Prerequisites and reading order

Contents · Programme reading order

Every result used in these lessons is proved in the lesson or in a completed programme lesson read beforehand. The links below identify those prerequisite proofs. Free external references provide attribution and comparison material; they never replace a programme proof.

• Read the commutative algebra foundations before their scheme applications: localization, Noetherian rings, associated primes, integral extensions, flatness and dimension. Exact theorem locators appear at each use.
• Before the sheaf and relative Spec arguments, read Affine schemes, Theorems 3.2 and 4.1 and Quasi-coherent sheaves on schemes, Sections 1–4. The latter proves gluing, localization and finite-type subsheaf extension.
• Before AG-MO-03 Theorem 4.3, read Flatness criteria, Lemmas 6.1–6.2 and Theorem 5.2. Their local criterion and generic-freeness dependencies are specified and proved in the named earlier algebra and flatness lessons. These arguments do not use AG-MO-03 Theorem 4.3 or absolute approximation. The smooth and étale finite-identity argument and constructible-image argument are written in AG-MO-03 itself.
• After AG-MO-06 Section 1, read Appendix Z of The theorem on formal functions, Lemmas Z.1–Z.11 and Lemma 0.1, before AG-MO-06 Section 2A. Appendix Z uses the pointwise quasi-finite criterion, not openness or Chevalley’s theorem. AG-MO-06 Section 2A then proves openness before its later use in the fibre-dimension lesson.
• Before projective-space constructions, read Projective space, relative Proj and maps to projective space, Sections 1–4, including coordinate maps, degree-one Proj, arbitrary base change and invertible quotients.
• Before qcqs sheaf extension and schematic closures, read Affine cohomology and Serre’s criterion, Theorem 4.3 and Lemma 4.4. AG-MO-10 Section 5 proves the separated finite-presentation ambient construction and both general-base Chow arguments, using the earlier full limit proofs in AG-MO-03.
• Before AG-MO-12 Theorem 4.1, read AG-RG-S04 Sections D2–D5, through Theorem D5.2, together with their stated preceding proof dependencies. These supply the finite-piece argument, étale integral-closure comparison and descent of the local isomorphism over arbitrary scheme bases. AG-MO-12 then proves its finite-subalgebra and finite-factorization steps.
• Before AG-MO-13 Theorem 5.2, read the complete commutative algebra proofs of completion, normality and depth named at each use, and Coefficient rings and the Cohen structure theorem, Sections 1–5. The required polynomial-normality proof is AG-RG-S04 Corollary B1.2; the separable field input is proved locally in AG-MO-13 Lemma 4.0.

The lesson number of another course does not establish dependency order. Follow the specific proof scopes above and the locators in each lesson before using a result.