# Appendix: black boxes

Exposé I and Exposé II quote several theorems without proof, citing later parts of SGA or EGA. The gentle reading treats
them the same way. This appendix collects every such black box in one place. For each, we record a one-sentence
statement, what the gentle reading uses it for, the files where it is cited, and the source.

Body files refer to a row by name in prose — e.g. "by Zariski's Main Theorem" — with the row's short slug given in the
first column for grep-ability.

The rows are ordered by topic. Reading top to bottom: definitional / categorical, then finite and quasi-finite
morphisms, then lifting, then flatness machinery, then local-ring properties, then topological invariance, then standard
commutative-algebra tools.

| Anchor                                            | Black box                                                            | Statement                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | What we use it for                                                                                                                                                                                                                                                                                                                                                                                                 | Used in                    | Source                                                                                                                                                       |
| ------------------------------------------------- | -------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | -------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| `formally-etale-lifting`                          | Lifting characterization of formal étaleness                         | A morphism `f : X → Y` is **formally étale** iff for every closed subscheme `Y₀′ ⊂ Y′` of an affine scheme defined by a nilpotent ideal of `𝒪_{Y′}`, every morphism `Y₀′ → X` over `Y` extends uniquely to a morphism `Y′ → X` over `Y`. The morphism `f` is **étale** iff it is formally étale and locally of finite presentation.                                                                                                                                                                                                                                                                                                                                                      | Equivalent modern definition of étale; the bridge between flat-and-unramified (gentle definition) and the lifting property is the conjunction of theorems I.5.5 and I.8.3.                                                                                                                                                                                                                                         | 04, 05, 11                 | EGA IV 17.1.1; EGA IV 17.3.1                                                                                                                                 |
| `quasi-finite-equivalences`                       | Quasi-finite module characterizations                                | For a local homomorphism `A → B` of noetherian local rings, the following are equivalent: (1) `B/𝔪B` is finite-dimensional over `k = A/𝔪`; (2) `𝔪B` is an ideal of definition of `B` and `κ(B)/k` is finite; (3) the completion `B̂` is finite as an `Â`-module.                                                                                                                                                                                                                                                                                                                                                                                                                          | Underwrites the three definitions of quasi-finite in file 04; (1) is taken as the primary definition, (2) and (3) follow.                                                                                                                                                                                                                                                                                          | 04                         | EGA 0 §0.7.4.2                                                                                                                                               |
| `zmt`                                             | Zariski's Main Theorem                                               | Let `f : X → Y` be quasi-finite at a point `x ∈ X`, with `Y` noetherian. There exists an open neighbourhood `U ∋ x` and a factorization `U → X′ → Y` with `U → X′` an open immersion and `X′ → Y` finite.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | Identifies a net local algebra with `B_𝔫` for `B` finite over the base; backbone of the local-form theorem I.7.6.                                                                                                                                                                                                                                                                                                  | 05, 11, 12                 | EGA IV §18.12.1; cf. Chevalley's seminar                                                                                                                     |
| `chevalley-proper`                                | Chevalley's theorem on proper morphisms                              | A proper quasi-finite morphism is finite.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | First proof of theorem I.5.1 (étale + radicial = open immersion): proper + flat + radicial + finite type collapses to rank-1 finite locally free.                                                                                                                                                                                                                                                                  | 05, 11                     | EGA III/IV; cf. EGA IV 8.11.1                                                                                                                                |
| `hensel`                                          | Hensel's lemma                                                       | **Algebra form.** For `A` a complete noetherian local ring with residue field `k`, every finite étale `k`-algebra `B̄` lifts uniquely to a finite étale `A`-algebra `B` with `B ⊗_A k ≃ B̄`. **Polynomial form (corollary).** A coprime factorization of a monic polynomial over `k` lifts to a factorization over `A`.                                                                                                                                                                                                                                                                                                                                                                    | Algebra form is the engine of theorem I.6.1 (étale extensions of complete local rings); polynomial form is the classical statement illustrated in file 05.                                                                                                                                                                                                                                                         | 05, 11                     | EGA IV §18.5.13 (algebra form); §17.7.3 (polynomial form)                                                                                                    |
| `flatness-sorites`                                | Flatness sorites                                                     | Composition, base change, and openness of flat morphisms of finite type; openness of the flat locus.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | Closes proposition I.4.5 (étale locus is open) and underpins the sorites of proposition I.4.6.                                                                                                                                                                                                                                                                                                                     | 04, 05, 11, 12             | EGA IV                                                                                                                                                       |
| `local-flatness-criterion`                        | Local flatness criterion                                             | For `M` finitely generated over a noetherian local ring `(A, 𝔪)`, `M` is flat over `A` iff `M` is free over `A` iff `Tor₁^A(M, A/𝔪) = 0`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | Justifies "finite + flat over a local ring = free" used in the étale story; e.g. the rank-one collapse in the first proof of I.5.1 and the construction of finite locally free algebras.                                                                                                                                                                                                                           | 04, 05, 11                 | EGA 0 §0.10.1.3                                                                                                                                              |
| `completion-flat`                                 | Faithful flatness of `A → Â`                                         | The `𝔪`-adic completion of a noetherian local ring is faithfully flat over it: `A → Â` is flat, and `M ⊗_A Â = 0` implies `M = 0` for any `A`-module `M`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | Reduces local statements to the complete case and lets conclusions propagate back; e.g. the second proof of theorem I.5.1 and the lifting/equivalence story of section I.5.                                                                                                                                                                                                                                        | 05, 06, 11, 12             | EGA 0 §0.10.2.3, §0.7.6.16                                                                                                                                   |
| `completion-test-for-flatness`                    | Flatness lifts/descends along completion                             | For `M` finitely generated over a noetherian local ring `(A, 𝔪)`, `M` is flat over `A` iff its completion `M̂` is flat over `Â`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | Lets one check étaleness on the complete local rings; used implicitly in the second proof of I.5.1 ("reduce to `A` complete") and in the proof of I.6.1.                                                                                                                                                                                                                                                           | 05, 11                     | EGA 0 §0.10.2.3                                                                                                                                              |
| `dim-depth-transfer`                              | Dimension and depth under quasi-finite flat                          | A local quasi-finite flat homomorphism of noetherian local rings preserves Krull dimension and depth.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | Opening of section I.9; routes the Cohen-Macaulay transfer and the Serre-criterion proof of I.9.5(i).                                                                                                                                                                                                                                                                                                              | 06, 12                     | EGA IV                                                                                                                                                       |
| `cm-transfer`                                     | Cohen-Macaulay transfer                                              | `A` is Cohen-Macaulay iff `B` is, for any local quasi-finite flat extension `A → B`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | Cited at the opening of section I.9 as one of the permanence properties already given by quasi-finite flatness.                                                                                                                                                                                                                                                                                                    | 06, 12                     | Corollary of `dim-depth-transfer`                                                                                                                            |
| `serre-normality`                                 | Serre's criterion for normality (R₁ + S₂)                            | For a noetherian ring `A`, `A` is normal iff every prime of height 1 gives a regular `A_𝔭` and every prime of height ≥ 2 has `depth A_𝔭 ≥ 2`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | First proof of theorem I.9.5(i): normality transfers along étale.                                                                                                                                                                                                                                                                                                                                                  | 06, 12                     | EGA IV 5.8.6                                                                                                                                                 |
| `nagata-finite`                                   | Nagata's theorem                                                     | The normalization of a noetherian Nagata ring `A` in a finite extension of `Frac(A)` is finite over `A`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | Section I.11 example (b): a complete integral local ring has finite normalization, hence its normalization is local.                                                                                                                                                                                                                                                                                               | 06, 12                     | Standard; cf. Nagata's _Local Rings_                                                                                                                         |
| `univ-homeo-equiv`                                | Universal-homeomorphism invariance of étale                          | If `Y′ → Y` is finite, radicial, and surjective (a universal homeomorphism), the functor `X ↦ X ×_Y Y′` is an equivalence on étale schemes. _Significance:_ étale-topological invariants are insensitive to inseparable phenomena like Frobenius.                                                                                                                                                                                                                                                                                                                                                                                                                                        | Section I.11 closing theorem; in particular Frobenius in characteristic `p` is invisible to the étale category.                                                                                                                                                                                                                                                                                                    | 06, 12                     | SGA 1 IX 4.10; EGA IV 18.10.3                                                                                                                                |
| `krull-intersection`                              | Krull's intersection theorem                                         | For a noetherian local ring `(A, 𝔪)` and a finitely generated `A`-module `M`, `⋂_n 𝔪^n M = 0`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | Step (ii) ⟹ (iii) of proposition I.3.1 (the differential criterion ⟹ diagonal criterion of unramifiedness).                                                                                                                                                                                                                                                                                                        | 03, 10                     | Standard                                                                                                                                                     |
| `nakayama`                                        | Nakayama's lemma                                                     | For `M` finitely generated over a local ring `(A, 𝔪)`, `M = 𝔪M` implies `M = 0`.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | Used throughout to lift information from the residue field to the local ring. Cited explicitly in the proofs of I.3.1, I.7.4, I.7.5, I.7.10.                                                                                                                                                                                                                                                                       | 03–05, 10, 11, 12          | Standard                                                                                                                                                     |
| `hironaka-multiplicity`                           | Hironaka's multiplicity criterion                                    | For a dominant morphism `f : X → Y` of finite type of reduced preschemes, `y ∈ Y` with `𝒪_y` regular: if every component of `f⁻¹(y)` has multiplicity `1` and `f⁻¹(y)_red` is normal, then `f⁻¹(y)` is reduced (hence normal), `X` is normal at all points of `f⁻¹(y)`, and `f` is flat over `Y` at all points of `f⁻¹(y)`.                                                                                                                                                                                                                                                                                                                                                              | Used for the Hironaka smoothness criterion II.2.5–II.2.6 (file 14): converts multiplicity-1 fiber data into smoothness of `f`.                                                                                                                                                                                                                                                                                     | 14                         | EGA IV §5.12.10                                                                                                                                              |
| `nagata-open-regular`                             | Nagata's open-regularity theorem                                     | The set of regular points of a (sufficiently nice — e.g. excellent or Nagata) noetherian scheme is open.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | Cited in II.5 (file 16) as the non-differential proof, due to Nagata, that breaks the circularity in the original differential-methods proof of II.5.5–II.5.6.                                                                                                                                                                                                                                                     | 16                         | Nagata, _Local Rings_; EGA IV §6.12                                                                                                                          |
| `mac-lane-separability`                           | Mac Lane's separability criterion via `Ω¹`                           | For a field extension `K / k`: `K` is separable over `k` iff `Ω¹_{K/k}` is generated by the differentials of a transcendence basis whose elements are linearly independent. Equivalently: every collection `f₁, …, f_n ∈ K` whose `df_i` are linearly independent in `Ω¹_{K/k}` is algebraically independent over `k`.                                                                                                                                                                                                                                                                                                                                                                   | Closes the proof of II.5.6 (file 16) without invoking Zariski's open-regularity theorem; the bridge between separability and the differential characterization of smooth field extensions.                                                                                                                                                                                                                         | 16                         | Bourbaki, _Algèbre_, Chapter V §16; Mac Lane, _Algebra_                                                                                                      |
| `serre-affine-vanishing`                          | Serre vanishing on affines                                           | For any affine scheme `X = Spec A` and any quasi-coherent `𝒪_X`-module `𝓕`, `H^i(X, 𝓕) = 0` for all `i ≥ 1`. The engine that makes Čech cohomology of quasi-coherent sheaves on a separated scheme computable on a single affine cover.                                                                                                                                                                                                                                                                                                                                                                                                                                                  | Throughout file 17; legitimizes every cocycle calculation in file 21 by ensuring the affine pieces of an affine cover are acyclic. Combined with Leray gives `Ȟ^p(𝔘, 𝓕) = H^p(X, 𝓕)`.                                                                                                                                                                                                                              | 17, 21, 22                 | Serre, _Faisceaux algébriques cohérents_ III.3; EGA III §1.3.1                                                                                               |
| `leray-acyclic-cover`                             | Leray's acyclic-cover theorem                                        | If `𝔘 = (U_i)` is an open cover of `X` such that every finite intersection `U_{i₀…i_p}` is `F`-acyclic (i.e. `H^q(U_{i₀…i_p}, F) = 0` for `q ≥ 1`), then the natural map `Ȟ^p(𝔘, F) → H^p(X, F)` is an isomorphism for every `p`.                                                                                                                                                                                                                                                                                                                                                                                                                                                        | Combined with `serre-affine-vanishing`, lets one compute sheaf cohomology of any quasi-coherent sheaf on a separated noetherian scheme from a single affine open cover. Used everywhere in file 17 and file 21.                                                                                                                                                                                                    | 17, 21                     | Godement, _Topologie algébrique et théorie des faisceaux_ II.5; EGA 0_III §12.4                                                                              |
| `formally-smooth-ega`                             | EGA lifting-property definition of formally smooth                   | EGA 0_IV §19.3.1 defines: a topological `A`-algebra `B` is **formally smooth** over `A` if for every discrete topological `A`-algebra `C` and every nilpotent ideal `J ⊂ C`, every continuous `A`-homomorphism `B → C/J` lifts to a continuous `A`-homomorphism `B → C`. Drops SGA III.1.1's finite-residual hypothesis.                                                                                                                                                                                                                                                                                                                                                                 | Cited from file 19 as the modern conceptual framing of III.1.1; the lifting property is what III.2.1 recovers from the formal-power-series definition. EGA's definition is the one used throughout EGA 0_IV §19 onwards.                                                                                                                                                                                           | 19                         | EGA 0_IV §19.3.1                                                                                                                                             |
| `cohen-ring`                                      | Cohen ring and structure theorem                                     | A **Cohen ring** is a complete noetherian local ring with maximal ideal `(p)` (or `(0)`) and perfect residue field. Cohen's structure theorem (EGA 0_IV §19.8.6): every complete noetherian local ring `(A, 𝔪, k)` is a quotient of `W[[T₁, …, T_n]]` for some Cohen ring `W` with the same residue field. For perfect `k` of char `p > 0`, `W = W(k)` is the ring of Witt vectors.                                                                                                                                                                                                                                                                                                      | Cited from file 22 as the foundation for the algebraization story over a complete local ring with residue field of mixed characteristic. The Cohen ring is the universal mixed-characteristic lift of the residue field.                                                                                                                                                                                           | 22                         | EGA 0_IV §19.8.6; Serre, *Local Fields* II §6 (Witt vectors)                                                                                                 |
| `coefficient-field`                               | Coefficient field theorem                                            | For an equicharacteristic complete local ring `(A, 𝔪, k)` (i.e. `char A = char k`), there exists a subfield `k' ⊂ A` with `A → k` inducing an isomorphism `k' ≃ k`. The "coefficient field." This realizes `A` as a quotient of `k[[T₁, …, T_n]]` (the equicharacteristic case of Cohen's structure theorem).                                                                                                                                                                                                                                                                                                                                                                            | Cited from file 22 as the foundation for the trivial lift `X = X₀ ⊗_k A` of any `k`-scheme in equicharacteristic; first obstruction to lifting in `H²(𝔤) ⊗ 𝔪/𝔪²` automatically vanishes when a coefficient field exists.                                                                                                                                                                                           | 22                         | EGA 0_IV §19.6                                                                                                                                               |
| `grothendieck-existence`                          | Grothendieck existence theorem                                       | EGA III §5.4.1: for `X → Spec R` proper with `R` a complete noetherian ring with ideal of definition `𝓘`, the functor `𝓕 ↦ 𝓕̂` from coherent sheaves on `X` to coherent sheaves on the formal completion `X̂` is an **equivalence of categories**. Algebraic and formal coherent sheaves agree.                                                                                                                                                                                                                                                                                                                                                                                            | Cited from files 18, 21, 22 as the algebraization theorem for coherent sheaves. Combined with `algebraization-via-ample`, upgrades formal smooth lifts to algebraic smooth lifts (Theorem III.7.3).                                                                                                                                                                                                                | 18, 21, 22                 | EGA III §5.4.1                                                                                                                                               |
| `algebraization-via-ample`                        | Algebraization criterion via ample sheaf                             | EGA III §4.7.1: a formal scheme `𝔛` proper over `Ŝ` (with `S` complete noetherian) equipped with a coherent formal sheaf `ℒ̂` whose restriction `ℒ₀ = ℒ̂\|_{X₀}` is an ample line bundle on the closed fiber is **algebraizable**: there exists a projective scheme `X / S` with `X̂ ≃ 𝔛` and an ample line bundle on `X` inducing `ℒ̂`.                                                                                                                                                                                                                                                                                                                                                     | Cited from file 22 as the algebraization criterion for formal schemes themselves. Combined with `grothendieck-existence` (algebraization of sheaves), upgrades III.6.10's formal smooth lift to III.7.3's algebraic smooth lift.                                                                                                                                                                                   | 22                         | EGA III §4.7.1                                                                                                                                               |
| `giraud-torsor`                                   | H¹ classifies torsors                                                | For a sheaf of (commutative) groups `𝒢` on a topological space (or scheme) `T`, the set of isomorphism classes of **torsors** (= principal homogeneous sheaves) under `𝒢` is in canonical bijection with `H¹(T, 𝒢)`. The trivial class corresponds to torsors admitting a global section.                                                                                                                                                                                                                                                                                                                                                                                                | Cited from file 17 as the classification theorem for torsors; underwrites the H¹ obstruction in §III.5 (file 21). The non-abelian generalization (Giraud, _Cohomologie non abélienne_) is what Grothendieck's grumpy footnote III.5.0 alludes to.                                                                                                                                                                  | 17, 21                     | Giraud, _Cohomologie non abélienne_ (Springer 1971); Milne, _Étale Cohomology_ III §4                                                                        |
| `tor-derived-functor`                             | Tor as left-derived tensor product                                   | For a commutative ring `A` and `A`-modules `M, N`, the Tor functor `Tor^A_i(M, N)` is the `i`-th left-derived functor of `M ⊗_A −`, computable from any free resolution of either argument. Independence of the resolution, symmetry `Tor^A_i(M, N) ≅ Tor^A_i(N, M)`, and the long exact sequence in either argument hold.                                                                                                                                                                                                                                                                                                                                                               | Cited from file 23 as the framework for the entire flatness story of files 24–27. The vanishing characterization of flatness (`M` flat ⟺ `Tor^A₁(M, N) = 0` for all `N`) is the bridge between the algebraic definition and the Tor calculus.                                                                                                                                                                      | 23, 24, 25, 26             | Cartan-Eilenberg, _Homological Algebra_ Ch. VI; Weibel, _Introduction to Homological Algebra_ §3.2–§3.3                                                      |
| `spectral-sequence-composite-functors`            | Grothendieck spectral sequence for composite functors                | For composable additive right-exact functors `F : 𝓐 → 𝓑` and `G : 𝓑 → 𝓒` between abelian categories with enough projectives, with `F` carrying projectives to `G`-acyclics: `E²_{p, q} = (L_p G)((L_q F)(M)) ⇒ L_{p+q}(G ∘ F)(M)`. The five-term exact sequence in low degrees is the form most often used.                                                                                                                                                                                                                                                                                                                                                                              | Cited from file 23 (where the construction is collected) and used in file 26's proof of IV.5.2 (Serre's lemma extending flatness through `A → B`). Cited exactly once in the Exposé IV rewrite.                                                                                                                                                                                                                    | 23, 26                     | Cartan-Eilenberg, _Homological Algebra_ XV.5; Weibel §5.8                                                                                                    |
| `noether-normalization`                           | Noether's normalization lemma                                        | Every finitely generated `K`-algebra `B` over a field `K` is finite over a polynomial subring `K[t₁, …, t_n]` for some algebraically independent `t₁, …, t_n ∈ B`. The integer `n` equals the Krull dimension of `B`. Geometrically: every affine variety of dimension `n` over a field admits a finite map to `𝔸^n`.                                                                                                                                                                                                                                                                                                                                                                    | Cited from file 23 (where the geometric reading is unpacked) and used in file 27's proof of IV.6.7 (the generic-freeness lemma). The inductive base of the generic-flatness story.                                                                                                                                                                                                                                 | 23, 27                     | Atiyah-Macdonald §5.1; Bourbaki AC V §3.1                                                                                                                    |
| `chevalley-constructibility`                      | Chevalley's constructibility theorem                                 | For `f : X → Y` of finite type with `Y` noetherian, the image `f(X)` is a constructible subset of `Y`. The fundamental constructibility theorem of algebraic geometry; the proof routes through Noether normalization and the dichotomy of IV.6.1.                                                                                                                                                                                                                                                                                                                                                                                                                                       | Cited from file 27 (used in IV.6.6's proof to license the application of IV.6.3 to `f(X)`). Without Chevalley, the open-image theorem reduces to a much weaker statement.                                                                                                                                                                                                                                          | 27                         | EGA IV §1.8.4; Stacks tag 054K                                                                                                                               |
| `bourbaki-flat-modules`                           | Bourbaki's exposition of flat modules                                | Bourbaki, _Algèbre Commutative_, Chap. I (Modules plats): the canonical reference for the sorites of flat modules, including the case of a non-commutative base ring (where some statements need adjustment). Develops flatness, faithful flatness, and base change in a categorically clean way.                                                                                                                                                                                                                                                                                                                                                                                        | Cited at the opening of file 24 as the canonical alternative source for §§1–2 of Exposé IV. SGA 1 IV explicitly cites it in its opening footnote IV.0.3.                                                                                                                                                                                                                                                           | 24                         | Bourbaki, _Algèbre Commutative_, Chap. I, Hermann 1961                                                                                                       |
| `ega-fibers-systematic`                           | EGA IV §§11–12: systematic study of fibers                           | EGA IV §§11–12 carries out the systematic study of fiber dimension, cohomological dimension, and depth for `X → Y` locally of finite type, equipped with a coherent sheaf `F` flat over `Y`. Includes Bertini- and Seidenberg-type theorems for hyperplane sections and the comparison of properties of `Y`, `F`, and the fibers.                                                                                                                                                                                                                                                                                                                                                        | Cited at the opening of file 24 as the long-form sequel to SGA 1 IV. The fiber-dimension content is referenced in passing in Grothendieck's grumpy footnote to IV.6.6 (file 27); the answer to the equidimensionality question lives there.                                                                                                                                                                        | 24, 27                     | EGA IV §11, §12                                                                                                                                              |
| `bourbaki-sem-195`                                | Strict pro-representability of left-exact functors                   | Let `𝒞` be a category in which every object is **artinian** (every descending chain of subobjects stabilises), and let `F : 𝒞 → Set` be a left-exact functor with finite values. Then `F` is **strictly pro-representable**: there exists a normalised pro-object `P = (P_i)_{i ∈ I}` (transition maps epimorphisms; every epi `P_i → P′` equivalent to one `φ_{ji}`) such that `F(X) ≅ colim_i Hom(P_i, X)`. The `P` is determined up to unique isomorphism.                                                                                                                                                                                                                            | Cited from file 29 (where the construction is collected) and used in file 31 step (c) of the proof of V.4.1 to build the pro-object `P` representing the fibre functor `F`. The output drives the construction of `π = Aut(F)`.                                                                                                                                                                                    | 29, 31                     | Grothendieck, _Technique de descente et théorèmes d'existence en géométrie algébrique_ II, Séminaire Bourbaki **195**, Feb 1960                              |
| `bourbaki-sem-190`                                | Faithfully flat descent (Grothendieck)                               | Faithful flat descent: a finite faithfully flat morphism is a strict epimorphism in the category of preschemes; properties such as "finite", "étale", and the existence of a quotient `X / G` for an admissible action are invariant under faithfully flat quasi-compact descent. **Now unpacked in files 40–43 (gentle Exposé VIII): VIII.1.1 (descent of quasi-coherent modules), VIII.2.1 (affine descent), VIII.5.2 (descent of morphisms in `Sch`), VIII.5.7 (descent of finite locally free), and VIII.7.5–8 (effectivity criteria) collectively cover the cited content.** The earlier files 30, 31, 33 still cite this entry by name; the substantive proofs are in files 40–43. | Originally cited from file 30 (V.2.6's equivalence (i) ⟺ (ii) ⟺ (iii) for principal coverings), file 31 (V.3.6's strict-epi statement for surjective finite étale morphisms), and file 33 (V.7's "principal covering ⟺ principal homogeneous in `𝒞(S)`"). Now also the conceptual umbrella for files 40–43; specific results are cross-referenced from there.                                                      | 30, 31, 33, 40, 41, 42, 43 | Grothendieck, _Technique de descente et théorèmes d'existence en géométrie algébrique_, Séminaire Bourbaki **190**, May 1959                                 |
| `ega-ii-affine-morphisms`                         | Affine morphisms as `Spec` of quasi-coherent algebras                | An `S`-prescheme `X → S` is **affine** (i.e. `f⁻¹(U)` is affine for every affine open `U ⊂ S`) iff `X = Spec_S(𝒜)` for a unique quasi-coherent `𝒪_S`-algebra `𝒜`. The functor `𝒜 ↦ Spec_S(𝒜)` is an anti-equivalence between quasi-coherent `𝒪_S`-algebras and affine `S`-preschemes. Tensor operations on the algebra side correspond to fibre products and base change on the prescheme side.                                                                                                                                                                                                                                                                                          | Cited from files 41 (§VIII.2 affine descent reduces to descent of quasi-coherent algebras) and 42 (§VIII.5.6: affine morphisms descend). Used implicitly throughout files 40–43 whenever a prescheme is replaced by its quasi-coherent algebra of functions.                                                                                                                                                       | 41, 42                     | EGA II §1.4                                                                                                                                                  |
| `ega-ii-proj-and-ample`                           | `Proj`-construction and ample invertible sheaves                     | For a quasi-coherent graded `𝒪_S`-algebra `𝒮 = ⊕_{n ≥ 0} 𝒮_n`, the **`Proj`-construction** `Proj_S(𝒮) → S` is a separated morphism whose fibers are the usual `Proj` of fibers. An invertible sheaf `𝓛` on `X` is **ample relative to `f : X → S`** iff for every `s ∈ S` there is an affine open neighbourhood `U ∋ s` such that `f⁻¹(U)` embeds as a closed subscheme into a projective `U`-space via a power `𝓛^{⊗n}\|_{f⁻¹(U)}`. Equivalently, `X` is `S`-isomorphic to an open of `Proj_S(⊕ f_*(𝓛^{⊗n}))`. Ampleness is preserved by base change.                                                                                                                                   | Cited from files 42 (§VIII.5.8 ampleness descends; §VIII.5.9 quasi-affineness via `𝓛 = 𝒪_X`) and 43 (§VIII.7.8 ample-invertible-sheaf effectivity criterion). The norm-of-an-ample-sheaf construction in §VIII.7.7 also leans on EGA II 6.6.4.                                                                                                                                                                     | 42, 43                     | EGA II §3, §4 (Proj construction and ample sheaves); §5.1 (quasi-affine via `𝒪_X` ample)                                                                     |
| `ega-ii-norm-finite-locally-free`                 | Norm of a section under a finite locally free morphism               | For a finite locally free morphism `f : X → S` of constant rank `r`, multiplication by a local section `s ∈ Γ(X, 𝒪_X)` defines an `𝒪_S`-linear endomorphism of `f_*(𝒪_X)` (locally free of rank `r`); its **determinant** is a section `N_f(s) ∈ Γ(S, 𝒪_S)`, the **norm** of `s`. Properties: (a) compatible with base change `Y → S`, hence `N_{f_Y}(s_Y) = N_f(s)\|_Y`; (b) `f^*(N_f(s))` divides `s^r` in `𝒪_X`; (c) the vanishing locus of `N_f(s)` is the image `f(Z(s))` of the vanishing locus of `s`. Consequently `S_{N_f(s)} = S − f(Z(s))`.                                                                                                                                   | Cited from file 43 (§VIII.7.6 finite locally free effectivity criterion; the norm of a section produces a saturated open of the upstairs scheme).                                                                                                                                                                                                                                                                  | 43                         | EGA II §6.5 (norm map for finite locally free morphisms); §6.6.4 (norm of an ample invertible sheaf)                                                         |
| `ega-iv-constructibility-via-finite-presentation` | Reductions of finite-presentation morphisms to noetherian situations | Every finite-presentation morphism `f : X → Y` of preschemes is obtained from a finite-type morphism `f₀ : X₀ → Y₀` between *noetherian* preschemes by base change along `Y → Y₀`. Concretely, when `Y = Spec A` is affine, one can find a finite-type subring `A₀ ⊂ A` and an `A₀`-scheme `X₀` of finite type with `X = X₀ ×_{A₀} A`. This is the standard "reduce to noetherian" device used to drop noetherian hypotheses in EGA-style proofs.                                                                                                                                                                                                                                        | Cited from file 42 (§VIII.6.6 remark on dropping noetherian hypotheses from VIII.6.1–6.2 for finite-presentation morphisms) and file 43 (§VIII.7.9 generalisation of the quasi-affine effectivity criterion to non-noetherian bases via finite presentation).                                                                                                                                                      | 42, 43                     | EGA IV §8 (passage à la limite for noetherian reductions); EGA IV §17–18 (étale and smooth versions)                                                         |
| `minkowski-discriminant`                          | Minkowski's discriminant inequality                                  | The discriminant of any non-trivial number field `K ≠ ℚ` has absolute value `> 1`. Equivalently: every non-trivial finite extension `K / ℚ` is ramified at some prime. Geometrically: `Spec ℤ` admits no non-trivial finite étale covering, hence `π₁^ét(Spec ℤ) = (1)`.                                                                                                                                                                                                                                                                                                                                                                                                                 | Cited from file 33 in the worked example `S = Spec ℤ` of the étale fundamental group. The "no unramified extensions" content is one of the most striking arithmetic facts attached to the étale `π₁`.                                                                                                                                                                                                              | 33                         | Minkowski, _Geometrie der Zahlen_ (1896); Neukirch, _Algebraic Number Theory_ III.2.18                                                                       |
| `lenstra-galois-theory-schemes`                   | Lenstra: Galois Theory for Schemes (lecture notes)                   | H. W. Lenstra, _Galois Theory for Schemes_, online lecture notes (Universiteit Leiden / Berkeley). Modern long-form alternative to SGA 1 V: Galois categories axiomatised in Ch. 3 (with a streamlined axiom system that drops the "strict" qualifier on epimorphisms), finite étale algebras in Ch. 4, finite étale morphisms in Ch. 5, complements (Spec of a field, infinite Galois extensions) in Ch. 6.                                                                                                                                                                                                                                                                             | Cited as the modern reader's-companion reference throughout files 28–33. Particular cross-references: Ch. 3 for the alternative axiomatic of file 31; Ch. 5 for the étale-cover side of files 30–31; Ch. 6 for the file 33 Spec(k) example.                                                                                                                                                                        | 28, 29, 30, 31, 32, 33     | Lenstra, _Galois Theory for Schemes_, lecture notes (Leiden / Berkeley)                                                                                      |
| `murre-cadoret-galois-categories`                 | Cadoret (after Murre): Galois Categories                             | A. Cadoret (after J. P. Murre), _Galois Categories_. Modern reference for Exposé V §§4–6: Ch. 2 develops the axiomatics; Ch. 3 carries out the proof of the main theorem (V.4.1) cleanly; Ch. 4 the functoriality (V.6); Ch. 5 the étale-cover side (V.3); Ch. 6 examples (Spec of a field, the first homotopy sequence, comparison theorems including the `π₁^{ét} = π̂₁^{top}` for proper varieties over `ℂ`).                                                                                                                                                                                                                                                                          | Cited as the cleanest modern walk-through of V.4.1 (file 31 leans on Ch. 3) and as the source of the worked examples in file 33. The first-homotopy-sequence material in Ch. 6 is the natural follow-up to file 33's Spec(𝔽_p) and elliptic-curve examples.                                                                                                                                                        | 28, 30, 31, 32, 33         | Cadoret, _Galois Categories_; Murre, _Lectures on an Introduction to Grothendieck's Theory of the Fundamental Group_ (Tata IFR, 1967)                        |
| `topological-covering-space-classification`       | Topological covering-space classification (Galois)                   | For a connected, locally arcwise connected, locally simply connected topological space `B` with base point `b₀ ∈ B`, the universal cover `B̃ → B` exists and the deck transformation group satisfies `Aut(B̃ / B) ≅ π₁^top(B, b₀)`. There is an equivalence between the category of connected coverings of `B` (up to iso) and the category of subgroups of `π₁^top(B, b₀)` (up to conjugacy). Restricting to _finite_ coverings gives an equivalence with open subgroups of `π̂₁^top(B, b₀)`.                                                                                                                                                                                              | Cited from file 28 as the topological prototype of the algebraic Galois theory. The finite-cover statement is the geometric template that file 31's Theorem V.4.1 generalises to schemes.                                                                                                                                                                                                                          | 28                         | Hatcher, _Algebraic Topology_, Ch. 1; May, _A Concise Course in Algebraic Topology_, Ch. 3                                                                   |
| `vistoli-fga-explained`                           | Vistoli: Notes on Grothendieck topologies (FGA Explained)            | A. Vistoli, _Notes on Grothendieck topologies, fibered categories and descent theory_, in _FGA Explained_ (Fantechi–Göttsche–Illusie–Kleiman–Nitsure–Vistoli), AMS 2005; arXiv math/0412512. Modern long-form treatment of SGA 1 VI + VIII + IX. Ch. 2 sites and sheaves; Ch. 3 fibered categories with full proofs of cleavage / pseudofunctor equivalences; Ch. 4 descent for quasi-coherent sheaves and stacks.                                                                                                                                                                                                                                                                       | Cited as the modern reader's-companion reference throughout files 34–38. Particular cross-references: Vistoli §2.3 for Grothendieck topologies (file 34 Module A); §3.1.3–§3.1.4 for the cleavage-pseudofunctor 2-equivalence (file 37 §VI.8); §3.2 for the QCoh fibred category (file 34 Module C example, file 38 closing pointer); Ch. 4 for the fpqc-descent statements deferred to a future Exposé VIII pass. | 34, 35, 36, 37, 38         | Vistoli, _Notes on Grothendieck topologies, fibered categories and descent theory_, arXiv math/0412512v4 (May 2007); FGA Explained, AMS 2005                 |
| `mac-lane-coherence`                              | Mac Lane's coherence theorem                                         | For a pseudofunctor (or, more generally, any monoidal-category-like or 2-categorical structure with associator and unitor data), the pentagon identity plus the unit conditions force *every* diagram of coherence isomorphisms to commute. In particular, every two parallel pasting compositions of coherence 2-cells in a pseudofunctor agree.                                                                                                                                                                                                                                                                                                                                        | Cited from file 34 Module C and file 37 §VI.8 as the input that licenses "we may treat the coherence data informally — every diagram of `c`'s commutes". Removes the need to verify higher-order coherences explicitly.                                                                                                                                                                                            | 34, 37                     | Mac Lane, _Natural associativity and commutativity_, Rice University Studies **49** (1963); Mac Lane, _Categories for the Working Mathematician_, Ch. VII §2 |
| `strictification`                                 | Strictification of pseudofunctors                                    | Every pseudofunctor `ϕ : ℰ° → Cat` is equivalent (in the 2-category of pseudofunctors) to a strict 2-functor `ϕ̃ : ℰ° → Cat`, after replacing each `ϕ(S)` with an equivalent category. Equivalently: every fibred `ℰ`-category is `ℰ`-equivalent to a split category. The strictification is constructive but enlarges the source data, so working with the un-strictified pseudofunctor is often preferable in practice.                                                                                                                                                                                                                                                                 | Cited from file 37 §VI.9 as the categorical-coherence answer to "do splittings of fibred categories always exist?" — they do up to equivalence, but not on the nose. The result underwrites the slogan "every fibred category is equivalent to a split one".                                                                                                                                                       | 37                         | Power, _A general coherence result_, J. Pure Appl. Algebra **57** (1989); Riehl, _Categorical Homotopy Theory_, §10                                          |

## How to cite

In body prose, name the black box at first mention in each section ("By Zariski's Main Theorem, …", "Serre's criterion
transfers across the étale extension"), then refer to it by name thereafter. Body files use file-level links of the form
`[name](appendix-black-boxes.md)` to point readers here; readers then locate the row by searching for the slug or the
black-box name. The glossary entries for Zariski's Main Theorem, Cohen-Macaulay, Serre's criterion, Nagata ring,
Hensel's lemma, and universal homeomorphism point at the corresponding rows here.

## On EGA citations

Several rows cite parts of EGA that are still being translated. The in-progress English translation lives at
`~/Code/ega/`. As of this writing the chapters most directly relevant to Exposé I — EGA IV §16 (differentials), EGA IV
§17 (smooth, unramified, étale), and EGA 0 §10 (flat-module criteria) — are translated; EGA IV §18 (Zariski's Main
Theorem) is mostly outstanding. For black boxes whose source is outside the translated set we give the EGA reference in
the form Grothendieck himself would; the gentle reading treats them as black boxes regardless.
