A topos-theoretic view of difference algebra Ivan Tomašić Ivan Tomašić, School of Mathematical Sciences, Queen Mary Uni- versity of London, London, E1 4NS, United Kingdom E-mail address:
[email protected] arXiv:2001.09075v1 [math.AG] 24 Jan 2020 2000 Mathematics Subject Classification. Primary . Secondary . Key words and phrases. difference algebra, topos theory, cohomology, enriched category Contents Introduction iv Part I. E GA 1 1. Category theory essentials 2 2. Topoi 7 3. Enriched category theory 13 4. Internal category theory 25 5. Algebraic structures in enriched categories and topoi 41 6. Topos cohomology 51 7. Enriched homological algebra 56 8. Algebraicgeometryoverabasetopos 64 9. Relative Galois theory 70 10. Cohomologyinrelativealgebraicgeometry 74 11. Group cohomology 76 Part II. σGA 87 12. Difference categories 88 13. The topos of difference sets 96 14. Generalised difference categories 111 15. Enriched difference presheaves 121 16. Difference algebra 126 17. Difference homological algebra 136 18. Difference algebraic geometry 142 19. Difference Galois theory 148 20. Cohomologyofdifferenceschemes 151 21. Cohomologyofdifferencealgebraicgroups 157 22. Comparison to literature 168 Bibliography 171 iii Introduction 0.1. The origins of difference algebra. Difference algebra can be traced back to considerations involving recurrence relations, recursively defined sequences, rudi- mentary dynamical systems, functional equations and the study of associated dif- ference equations. Let k be a commutative ring with identity, and let us write R = kN for the ring (k-algebra) of k-valued sequences, and let σ : R R be the shift endomorphism given by → σ(x0, x1,...) = (x1, x2,...). The first difference operator ∆ : R R is defined as → ∆= σ id, − and, for r N, the r-th difference operator ∆r : R R is the r-th compositional power/iterate∈ of ∆, i.e., → r r ∆r = (σ id)r = ( 1)r−iσi.