiguities. Clearly, for every pair of functorsF1 : A B and F2 : B C we have

(F2 F1) = F1 F2 .Moreover, in the situation of (1.1.10), remark 1.1.19(i,iii) yields the identities

(1.2.3) ( ) = and ( ) = .

1.2.4. If C has small Hom-sets (see (1.1.1)), there is a natural functor

hC : C C

called the Yoneda embedding, which assigns to every X Ob(C ) the functorhX : C

o Set Y 7 HomC (Y,X) for every Y Ob(C )and to any morphism f : X X in C , the natural transformation hf : hX hX such that

hf,Y (g) := f g for every Y Ob(C ) and every g HomC (Y,X).

Definition 1.2.5. Let C be a category, and V any universe such that C has V-small Hom-sets.We say that an object F of C V is representable in C , if there exists an isomorphism

hX F in C V

for someX Ob(C ), in which case we also say thatX represents F . From the full faithfulnessof (1.2.1), it follows that the representability of a presheaf is independent of the universe V.


Proposition 1.2.6 (Yonedas lemma). With the notation of (1.2.4), we have :(i) The functor hC is fully faithful.

(ii) Moreover, for every F Ob(C ) and every X Ob(C ) there is a natural bijection

F (X) HomC(hX , F )

functorial in both X and F .

Proof. Clearly it suffices to check (ii). However, the sought bijection is obtained explicitly asfollows. To a given a F (X), we assign the natural transformation a : hX F such that

a,Y (f) := Ff(a) for every Y Ob(C ) and every f hX(Y ).

Conversely, to a given natural transformation : hX F we assign X(1X) F (X). Itis easily seen that these rules establish mutually inverse bijections. The functoriality in F isimmediate, and the functoriality in the argumentX amounts to the commutativity of the diagram

HomC(hX , F ) //

HomC (h,F )

F (X )

F ()

HomC(hX , F )

// F (X)

for every morphism : X X in C : the verification shall also be left to the reader.

Example 1.2.7. (i) Every representable presheaf on the category Set admits a natural repre-senting object. Indeed, let 1 denote any set with one element (a canonical choice for 1 is the set{}); then for every representable F Ob(Set) we get a natural isomorphism

F hF (1)

as follows. For any set S, we have a natural bijection S HomSet(1, S) that assigns to everys S the unique map 1s : 1 S whose image is {s}. We deduce a map

S F (S) F (1) (s, a) 7 F (1s)(a)

which is the same as a map F (S) HomSet(S, F (1)) that realizes the sought isomorphism.(ii) We may apply Yonedas lemma to prove the uniqueness of the (left or right) adjoint for

a given functor. Indeed, let G : B A be a functor between any two categories, suppose thatF and F are both right adjoint to G, and fix adjunctions and for F and respectivelyF . Choose also a universe U such that both A and B are U-small. Then and can beregarded as isomorphisms from the functor G hA to hB F and respectively to hB F . Inthis situation, proposition 1.2.6(i) implies that there exists a unique isomorphism : F F which makes commute the diagram

G hA

yyrrrrrr

rrrr

&&MMMMM

MMMMM

hB FhB // hB F .

Taking into account remark 1.1.19(iv), a dual argument yields a corresponding uniqueness as-sertion for left adjoints.

Remark 1.2.8. (i) Let C be any category, and F1, F2 two representable presheaves on C ,and pick X1, X2 Ob(C ) with isomorphisms i : hXi

Fi (i = 1, 2). It follows from


proposition 1.2.6(i) that for every morphism f : F1 F2 in C there exists a unique morphism : X1 X2 in C that makes commute the diagram

hX1h //

hX2

F1

f // F2.

In this situation, we say that represents the morphism f .(ii) Likewise, suppose that G : B C is any functor. We say that G is representable if

there exists a functor : B C with an isomorphism G hC . It follows easily from(i) that G is representable if and only if G(B) is representable in C for every B Ob(B).Moreover, any two representatives for G are isomorphic in Fun(B,C ). Furthermore, if G,G :B C are any two representable functors, and , : B C two corresponding representingfunctors, then any natural transformation t : G G is represented uniquely by a naturaltransformation : , by which we mean that t = hC .

1.2.9. Limits and colimits. We wish to explain some standard constructions of presheaves thatare used pervasively throughout this work. Namely, let I be a small category, C a category andX any object of C . We denote by cX : I C the constant functor associated with X :

cX(i) := X for every i Ob(I) cX() := 1X for every Morph(I).Any morphism f : X X induces a natural transformation

cf : cX cX by the rule : (cf )i := f for every i I.If F : I C is any functor, a cone of vertex X and basis F is any natural transformationcX F . Dually, a cocone with vertex X and basis F is a natural transformation F cX .

Definition 1.2.10. With the notation of (1.2.9), let V be any universe with U V, such that Chas V-small Hom-sets, and let F : I C be any functor.

(i) The limit of F is the V-presheaf on C denoted

limIF : C o V-Set

that assigns to every X Ob(C ), the set limI F (X) of all cones cX F . Anymorphism f : X X of C induces the maplimIF (f) : lim

IF (X) lim

IF (X ) 7 cf for every : cX F .

Then we also say that I is the indexing category for the limit of F .(ii) Dually, the colimit of F is the V-presheaf on C o

colimI

F := limIoF o.

Remark 1.2.11. Let C ,C be any two categories with V-small Hom-sets for some universe Usuch that U V, and I, I two small categories.

(i) Any diagram of functors

I I F C H C

induces a natural morphism

limH : lim

IF H(lim

IH F ) in C V

(notation of (1.2.2)) by ruling that

limH(X)() := H for every X Ob(C ) and every cone : cX F .


If : I I and K : C C are any two other functors, with I also small and C withV-small Hom-sets, the resulting diagram in C V commutes :

(1.2.12)

limI F

limH

limKH // H K(limI K H F )

H(limI H F ).H(limK

)

22ffffffffffffffffffffffff

The reader may spell out the corresponding assertions for colimits.(ii) Any diagram of functors and natural transformations

I 1&&

2

88 I

F1 ''

F2

77 g C

induces commutative diagrams

limI F1limI g //

lim11C

limI F2

lim2 1C

colimI F2colimI g //

colim2 1C

colimI F1

colim11C

limI(F11)limI (g) // limI(F22) colimI(F22)

colimI (g) // colimI(F11)

where the top (resp. bottom) arrow of the left diagram is given by the rule :

7 g ( resp. 7 (g ) )for every X Ob(C ) and every cone : cX F1 (resp. : cX F1 1). The top (resp.bottom) arrow of the right diagram is given by the rule

7 g ( resp. 7 (g ) )for every X Ob(C ) and every cocone : F2 cX (resp. : F2 1 cX).

(iii) Suppose that limI F is representable by L Ob(C ), so we have an isomorphism

(1.2.13) : hL lim

IF in C V .

Notice that the cone := L(1L) determines : indeed the latter assigns, to everyX Ob(C ),the bijection :

X : HomC (X,L) lim

IF (X) f 7 cf .

Conversely, we say that a given : cL F is a universal cone, if the induced map X is abijection for every X Ob(C ), in which case the limit of F is representable by L. Clearly, theuniversal property for a cone is independent of the choice of auxiliary universe V.

Likewise, if colimI F is representable by Co Ob(C o), the choice of an isomorphism

(1.2.14) hCo colim

IF in (C o)V

induces a universal cocone : F cC , which in turns determines (1.2.14) by the rule :

HomC (C,X) colim

IF (Xo) f 7 cf .

Moreover, with the notation of (ii), suppose that the limits (resp. colimits) of F1 and F2 arerepresentable by objects L1 and L2 of C (resp. Co1 and C

o2 of C

o), and let us fix isomorphismsas (1.2.13) and (1.2.14) for F1 and F2. Then :

The limit of g is represented by a morphism L1 L2 in C . The colimit of g is represented by a morphism Co2 Co1 in C o, i.e. by a morphismC1 C2 in C (see remark 1.2.8(ii)).


(iv) Suppose that the universe V is sufficiently large, so that both C and C U have V-smallHom-sets. Every i I induces a morphism of presheaves

ti : limIF hFi

that assigns to every X Ob(C ) and every cone : cX F the morphism i : X Fi,regarded as an element of hFi(X). We obtain in this way a natural cone

(1.2.15) climI F hC F i 7 tiwhere hC : C C V is the Yoneda embedding. We call (1.2.15) the tautological cone associ-ated with F . We claim that the tautological cone is universal. Indeed, say that G is any presheafon C , and : cG hC F any cone. We define a morphism f : G limI F as follows.To every i Ob(I), the cone attaches a morphism of presheaves i : G hFi, which is thedatum, for every X Ob(C ) of a map Xi : G(X) HomC (X,Fi), and for fixed s G(X),the system

X (s) := (Xi (s) : X Fi | X Ob(C ))

is a cone cX F . Then we let f (X) : G(X) limI F (X) be the map such that s 7 X (s)for every X Ob(C ) and every s G(X). The rule 7 f yields an inverse for the map

G : HomC(G, limIF ) lim

IhC F

associated, as in (iii), with the cone (1.2.15), whence the claim : details left to the reader.Dually, with F we may also associate a tautological cocone

hC o F o hcolimI F in (C o)Vwhich is a universal cocone.

Example 1.2.16. Let V be a universe containing U, and C any category with V-small Hom-sets.(i) For i = 1, 2, let fi : A Bi be two morphisms in C ; the push-out or amalgamated sum

of f1 and f2 is the colimit of the functor F : I C , defined as follows. The set Ob(I) consistsof three objects s, t1, t2 and Morph(I) consists of two morphisms

t11 s 2 t2

(in addition to the identity morphisms of the objects of I); the functor is given by the rule :Fs := A, Fti := Bi and Fi := fi (for i = 1, 2). For any C Ob(C ), a cocone F cCamounts to a pair of morphisms f i : Bi C (i = 1, 2) such that f 1 f1 = f 2 f2. If thecocone is universal (and thus, C represents the push-out of f1 and f2), we say that the resultingcommutative diagram :

Af1 //

f2

B1

f 1

B2f 2 // C

is cocartesian. Dually one defines the fibre product or pull-back of two morphisms gi : Ai B(i = 1, 2). If D Ob(C ) represents this fibre product, a cone with vertex D is given by a pairof morphisms gi : D Ai (i = 1, 2) such that g1 g1 = g2 g2, and we say that the diagram :

Dg1 //

g2

A1

g1

A2g2 // B


is cartesian if this cone is universal. The push-out of f1 and f2 is a V-presheaf, and is usuallyjust called the amalgamated sum of B1 and B2 over A, denoted

B1 q(f1,f2) B2 or simply B1 qA B2unless the notation gives rise to ambiguities. Likewise one writes

A1 (g1,g2) A2 or just A1 B A2for the fibre product of g1 and g2, which is a V-presheaf as well.

(ii) Similarly, consider the category I with Ob(I ) = {s, t}, and with Morph(I ) consistingof two morphisms

s1 //

2

// t

(in addition to 1s and 1t). If A,B Ob(C ) are any two objects and f1, f2 : A B any twomorphisms, we get a functor F : I C by the rule : s 7 A, t 7 B and i 7 fi for i = 1, 2.Then, the colimit of F is also called the coequalizer of f1 and f2, and is sometimes denoted

Coequal(f1, f2).

Dually, the limit of F is also called the equalizer of f1 and f2, sometimes denoted

Equal(g1, g2).

(iii) Let I U be any small set, and B := (Bi | i I) any family of objects of C . We mayregard I as a discrete category (see example 1.1.6(ii)), and then the rule i 7 Bi yields a functorI C , whose limit (resp. colimit) is called the product (resp. coproduct) of the family B, andis denoted

iI

Bi (resp.iI

Bi ).

If I = {1, 2} is a set with exactly two elements, we also write B1 B2 (resp. B1 qB2) for thislimit (resp. colimit), and we call it sometimes a binary product (resp. a binary coproduct).

(iv) Let B be any category, and suppose that a given X Ob(B) represents the emptyproduct in B, i.e. the product of an empty family of objects of B (this product is a V-presheaf,for any suitably large universe V). This means precisely that HomB(Y,X) consists of exactlyone element, for every Y Ob(B). Any such X is called a final object of B.

Dually, we say that X is an initial object of B, if HomB(X, Y ) consists of a single element,for every Y Ob(B). Then an object of B is initial if and only if it represents the emptycoproduct in B.

Moreover, we shall say that an object X of B is disconnected, if there exist A,B Ob(B),neither of which is an initial object of B, and such that X represents the coproduct A q B(which, again, is well defined in any sufficiently large universe V). We say that X is connected,if X is not disconnected and is not an initial object of B.

(v) For instance, the initial object of Set is the empty set, and any set 1 with one element is afinal object of Set. More generally, if B is any category, the initial object of B is the presheafB such that B(X) = for every X Ob(B), and the presheaf 1B such that 1B(X) = 1for every X Ob(B) is a final object.1.2.17. Let C be a category, f : A B a morphism in C ; as explained in example 1.2.16(i),the pair of morphisms B 1B B 1B B can be regarded as an element (B q(f,f) B)(B),which, by Yonedas lemma (proposition 1.2.6(ii)) corresponds to a morphism

of : hB B qA B in (C o).If B qA B is representable in C , this is the same as a morphism

f : B qA B B in C


called the codiagonal of f . Dually, we have a natural morphism :

f : hA AB A in C

which in case AB A is representable in C is the same as a diagonal morphism of f

f : A AB A in C .

Proposition 1.2.18. (i) With the notation of (1.2.17), the following conditions are equivalent :(a) f : A B is a monomorphism in C(b) f : hA AB A is an isomorphism in C .

In case AB A is representable in C , denote by Ap1 AB A

p2 A the universal cone of thisfibre product. Then (a) and (b) are moreover equivalent to :

(c) f : A AB A is an isomorphism in C .(d) f : A AB A is an epimorphism in C .(e) p1 = p2.

(ii) Dually, the following conditions are equivalent :(a) f : A B is an epimorphism in C .(b) of : hB B qA B is an isomorphism in (C o).

In case B qA B is representable in C , denote by Be1 B qA B

e2 B the universal cocone ofthis amalgamated sum. Then (a) and (b) are moreover equivalent to :

(c) f : B qA B B is an isomorphism in C .(d) f : B qA B B is a monomorphism in C .(e) e1 = e2.

Proof. (i.a)(i.b): For every X Ob(C ) the map f,X : hA(X) A B A(X) is given bythe rule : (X

g A) 7 (A g X g A). Hence, f is an isomorphism if and only if for everyX Ob(C ) and every pair of morphisms g1, g2 : X A such that f g1 = f g2, we haveg1 = g2. This means precisely that f is a monomorphism.

(i.b)(i.c) is clear, since the Yoneda imbedding hC is fully faithful.(i.c)(i.d) is trivial.(i.d)(i.e): By definition, p1 f = 1A = p2 f . If f is an epimorphism, then p1 = p2.(i.e)(i.c): Set p := p1; since p f = 1A, it suffices to show that f p = 1ABA. Due

to the universality of the cone Ap1 A B A

p2 A, we are then reduced to checking thatp1 f p = p1 and p2 f p = p2, which is clear, since by assumption p = p2.

Assertion (ii) follows from (i) by duality.

Some special kinds of limits occur frequently in applications; we gather some of them in thefollowing :

Definition 1.2.19. Let I be a category.(i) We say that I is finite if both Ob(I) and Morph(I) are finite sets.(ii) We say that I is connected, if Ob(I) 6= and every two objects i, j can be connected by

a finite sequence of morphisms in I :

(1.2.20) i k1 k2 kn j.

(iii) We say that I is directed, if for every i, j Ob(I) there exist k Ob(C ) and morphismsi k j in I . We say that I is codirected, if Io is directed.

(iv) We say that I is locally directed if, for every i Ob(I), the category i/I is directed. Wesay that I is locally codirected, if Io is locally directed.


(v) We say that I is pseudo-filtered if it is locally directed, and the following coequalizingcondition holds. For any i, j Ob(I), and any two morphisms f, g : i j, there existk Ob(I) and a morphism h : j k such that h f = h g.

(vi) We say that I is filtered, if it is pseudo-filtered and connected. We say that I is cofilteredif Io is filtered.

(vii) Let F : I C be a functor to any category C . Then limI F is a well defined V-presheaf, for any sufficiently large universe V, and we say that the limit of F is small (resp.connected, resp. codirected, resp. locally codirected, resp. cofiltered, resp. finite), if I is small(resp. connected, resp. codirected, resp. locally codirected, resp. cofiltered, resp. finite).Dually, one defines small, connected, directed, locally directed, filtered and finite colimits.

(viii) We say that a category C is complete (resp. finitely complete) if every small limit (resp.finite limit) of C is representable in C . We say that C is cocomplete (resp. finitely cocomplete)if every small colimit (resp. finite colimit) of C is representable in C o.

Remark 1.2.21. Let I be any category.(i) If I is connected and locally directed, then I is directed. Indeed, let i, j be any two objects

of I; we have to show that there exist k Ob(I) and morphisms a : i k and b : j k.However, by assumption there exists a sequence (1.2.20), for some n N and objects k1, . . . , knof I . Then, a simple induction reduces the assertion to the case where n = 2. But since I islocally directed, we may then find k Ob(I) and morphisms c : k1 k, b : j k, so itsuffices to take a := c d, where d : i k1 is the morphism appearing in (1.2.20). Dually, if Iis connected and locally codirected, then I is codirected.

It follows that a category I is filtered if and only if it is directed, satisfies the coequalizingcondition of definition 1.2.19(v), and its set of objects is non-empty.

(ii) Let I be a small category. It is easily seen that I is connected if and only if it is aconnected object of the category Cat (see example 1.2.16(iv)). In general, there is a naturaldecomposition in Cat :

IsS

Is

where each Is is a connected category, and S is a small set. We call {Is | s S} the set ofconnected components of I , and we denote it

0(I).

Especially, every small pseudo-filtered category is a coproduct of a (small) family of filteredcategories. This decomposition induces natural isomorphisms in A :

colimI

FsS

colimIs

F es limIFsS

limsS

F es

for any functor F : I A , where es : Is I is the natural inclusion functor, for every s S.The details shall be left to the reader.

Proposition 1.2.22. For any category C we have :(i) C is complete (resp. finitely complete) if and only if the following conditions hold :

(a) The product of every small (resp. finite) family of objects is representable in C .(b) All equalizers are representable in C .

(ii) Dually, C is cocomplete (resp. finitely cocomplete) if and only if the following holds :(a) The coproduct of any small (resp. finite) family of objects is representable in C .(b) All coequalizers are representable in C .

Proof. Clearly it suffices to show (i). Now, conditions (a) and (b) obviously hold if C is com-plete. Conversely, suppose that the conditions hold, let I be any small (resp. finite) category,


and F : I C any functor. We regard Ob(I) (resp. Morph(I)) as a discrete subcategory of I(resp. of Morph(I) : see example 1.1.6(ii)), and we consider the diagram

Morph(I)s //

t// Ob(I)

ob // I

where s and t are respectively the restrictions of the functors s and t from (1.1.30), and ob is theinclusion functor. We let as well m : ob s ob t be the restriction of the natural transfor-mation m of (1.1.30). By assumption, we may find objects P,Q Ob(C ) with isomorphisms

P : hP L1 := lim

Ob(I)F ob Q : hQ

L2 := limMorph(I)

F ob t in C V

for any universe V such that C has V-small Hom-sets. We define two morphisms of presheavesa, b : L1 L2 by the rules :

a() := t and b() := (F m) ( s)for every X Ob(C ) and every cone : cX F ob. Then a and b are represented by uniquemorphisms , : P Q in C (see example 1.2.8(i)) such that

a P = Q h and b P = Q h.Lastly, let E := Equal(, ) (notation of example 1.2.16(ii)), and notice that the tautologicalcone for E yields a monomorphism E hP , so we may regard E as a subobject of hP ; itfollows easily that P restricts to an isomorphism between E hP and the presheaf E L1such that, for every X Ob(C ), the set E (X) consists of all the cones : cX F ob witha() = b(). A simple inspection shows that the latter condition describes precisely the conescX F ; on the other hand, by assumption E is representable in C , so the same holds for thelimit of F .

Example 1.2.23. (i) The category Set is complete and cocomplete. More precisely, for anysmall family of sets S := (Si | i I) we have natural representatives for the product andcoproduct of S : namely, for the product one may take the usual cartesian product of the setsSi, and for the coproduct one may take the disjoint union

iI

Si :=iI

({i} Si).

Likewise, we have natural representatives for the equalizer and coequalizer of any pair of mapsf, g : S S : namely, the equalizer of f and g is represented by the subset

Equal

foundations for almost ring theory november 4, 2016

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