Since the last update of October 2014 we have added the following material:

  1. Structure modules over PIDs following Warfield Tag 0ASL
  2. Correct proof of Lemma Tag 05U9 thanks to Ofer Gabber
  3. A flat ring map which is not a directed colimit of flat finitely presented ring maps Tag 0ATE
  4. Glueing dualizing complexes Tag 0AU5
  5. Trace maps Tag 0AWG
  6. Duality for a finite morphism Tag 0AWZ
  7. Grauert-Riemenschneider for surfaces Tag 0AX7
  8. Torsion free modules Tag 0AVQ
  9. Reflexive modules Tag 0AVT
  10. Finiteness theorem for f_* Tag 0AW7
  11. Fix definition of depth (thanks to Burt) Tag 00LE and Tag 0AVY
  12. Characterizing universally catenary rings Tag 0AW1
  13. Improved section on Jacobson spaces thanks to Juan Pablo Acosta Lopez Tag 005T
  14. Faithfully flat descent of ML modules thanks to Juan Pablo Acosta Lopez Tag 05A5
  15. Improvements to the chapter on Chow homology discussed here
  16. Degrees of vector bundles on curves Tag 0AYQ
  17. Degrees of zero cycles Tag 0AZ0 and how this relates to degrees of vector bundles and with numerical intersections
  18. Quotient by category of torsion modules thanks to Ingo Blchschmidt Tag 0B0J
  19. New chapter on intersection theory discussed here
  20. Example of different colimit topologies Tag 0B2Y
  21. Section on topological groups, rings, modules Tag 0B1Y
  22. Section on tangent spaces Tag 0B28
  23. A bunch of material on (quasi-)projectivity, for example Tag 0B41 and Tag 0B44
  24. Glueing in a modification Tag 0B3W at a point of a scheme
  25. Improved material on sober spaces thanks to Fred Rohrer Tag 004U
  26. Riemann-Roch and duality for curves Tag 0B5B
  27. Fix idiotic mistake about graded projective modules, thanks to Rishi Vyas read his explanation on github
  28. Base change map in duality Tag 0AA5 is often an isomorphism (Tag 0AA8) and commutes with base change Tag 0AWG
  29. Bunch of changes thanks to Darij Grinberg
  30. Material on group schemes over fields Tag 047J
  31. Material on (locally) algebraic group schemes over fields Tag 0BF6
  32. Thickenings of quasi-affine schemes are quasi-affine Tag 0B7L
  33. Minimal closed subspaces which aren’t schemes Tag 0B7X
  34. Monomorphisms of algebraic spaces Tag 0B89
  35. Change of base field and schematic locus Tag 0B82
  36. Separated group algebraic spaces over fields are schemes Tag 0B8G
  37. Picard scheme of smooth projective curves over algebraically closed fields Tag 0B92
  38. Improved discussion of invertible modules… (too ashamed to put a link here)
  39. Jacobson algebraic spaces Tag 0BA2
  40. Nagata spaces Tag 0BAT
  41. For an algebraic space: locally Noetherian + decent => quasi-separated Tag 0BB6
  42. Various improvements on rational and birational maps Tag 01RR, Tag 01RN, and Tag 0BAJ
  43. Dimension formula for algebraic spaces Tag 0BAW
  44. Generically finite morphisms of algebraic spaces Tag 0BBA
  45. Birational morphisms of algebraic spaces Tag 0ACU
  46. Elementary etale neighbourhoods on algebraic spaces Tag 03IG
  47. Complements of affine opens have codimension 1 Tag 0BCQ
  48. Norms of invertible modules Tag 0BCX which allows us to descend ample invertible modules
  49. Descending (quasi-)projectivity through field extensions Tag 0BDB
  50. Section on splitting complexes Tag 0BCF for better handling of local structure of perfect complexes
  51. Section on stably free modules Tag 0BC2
  52. Jumping loci for perfect complexes on schemes Tag 0BDH
  53. Applications of cohomology and base change Tag 0BDM
  54. Theorem of the cube Tag 0BEZ
  55. Weil divisors on locally Noetherian schemes Tag 0BE0
  56. The Weil divisor class associated to an invertible module Tag 02SE
  57. K\”unneth formula for schemes over a field Tag 0BEC
  58. Algebraic group schemes are quasi-projective Tag 0BF7
  59. Numerical intersections Tag 0BEL
  60. Section on abelian varieties Tag 0BF9 containing just enough for our use later
  61. Tried to improve the exposition of convergence for spectral sequences using terminology mostly as in Weibel; still very far from perfect
  62. Long overdue characterization of algebraic spaces Tag 0BGQ
  63. Chapters on resolution of surface singularities one for schemes and one for algebraic spaces


Updated yet again

Just finished working through your comments once more. Please continue to leave these comments; I find them always to be helpful. In particular let us know if there is an argument that is just too quick and not clear, etc. Please also let us know what you tried…