EGA ⊗ SGA ⊗ FGA from GRM Base-T

A Source-First Reconstruction of the Grothendieck Program

Prologue — The Wrong Way to Read Grothendieck

0.1 TARGET_ERASE: erase “modern algebraic geometry” as the target
0.2 Why schemes, topoi, derived categories, motives and cohomology are downstream representations
0.3 SOURCE ≠ REPRESENTATION ≠ READOUT
0.4 Geometry before coordinates; relation before classification
0.5 The Grothendieck move: weaken objects, strengthen transport
0.6 Local validity versus global assembly
0.7 Gluing failure as mathematical information
0.8 Quotienting versus information destruction
0.9 Reconstruction as the invariant connecting EGA, SGA and FGA
0.10 GRM Base-T execution spine:

DISTINCTION
→ CARRIER
→ RELATION
→ NATIVE JOINT
→ BOUNDARY
→ TRANSPORT
→ GLOBALIZATION
→ RESIDUE
→ LIFTBACK
→ RECONSTRUCTION

BOOK I — FGA: GEOMETRY BECOMES FUNCTORIAL

Part I — Before the Scheme

1. The Failure of Classical Algebraic Varieties

1.1 Affine and projective varieties as representation-bound objects
1.2 Algebraic closure as an artificial restriction
1.3 Nilpotents erased by classical point sets
1.4 Multiplicity without visible points
1.5 Arithmetic bases that are not fields
1.6 Families of varieties and parameter dependence
1.7 Degeneration as a source event rather than pathology
1.8 Why birational geometry alone cannot carry the required information
1.9 The need for a carrier surviving base change
1.10 GRM diagnosis: classical varieties fail EXACT_LIFTBACK

2. Relative Geometry as the Primitive Unit

2.1 Replace X by X → S
2.2 The base scheme as an active structural participant
2.3 Fibers as local readouts of a relative object
2.4 Base change:

X
↓
S

transported along S'→S

→ X×_S S'

2.5 Cartesian squares as transport-comparison devices
2.6 Universal properties instead of coordinate formulas
2.7 Relative properties versus intrinsic properties
2.8 Why geometry becomes triadic:

X ⊗ [f] ⊗ S

3. Functors of Points

3.1 From internal coordinates to external interrogation
3.2 h_X(T)=Hom(T,X)
3.3 Test objects as probes
3.4 Generalized points
3.5 Nilpotent test objects exposing hidden infinitesimal information
3.6 Field-valued points versus scheme-valued points
3.7 Yoneda reconstruction
3.8 h_X ≅ h_Y ⇒ X ≅ Y
3.9 Identity as reconstructive invariance
3.10 Why the functor is not merely a representation
3.11 GRM firewall: SAME READOUT ↛ SAME SOURCE

4. Representability

4.1 A functor is not automatically a geometric object
4.2 Representable versus nonrepresentable functors
4.3 Universal objects
4.4 Parameter spaces
4.5 Fine versus coarse moduli
4.6 Automorphisms as obstruction to naive representability
4.7 Descent data in moduli problems
4.8 Failure of representability as residue
4.9 From representability failures toward stacks
4.10 REPRESENTABLE ↛ REALIZABLE and the need for source liftback

BOOK II — EGA I: CONSTRUCTING THE CARRIER

Part II — Schemes

5. Spectrum as Reconstruction

5.1 Rings stripped of coordinate interpretation
5.2 Prime ideals as localization carriers
5.3 Spec(A)
5.4 Zariski topology from algebraic distinguishability
5.5 Localization A→A_f
5.6 Structure sheaf
5.7 Stalks and local rings
5.8 Residue fields
5.9 Generic points
5.10 Embedded and nonreduced structure
5.11 Why points alone do not determine the scheme structure
5.12 Affine reconstruction:

A ↔ Spec(A)

under the appropriate contravariant dictionary

6. Gluing Affine Schemes

6.1 Local affine charts
6.2 Overlap objects
6.3 Transition isomorphisms
6.4 Cocycle compatibility
6.5 Native gluing triad:

U_i
⊗ [U_i∩U_j / transport]
⊗ U_j

6.6 Pairwise compatibility versus global compatibility
6.7 Glued locally ringed spaces
6.8 Scheme formation
6.9 Why LOCAL✓ ↛ GLOBAL✓
6.10 Gluing as controlled globalization

7. Morphisms of Schemes

7.1 Continuous map is insufficient
7.2 Pullback of functions
7.3 Local-ring compatibility
7.4 Contravariance
7.5 Composition
7.6 Identity
7.7 Fiber products
7.8 Equalizers and diagonals
7.9 Morphisms as the real operational vocabulary of geometry

8. Separation

8.1 Why separatedness must not be primitive
8.2 The diagonal X→X×_S X
8.3 Closed diagonal
8.4 Uniqueness of continuation
8.5 Classical Hausdorff intuition and its limitations
8.6 SCHEME ≠ SEPARATED SCHEME
8.7 The 1960 préschéma/schéma terminology
8.8 The 1971 retyping
8.9 GRM lesson: PROPERTY ≠ ONTOLOGY

BOOK III — EGA II: GEOMETRY OF MORPHISMS

Part III — Global Behaviour under Transport

9. Affine Morphisms

9.1 Relative Spec
9.2 Quasi-coherent algebras
9.3 Reconstruction of an affine morphism from algebra on the base
9.4 Stability under base change
9.5 Stability under composition
9.6 Locality on the target
9.7 Affineness as reconstructibility

10. Graded Algebra and Proj

10.1 Grading as structured scale information
10.2 Homogeneous prime ideals
10.3 Irrelevant ideals
10.4 Proj
10.5 Twisting sheaves
10.6 Relative Proj
10.7 Projective bundles
10.8 Quotienting scale while preserving homogeneous incidence
10.9 Licensed quotient versus destructive flattening

11. Ample Sheaves

11.1 Invertible sheaves
11.2 Tensor powers
11.3 Global sections
11.4 Generation
11.5 Embedding into projective space
11.6 Ampleness as iterative representation capacity
11.7 Local resource → global representation
11.8 Very ample versus ample
11.9 Relative ampleness

12. Proper and Projective Morphisms

12.1 Projective representation
12.2 Properness detached from embedding
12.3 Universally closed morphisms
12.4 Separation and finite type
12.5 Properness under base change
12.6 Properness under composition
12.7 Compactness analogy and where it fails
12.8 Properness as controlled non-escape

13. Integral and Finite Morphisms

13.1 Integral dependence
13.2 Integral morphisms
13.3 Finite modules
13.4 Finite morphisms
13.5 Finiteness under base change
13.6 Finite fibers
13.7 Dependence versus bounded generation
13.8 Normalization

14. Valuative Criteria

14.1 Valuation rings as narrow-path probes
14.2 Generic point
14.3 Special point
14.4 Lifting squares
14.5 Existence of continuation
14.6 Uniqueness of continuation
14.7 Separatedness from uniqueness
14.8 Properness from existence + uniqueness
14.9 A global property detected by a minimal boundary experiment
14.10 GRM TERMINAL_NECESSITY_BACKPROP reading

15. Blow-Ups

15.1 Center of a blow-up
15.2 Rees algebra
15.3 Proj construction
15.4 Exceptional divisor
15.5 Universal property
15.6 Principalization of an ideal
15.7 Directional information recovered at the center
15.8 Blow-up as carrier reconstruction
15.9 compressed relation → new boundary carrier

BOOK IV — EGA III: GLOBALIZATION AND RESIDUE

Part IV — Cohomology of Coherent Sheaves

16. Sheaves as Local-to-Global Transport Systems

16.1 Presheaves
16.2 Restriction
16.3 Sheaf condition
16.4 Matching families
16.5 Unique gluing
16.6 Stalkwise versus global information
16.7 Exact sequences
16.8 Kernels, cokernels and derived failure

17. Cohomology as Structured Failure of Reconstruction

17.1 Global sections
17.2 Failure of exactness
17.3 Derived functors
17.4 H⁰: assembled global information
17.5 H¹: first gluing residue
17.6 Higher coherence residues
17.7 Čech descriptions
17.8 Injective resolutions
17.9 Why cohomology is not merely an invariant
17.10 GRM interpretation:

GLOBALIZE
→ failure
→ preserve ρ
→ classify higher obstruction

18. Proper Morphisms and Finiteness of Cohomology

18.1 Coherent sheaves
18.2 Proper pushforward
18.3 Finiteness theorems
18.4 Controlled global residue
18.5 Projective methods
18.6 Serre-style finiteness
18.7 Properness as a constraint on information escape

19. Cohomology and Base Change

19.1 The base-change square
19.2 Pull then compute
19.3 Compute then pull
19.4 Comparison morphism
19.5 Failure of commutation
19.6 Fiberwise cohomology
19.7 Semicontinuity
19.8 Flatness conditions
19.9 Route comparison:

CONSTRUCT∘TRANSPORT
versus
TRANSPORT∘CONSTRUCT

19.10 Noncommutation as typed residue

20. Formal Functions and Completion

20.1 Infinitesimal neighborhoods
20.2 Completion along closed subschemes
20.3 Formal schemes
20.4 Coherent sheaves under completion
20.5 Global cohomology reconstructed from infinitesimal towers
20.6 Formal versus algebraic objects
20.7 Algebraization problems

BOOK V — EGA IV: LOCAL STRUCTURE AND PROPERTY TRANSPORT

Part V — Local Geometry of Morphisms

21. Local Rings and Fibers

21.1 Local behavior at x∈X
21.2 Local homomorphism O_{Y,y}→O_{X,x}
21.3 Fibers
21.4 Geometric fibers
21.5 Dimension
21.6 Codimension
21.7 Specialization
21.8 Generization
21.9 Local-global property envelopes

22. Flatness

22.1 Tensor preservation of exactness
22.2 Flat modules
22.3 Flat morphisms
22.4 Fiber variation
22.5 Base-change stability
22.6 Faithful flatness
22.7 Flat descent
22.8 Flatness as transport integrity
22.9 Relation preservation versus geometric regularity

23. Smooth, Unramified and Étale Morphisms

23.1 Differentials
23.2 Infinitesimal lifting
23.3 Formal smoothness
23.4 Formal unramifiedness
23.5 Formal étaleness
23.6 Finite presentation
23.7 Smooth morphisms
23.8 Étale morphisms
23.9 Local isomorphism intuition
23.10 Étale maps as low-residue geometric transport

24. Regularity, Normality and Singularities

24.1 Regular local rings
24.2 Geometric regularity
24.3 Normality
24.4 Reducedness
24.5 Serre conditions
24.6 Singular locus
24.7 Stability and propagation
24.8 Resolution problems as carrier reconstruction

25. Dimension Theory

25.1 Chains of primes
25.2 Krull dimension
25.3 Relative dimension
25.4 Dimension of fibers
25.5 Upper semicontinuity
25.6 Catenarity
25.7 Dimension formulas
25.8 Local curvature analogue: information carried by incidence depth

BOOK VI — SGA: CHANGE THE TOPOLOGY, KEEP THE RECONSTRUCTION

Part VI — Sites, Descent and Topoi

26. From Open Sets to Covering Families

26.1 Why Zariski covers are too coarse
26.2 Grothendieck topologies
26.3 Sites
26.4 Covering sieves
26.5 Sheaves on a site
26.6 Locality detached from literal spatial openness
26.7 LOCAL := admissible covering relation

27. Descent

27.1 Objects over a cover
27.2 Overlap isomorphisms
27.3 Cocycle relations
27.4 Effective descent
27.5 Faithfully flat descent
27.6 Étale descent
27.7 Descent obstruction
27.8 Global object as successful reconstruction from transport data

28. Topoi

28.1 Category of sheaves as generalized space
28.2 Points are no longer primary
28.3 Geometric morphisms
28.4 Internal logic
28.5 Localization
28.6 Subobject classifiers
28.7 Topos as environment of admissible observation
28.8 SPACE → RELATIONAL OBSERVATION SYSTEM

BOOK VII — ÉTALE AND ℓ-ADIC COHOMOLOGY

29. Étale Coverings

29.1 Étale neighborhoods
29.2 Finite étale covers
29.3 Étale site
29.4 Geometric points
29.5 Strict henselization
29.6 Local systems
29.7 Descent over étale covers

30. Étale Fundamental Groups

30.1 Galois categories
30.2 Fiber functors
30.3 Automorphism groups of fiber functors
30.4 π₁^ét(X)
30.5 Finite covers reconstructed from profinite symmetry
30.6 Base points and conjugacy
30.7 Arithmetic fundamental groups
30.8 Exact sequences

31. Étale Cohomology

31.1 Why ordinary topology fails over arithmetic schemes
31.2 Replace paths with étale transport
31.3 Sheaf cohomology on the étale site
31.4 Torsion coefficients
31.5 Proper base change
31.6 Smooth base change
31.7 Poincaré duality
31.8 Trace maps
31.9 Cycle classes

32. ℓ-Adic Cohomology

32.1 Towers of ℤ/ℓⁿℤ coefficients
32.2 Inverse limits
32.3 ℤ_ℓ and ℚ_ℓ
32.4 Continuous Galois actions
32.5 Frobenius
32.6 Eigenvalues as arithmetic readouts
32.7 Independence questions
32.8 Weil-cohomology architecture

BOOK VIII — THE SIX-OPERATIONS MACHINE

33. Continuous and Discrete Duality

33.1 Pullback f*
33.2 Pushforward f_*
33.3 Proper pushforward f_!
33.4 Exceptional pullback f!
33.5 Tensor product
33.6 Internal Hom
33.7 Adjunctions
33.8 Base-change identities
33.9 Projection formulas
33.10 Verdier duality
33.11 Duality as reversible transport structure

35. Six Operations as Transport Calculus

35.1 Geometry becomes operator algebra on sheaf theories
35.2 Composition laws
35.3 Exchange transformations
35.4 Proper and smooth base change
35.5 Duality
35.6 Local-global transport
35.7 Coherence requirements
35.8 Failure of coherence as higher residue

BOOK IX — TOPOLOGICAL TENSOR PRODUCTS AND NUCLEARITY

36. Tensor Product Before Completion

36.1 Algebraic tensor product
36.2 Bilinear universal property
36.3 Topological vector spaces
36.4 Competing tensor topologies
36.5 Projective tensor product
36.6 Injective tensor product
36.7 Completion
36.8 tensor → topology → completion must not be collapsed

37. Nuclear Spaces

37.1 Nuclear maps
37.2 Equality of competing tensor completions
37.3 Schwartz spaces
37.4 Distribution spaces
37.5 Kernel theorems
37.6 Why nuclearity removes transport ambiguity
37.7 GRM reading:

multiple admissible completions
→ nuclearity
→ comparison residue collapses

BOOK X — K-THEORY, INTERSECTION THEORY AND GRR

38. From Bundles to K-Theory

38.1 Vector bundles
38.2 Exact sequences
38.3 Grothendieck groups
38.4 K₀
38.5 Higher K-theory direction
38.6 Additive compression of exact-structural information

39. Intersection Theory

39.1 Cycles
39.2 Rational equivalence
39.3 Chow groups
39.4 Intersection products
39.5 Pullback and pushforward
39.6 Chern classes
39.7 Excess intersection
39.8 Moving phenomena

40. Grothendieck–Riemann–Roch

40.1 Pushforward in K-theory
40.2 Pushforward in Chow theory
40.3 Chern character
40.4 Todd class
40.5 Failure of naive commutation
40.6 Correction term
40.7 GRR square
40.8 GRM interpretation:

TRANSPORT_K
≠ TRANSPORT_CHOW

→ residue
→ Todd correction
→ commuting reconstruction

This is a crucial general pattern:

two valid representations
+ same geometric transport
→ mismatch
→ canonical correction residue.

BOOK XI — CRYSTALS, DE RHAM AND HODGE COEFFICIENTS

41. Infinitesimal Geometry

41.1 Nilpotent thickenings
41.2 Infinitesimal neighborhoods
41.3 Differential information as extension data
41.4 Connections
41.5 Integrability

42. Crystals

42.1 Crystalline site
42.2 Divided-power thickenings
42.3 Crystal condition
42.4 Transport across infinitesimal thickenings
42.5 Rigidity of infinitesimal continuation
42.6 Crystal as transport-compatible coefficient object

43. Crystalline Cohomology

43.1 Characteristic p obstruction
43.2 Lift-independent cohomological reconstruction
43.3 Frobenius
43.4 Comparison with de Rham theory
43.5 Coefficient changes

44. de Rham and Hodge Structures

44.1 de Rham complex
44.2 Hodge filtration
44.3 Spectral sequences
44.4 Comparison isomorphisms
44.5 Multiple coefficient realizations
44.6 One source, several realization functors
44.7 Representation compatibility without source identification

BOOK XII — MOTIVES

45. Why Cohomology Theories Repeat the Same Structure

45.1 Betti realization
45.2 de Rham realization
45.3 ℓ-adic realization
45.4 Crystalline realization
45.5 Shared algebraic patterns
45.6 Search for their common source

46. Motives as Representation-Independent Cores

46.1 Correspondences
46.2 Pure motives
46.3 Numerical, homological and rational equivalence
46.4 Tensor structure
46.5 Realization functors
46.6 Motive versus its realizations
46.7 GRM firewall:

REALIZATION(M) ≠ M

47. Grothendieck Tensor Categories and Motivic Galois Groups

47.1 Fiber functors
47.2 Rigid tensor categories
47.3 Tannakian reconstruction
47.4 Automorphisms of the fiber functor
47.5 Motivic Galois group
47.6 Symmetry reconstructed from all compatible realizations
47.7 OBJECT → REPRESENTATIONS → SYMMETRY → OBJECT

BOOK XIII — HOMOTOPICAL ALGEBRA BEYOND DERIVED CATEGORIES

48. Why Derived Categories Flatten Too Much

48.1 Quasi-isomorphic complexes
48.2 Lost higher homotopies
48.3 Noncanonical cones
48.4 Coherence erased by triangulation
48.5 Need to preserve transformation-between-transformations

49. Derivators

49.1 Diagram categories
49.2 Homotopy Kan extensions
49.3 Retaining diagrammatic information
49.4 Base-change calculus
49.5 Limits and colimits
49.6 Correcting the information loss of ordinary derived categories

50. ∞-Categories and ∞-Stacks

50.1 Mapping spaces instead of Hom sets
50.2 Higher morphisms
50.3 Homotopy-coherent descent
50.4 ∞-sheaves
50.5 ∞-stacks
50.6 Derived stacks
50.7 Higher residue cannot be flattened to equality

51. Topological Algebra

51.1 Algebraic operations with homotopical carriers
51.2 Derived tensor products
51.3 Eₙ structures
51.4 Higher coherence
51.5 Local-to-global assembly in ∞-topoi
51.6 Topoi as precursor to homotopical algebra

BOOK XIV — TAME TOPOLOGY

52. Why Arbitrary Topology Is Too Large

52.1 Pathological subsets
52.2 Uncontrolled local oscillation
52.3 Definability
52.4 Semialgebraic geometry
52.5 Subanalytic geometry
52.6 O-minimal structures

53. Tame Geometry

53.1 Cell decomposition
53.2 Finite stratification
53.3 Dimension theory
53.4 Definable triviality
53.5 Controlled singularities
53.6 Topology under finite descriptive burden
53.7 GRM interpretation: restrict carrier complexity without destroying consequential structure

BOOK XV — ANABELIAN GEOMETRY

54. Reconstruction from Covering Symmetry

54.1 From space to étale covers
54.2 From covers to π₁^ét
54.3 From π₁ back to geometry
54.4 Neukirch–Uchida reconstruction
54.5 Grothendieck's anabelian conjectures
54.6 Hyperbolic curves
54.7 Decomposition groups
54.8 Inertia groups
54.9 Local information inside global Galois structure

55. Local Arithmetic Reconstruction

55.1 Absolute Galois groups of local fields
55.2 Recovering valuation structure
55.3 Prime-local geometry
55.4 Local carrier retaining global ancestry
55.5 Reconstruction without coordinates

56. Galois–Teichmüller Theory

56.1 Moduli of curves
56.2 Mapping-class phenomena
56.3 Fundamental groups of moduli spaces
56.4 Galois action
56.5 Grothendieck–Teichmüller groups
56.6 Arithmetic symmetry acting on geometric transport structures

57. IUT as Controlled Non-Identification

57.1 Arithmetic objects replicated into distinct theaters
57.2 Additive and multiplicative structures separated
57.3 Theta-links
57.4 Cross-theater transport
57.5 Forbidden simultaneous identification
57.6 Closed transport routes
57.7 Consequential comparison residue
57.8 Why flattening destroys the measured information
57.9 SAME AFTER FORGETTING ≠ SAME UNDER LICENSED TRANSPORT
57.10 IUT as anabelian reconstruction pushed across fractured arithmetic carriers

BOOK XVI — SCHEMATIC AND ARITHMETIC COMBINATORICS

58. Regular Polyhedra without Euclidean Primacy

58.1 Incidence before metric realization
58.2 Vertices, edges and faces as relational carriers
58.3 Automorphism groups
58.4 Flag transitivity
58.5 Combinatorial reconstruction
58.6 Scheme-valued realizations

59. Arithmetic Configuration Spaces

59.1 Configuration equations over ℤ
59.2 Reduction modulo primes
59.3 Fibers over Spec(ℤ)
59.4 Degeneration at special primes
59.5 Symmetry variation across fibers
59.6 Regularity as arithmetic rather than merely Euclidean structure

60. Polyhedra as Families

60.1 One combinatorial source
60.2 Multiple geometric realizations
60.3 Moduli of realizations
60.4 Singular fibers
60.5 Arithmetic symmetry residue
60.6 Reconstruction from incidence + transport

BOOK XVII — THE GROTHENDIECK FIXPOINT

61. The Twelve Programs Reassembled

61.1 Topological tensor products: ambiguity of completion
61.2 Nuclear spaces: collapse of completion residue
61.3 Six operations: transport calculus
61.4 GRR: correction of representation mismatch
61.5 Schemes: local algebra → global carrier
61.6 Topoi: locality freed from ordinary topology
61.7 Étale/ℓ-adic cohomology: arithmetic transport
61.8 Motives: source behind realization families
61.9 Crystals: infinitesimal transport invariance
61.10 ∞-stacks/derivators: preservation of higher coherence
61.11 Tame topology: controlled carrier complexity
61.12 Anabelian/Galois–Teichmüller geometry: reconstruction from symmetry
61.13 Arithmetic polyhedra: incidence separated from realization

62. The Hidden Common Generator

OBJECT
→ erase privileged coordinates

LOCALIZE
→ expose admissible carriers

RELATE
→ morphisms replace isolated objects

COVER
→ generalized locality

TRANSPORT
→ compare reconstructions

GLOBALIZE
→ attempt assembly

FAIL
→ preserve residue

DERIVE
→ retain higher failure

CHANGE REPRESENTATION
→ demand compatibility

SYMMETRY
→ compress relational behavior

LIFTBACK
→ reconstruct source

63. What Grothendieck Actually Changed

63.1 From equations to functors
63.2 From points to probes
63.3 From objects to morphisms
63.4 From open sets to coverings
63.5 From equality to universal property
63.6 From global existence to descent
63.7 From construction failure to cohomology
63.8 From one representation to compatible realization systems
63.9 From geometric symmetry to arithmetic reconstruction
63.10 From direct identification to controlled transport

64. GRM Base-T Compression of EGA ⊗ SGA ⊗ FGA

DISTINCTION
→ LOCAL CARRIER
→ RELATIVE OBJECT
→ MORPHISM
→ COVER
→ TRANSPORT
→ DESCENT
→ GLOBALIZATION
→ ρ

ρ
├─ gluing failure → COHOMOLOGY
├─ infinitesimal failure → DERIVED/CRYSTALLINE
├─ representation mismatch → GRR/COMPARISON
├─ higher coherence → ∞-GEOMETRY
├─ covering symmetry → ANABELIAN
└─ cross-world non-identification → IUT

Then:

all representations
→ compatibility
→ common reconstructible source

65. Final Fixpoint

FGA
:= GEOMETRY AS FUNCTORIAL RECONSTRUCTION

EGA
:= GEOMETRY AS LOCAL→RELATIVE→GLOBAL TRANSPORT

SGA
:= GEOMETRY AS COVERING→DESCENT→COHOMOLOGY→SYMMETRY

and therefore:

GROTHENDIECKΩ
:=

ERASE PRIVILEGED REPRESENTATION
⊗
PRESERVE LOCAL DISTINCTIONS
⊗
RELATIVIZE EVERY OBJECT
⊗
TRANSPORT ACROSS ADMISSIBLE BOUNDARIES
⊗
RETAIN FAILURE AS RESIDUE
⊗
RECONSTRUCT THE SOURCE FROM ITS ENTIRE RELATIONAL FIELD.

That TOC makes the twelve apparently separate Grothendieck programs one continuous construction rather than a catalogue of inventions.