June 12 version

(Thanks to Brian Conrad for many suggestions!)

There will be 20 talks over 10 days, in MIT Rm 2-255:  Sun. June 6- Wed.
June 9, Fri. June 11, Mon. June 14 - Fri. June 18.  The talks will begin
at 1.  Each will be (very approximately) 1-1.5 hours, and there will
be a half hour break in between.

A tentative schedule for the talks (also listed in the summary):

Chapter 1.  Finish in 8 lectures (by this coming Wed.)  
Sunday:  1-3 Ravi; 4 Jason  (60 pages)
Monday:  5 Jason; 6 Tom (15 pages) 
Tuesday: EGA IV_1.1-1.6 Andreas; 7-8 Kalle; 9 Adam (25 pages)
Wednesday:  10 Ravi; 10 Benji (33 pages)

Chapter 2.  Finish in 11 lectures
Friday:  1 Andreas; 2 Jason (29 pages)
Monday:  2 Kalle; 3 Benji (37 pages)
Tuesday:   4 Adam; 4 Ravi (23 pages)
Wednesday:  5 Ana Maria; 5 Behrang (16 pages)
Thursday:  6 Benji; 7 Andreas (42 pages)
Friday:    musings on 8 Torsten (lots of pages)

EGA I.  The language of schemes

The pre-seminar (read on our own).

Introduction  5-9

Chapter 0 preliminaries 11-78, about 60 pages.  

1.  Rings of fractions.

1.0 Rings and algebras.
1.1 Radical of an ideal.  Nilradical and radical of a ring.
1.2 Modules and ring of fractions.
1.3 Functorial properties.
1.4 Change of multiplicative subset
1.5 Base change.
1.6 Identification of a module M_f with an inductive limit.
1.7 Support of a module

2. Irreducible and Noetherian spaces.

3. Inverse on sheaves.  

3.1 Sheaves with values in a category.
3.2 Presheaves on a base.
3.3 Gluing sheaves.
3.4 Direct images of presheaves.
3.5 Inverse images of presheaves.
3.6 Simple sheaves and locally simple sheaves.
3.7 Inverse images of sheaves of groups or rings.
3.8 Sheaves of pseudo-discrete spaces.

4. Ringed spaces.  Ends on p. 44.

4.1 Ringed spaces, A-modules, A-algebras.
4.2 Direct image of an A-module.
4.3 Inverse image of a B-module.
4.4 Relation between direct images and inverse images.

5. Quasi-coherent sheaves and coherent sheaves.  44-54, 10 pages.

5.1 Quasi-coherent sheaves.
5.2 Sheaves of finite type.
5.3 Coherent sheaves.
5.4 Locally free sheaves.
5.5 Sheaves on a locally ringed space.

6. Flatness.  54-60, 6 pages.

6.1 Flat modules.
6.2 Base change.
6.3 Localization.
6.4 Faithfully flat modules.
6.5 Restriction of scalars.
6.6 Faithfully flat rings.
6.7 Flat morphisms of ringed spaces.

7. Adic rings.  60-78, 18 pages.

7.1 Admissible rings.
7.2 Adic-rings and projective limits.
7.3 Pre-adic noetherian rings.
7.4 Quasi-finite modules on local rings.
7.5 Rings of formally restricted series.  
  (From Brian: e.g., the p-adic completion of Z_p[X] rather than the
  (p,X)-adic completion --- it is "formal power series with coeff --->
  0", which is usually called "restricted power series"; these are the
  kinds of rings you get when you consider the deformations (Z/p^n)[X]
  of F_p[X] and try to "pass to the limit").
7.6 Complete rings of fractions.
  (arise in study of local rings on formal schemes; don't skip)
7.7 Tensorial complete products.
7.8 Topologies on modules of homomorphisms.


----

The seminar.

Chapter 0.  1 lecture the week before seminar begins, concentrating on
(i) the little bit about the general concept of a coherent sheaf, as
would be applicable to other geometric theories like analytic spaces, etc;
(ii) stuff on adic rings (really important for formal schemes)
(iii) sheaves of topological rings


Chapter 1.  10 chapters, 79-213.  Finish in 8 lectures.

1 intermediate lecture on later fill-in from EGA IV_1.

Chapter 2.  8 chapters, pages 5-203.  Finish in 11 lectures?
See below for schedule.

1. Affine schemes.  80-97, 17 pages.

1.1 Spec of a ring.
1.2 Functorial properties.
1.3 Sheaf associated to a module.
1.4 Quasicoherent sheaves on Spec.
1.5 Coherent sheaves on Spec.
1.6 Functorial properties of quasicoherent sheaves on Spec.
1.7 Characterization of morphisms of affine schemes.

2.  Preschemes and morphisms of preschemes.  97-104, 7 pages.

2.1 Definition of preschemes.
2.2 Morphisms of preschemes.
2.3 Gluing preschemes.
2.4 Local schemes. 
2.5 Preschemes over a scheme.

3.  Products of preschemes. 104-119, 15 pages.

3.1 Sums of preschemes.
3.2 Product of preschemes.
3.3 Formal properties of the product; base change.
3.4 Points of a prescheme with values in a prescheme; geometric points.
3.5 Surjections and injections.
3.6 Fibers
3.7 Application:  reduction of a prescheme mod I.

4. Sub-preschemes and immersions. 119-127, 8 pages.

4.1 Subpreschemes.
4.2 Immersions.
4.3 Product of immersions.
4.4 Reciprocal image of a subscheme.
4.5 Local immersions and local isomorphisms.

5. Reduced preschemes; separatedness.  127-140, 13 pages.

5.1 Reduced preschemes.
5.2 Existence of a subprescheme having as its underlying (topological)
     space a given subspace
5.3 Diagonal; graph of a morphism.
5.4 Separated morphisms and separated preschemes.
5.5 Criterion for separatedness.

6. Finiteness conditions.  140-155, 15 pages.

6.1 Noetherian and locally noetherian preschemes.
6.2 Artinian preschemes.
6.3 Morphisms of finite type.
6.4 Algebraic preschemes.
6.5 Local determination of a morphism.
6.6 Quasi-compact morphisms and morphisms locally of finite type.

A brief foray to EGA IV_1.1

1.1 Quasicompact morphisms
1.2 Quasiseparated morphisms
1.3 Morphisms locally of finite type
1.4 Morphisms locally of finite presentation
1.5 Morphisms of finite type
1.6 Morphisms of finite presentation, 
1.7 Bettering earlier results (i.e. rewriting parts of EGA I and II
 with better definitions)

Back to EGA I:


7.  Rational maps. 155-164, 9 pages.

7.1 Rational maps and rational functions.
7.2 Domain of definition of a rational map.
7.3 Sheaf of rational functions.
7.4 Torsion sheaves and torsion-free sheaves.

8.  Chevalley schemes.  164-169, 5 pages.

8.1 Local rings "apparentes".
8.2 Local rings of an integral scheme.
8.3 Chevalley Schemes.

9.  Further information on quasicoherent sheaves.  169-180, 11 pages.

9.1 Tensor product of quasicoherent sheaves.
9.2 Direct image of a quasicoherent sheaf. 
9.3 Extension of sections of a quasicoherent sheaf.
9.4 Extension of quasicoherent sheaves.
9.5 Closed image of a prescheme.  Closure of a subscheme.
9.6 Quasicoherent sheaves of algebras; change of the structure sheaf.

10.  Formal schemes.  180-213, 33 pages.

10.1 Affine formal schemes.
10.2 Morphisms of formal affine schemes.
10.3 Ideals of definition of formal affine schemes.
10.4 Formal preschemes and morphisms of formal preschemes.
10.5 Ideals of definition of formal preschemes.
10.6 Formal preschemes as inductive limits of schemes.
10.7 Products of formal preschemes.
10.8 Formal completion of a prescheme along a closed part.
10.9 Extending a morphism to the formal completion.
10.10 Application of coherent schemes to formal affine schemes.
10.11 Coherent sheaves on formal schemes.
10.12 Adique morphisms of formal preschemes.
10.13 Morphisms of finite type.
10.14 Closed subpreschemes of formal preschemes.
10.15 Separated formal preschemes.

EGA IV_1.1

1.8 Morphisms of finite presentation and constructible sets
1.9 Pro-constructible and ind-constructible sets
if time:  1.10 Application to open morphisms


EGA II.  Global study of some classes of morphism.

1.  Affine morphisms.  5-19, 14 pages.

1.1 S-preschemes and O_S-algebras.
1.2 Affine preschemes over a prescheme.
1.3 Affine prescheme over S associated to an O_S algebra.
1.4 Quasi-coherent sheaves on an affine prescheme over S.
1.5 Base change.
1.6 Affine morphisms.
1.7 Vector bundle associated to a sheaf of modules.

2.  Proj ("Homogeneous Spec").  19-49, 30 pages.

2.1 Generalities on graded rings and modules.
2.2 Rings of fractions of a graded ring.
2.3 Proj of a graded ring.
2.4 Scheme structure on Proj(S).
2.5 Sheaf associated to a graded module.
2.6 Graded S-module associated to a sheaf on Proj(S).
2.7 Finiteness conditions.
2.8 Functorial "Comportements".
2.9 Closed subschemes of Proj(S).

3.  Proj of a graded sheaf of algebras.  49-71, 22 pages.

3.1 Proj of a graded quasicoherent O_X-algebra.
3.2 The sheaf on Proj(S) associated to a graded S-module. (Here S's are script-S's.)
3.3 The graded S-module associated to a sheaf on Proj(S).
3.4 Finiteness conditions.
3.5 Functorial "Comportements".
3.6 Closed subpreschemes of Proj(S).
3.7 Morphisms from a prescheme to a Spec.
3.8 Crteria for immersion in a proj.  

4.  Projective fibrations.  Ample sheaves.  71-94, 23 pages.

4.1 Definition of projective fibrations.
4.2 Morphisms from a prescheme to a projective fibration.
4.3 Segre morphisms.
4.4 Immersions in projective fibrations.  Very ample sheaves.
4.5 Ample sheaves.
4.6 Relatively ample sheaves.

5.  Quasi-affine morphisms; quasi-projective morphisms; proper morphisms; projective morphisms.  94-110, 16 pages.

5.1 Quasi-affine morphisms.
5.2 Serre's criterion.
5.3 Quasi-projective morphisms.
5.4 Proper morphisms and universally closed morphisms.
5.5 Projective morphisms.
5.6 Chow's Lemma.

6.  Integral morphisms and finite morphisms.  110-138, 28 pages.

6.1 Preschemes entire over another prescheme.
6.2 Quasifinite morphisms. 
6.3 Integral closure of a prescheme. 
6.4 Determinant of an endomorphism of O_X-modules.
6.5 Norm of an invertible sheaf.
6.6 Application:  criteria for ampleness.
6.7 Chevalley's Theorem.

7.  Valuative criteria.  138-152, 14 pages.

7.1 Further information on valuation rings.
7.2 Valuative criterion for separatedness.
7.3 Valuative criterion for properness.
7.4 Algebraic curves and function fields of dimension 1.
(4 pages missing here!) 

8.  Blown-up schemes; projective cones; projective closure.  152-203, 51 pages.
(We can end here.  Possibly skippable, or Johan could summarize.)

8.1 "Eclates" preschemes.
8.2 Preliminary results on localization in graded rings.
8.3 Projective cones.
8.4 Projective closure of a vector bundle.
8.5 Functorial "Comportements".
8.6 A canonical isomorphism for pointed cones.
8.7 Blowing up projective cones.
8.8 Ample sheaves and contractions (blow-downs?).
8.9 Grauert's criterion for ampleness:  enonce.
8.10 Grauert's criterion for ampleness:  proof.
8.11 Uniqueness of contractions.
8.12 Quasicoherent sheaves on projective cones.
8.13 Projective closures of subsheaves and closed subschemes.
8.14 Remarks on sheaves associated to graded S-modules.