Prerequisite readings

Algebra and Galois reading routes

Module prerequisites for Wedderburn–Artin

Schur’s lemma and complete reducibility for arbitrary unital modules, with finite-length arguments where explicitly required.

Schur’s lemma · Complete reducibility · Wedderburn–Artin and uniqueness

Regular trace and reduced trace

The regular-trace criterion is stated in characteristic zero. The positive-characteristic matrix example explains why reduced trace is a different invariant; it is not a characteristic-zero restriction on Brauer groups.

Regular trace and its positive-characteristic limitation · Reduced trace and reduced norm

Continuous Hilbert 90

Further reading for the profinite continuation: every field, continuous cocycles, and discrete separable-closure coefficients. The finite-extension descent results do not require this continuation as an extra hypothesis.

Finite quotient cochains, Proposition 3.1 · Hilbert 90, Theorem 4.1 · Editable source

Commutative-algebra reading routes

Prime separation, radicals, localization and minimal primes hold without a Noetherian hypothesis. Finite irreducible decomposition is stated with its Noetherian-topology condition. Polynomial dimension and normalization here concern algebras over a field.

Prime separation and radicals · Minimal primes and finite irreducible decomposition · Dimension and height over fields · Local dimension and Hilbert–Samuel growth

Alternative reader addresses

These addresses contain the same readings, not additional lessons.

• AG-CA-03 — Noetherian and Artinian rings · Same reading under its other filename
• AG-CA-05 — Integral extensions: lying over, going up and going down · Same reading under its other filename
• AG-CA-09 — Krull dimension and Noether normalization · Same reading under its other filename
• AG-CA-11 — Dimension theory of Noetherian local rings · Same reading under its other filename
• AG-CA-12 — Regular sequences, depth and Cohen–Macaulay modules · Same reading under its other filename
• AG-CA-14 — Regular local rings · Same reading under its other filename
• AG-CA-19 — Completion · Same reading under its other filename

Hilbert–Samuel growth and local dimension

For finite modules over Noetherian local rings and ideals of definition, the associated graded construction gives eventual polynomial growth. Artin-Rees gives top-coefficient additivity, not additivity of whole polynomials. Local dimension is identified in its separate theorem before use for completion.

• Hilbert-Serre with the multiplication kernel
• Hilbert-Samuel existence and generator bound, Theorem 4.2
• Degree and top-coefficient exact-sequence formula, Theorem 6.1
• Artin-Rees, Theorem 5.1
• Nakayama, Theorem 4.2
• Local dimension, Theorem 2.1
• Completion reading

Alternative reader address for the same Hilbert–Samuel lesson

Finite prime avoidance and module dimension

Finite prime avoidance supplies parameters over every residue field. A finite module over a Noetherian ring admits a finite prime filtration; filtration primes need not all be associated to the whole module. These are the stated inputs of local ring and module dimension.

• Finite prime avoidance, Solution 8.5
• Prime filtration and associated-prime finiteness, Theorem 2.2
• Associated-prime existence and zero divisors, Theorem 1.2
• Parameter-count and module-dimension consumers
• Minimal primes and finite irreducible decomposition

Alternative reader address for the same associated-primes lesson

Affine modules and pullback

For any ring, modules and quasi-coherent sheaves on its spectrum are equivalent. Along an arbitrary morphism, pullback on compatible affine opens is extension of scalars. No Noetherian or finite-generation hypothesis is imposed; the separate pushforward statement has its own hypotheses.

• Affine equivalence, Theorem 3.2
• Pullback, Theorem 3.1
• Quasi-coherent cohomology consumer, Proposition 4.2

Galois algebra, primary components and Sylow subgroups

Finite separable extensions in arbitrary characteristic, closed subgroups of Galois groups, primary decomposition of arbitrary torsion modules, split finite field algebras and Sylow quotient images. The tensor calculation uses the stated finite Galois splitting field itself.

• Profinite groups and infinite Galois theory, Section 0
• Section 3
• Extensions of complete valued fields, Proposition 0.1
• Hensel's lemma, squares and roots of unity, Lemma 4.0
• Normal integral bases, Lemma 5.23
• Integrality of characters and Burnside's theorem, Lemma 4.2
• Profinite groups and infinite Galois theory, Section 0
• Faithful flatness, Theorem 1.1 and Theorem 2.1
• same Section 0

Finite extensions of the complex Laurent-series field

Finite separable normalization over a Noetherian normal domain, simple-root lifting in complete local rings, local factors of a finite algebra over a henselian ring, dimension under integral inclusion, the one-dimensional normal local DVR criterion, and completeness of finite modules. These apply to the displayed integral closure over C[[t]].

• Normalization, Theorem 4.1
• Completion, Theorem 5.1
• Henselian local rings and henselization, Theorem 2.2
• Integral extensions, Theorem 5.1
• Discrete valuation rings, Theorem 1.2
• Completion, Theorem 3.1

Geometry and coherent finiteness for torsion on curves

Finite normalization, the separated quasi-finite factorization and coherent cohomology under proper morphisms. Projective finiteness and vanishing supply the stated preceding coherent input; the curve cohomology arguments themselves remain in the course.

• Normalization, Theorem 5.1
• Zariski's Main Theorem, Theorem 4.2
• Coherence of higher direct images, Theorem 4.1
• Corollary 5.1
• Serre's projective finiteness and vanishing, Theorem 2.1 and Corollary 2.3

Proper geometry and henselian comparison

General Stein factorization, proper approximation over an arbitrary base, coherent formal existence and the full finite-type Chow construction support the geometric reductions in proper base change.

• General Stein factorization, Theorem 6.1
• Proper approximation, Appendix A.4.1 and A.4.2
• Coherent formal existence, Theorems 3.2 and 6.1
• Chow's lemma over a qcqs base, Theorem 5.1
• Finite Zariski Main, Theorem 4.2
• Finite-presentation descent, Sections 3–4
• Henselian-pair idempotents and étale lifting, Lemmas 5.3 and 5.4