Topic Commutative Algebra
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Commutative Algebra

This is the foundational layer: the language and toolkit of commutative rings, modules, ideals, and their geometric counterpart via Spec.

Scope

The Langlands roadmap draws on the following commutative-algebra inputs:

Rings and ideals

  • Commutative rings, ideals (prime, maximal, primary), quotient rings.
  • The prime spectrum \(\operatorname{Spec} R\) and the Zariski topology.
  • Localisation: \(R[S^{-1}]\), \(R_{\mathfrak{p}}\), \(R[f^{-1}]\).
  • Noetherian hypotheses: Hilbert basis theorem, ACC on ideals.

Modules

  • \(R\)-modules: free, projective, flat, finitely generated, finitely presented.
  • Tensor products \(M \otimes_R N\).
  • Module-finite vs. algebra-finite (finite vs. finite type).

Algebras

  • \(R\)-algebras as ring maps \(R \to A\).
  • Finitely generated \(R\)-algebras (= \(R[X_1, \dots, X_n]/I\)).
  • Polynomial rings, Laurent polynomial rings, multivariate polynomials.

Hilbert's Nullstellensatz

For an algebraically closed field \(k\) and a finitely generated \(k\)-algebra \(A\):

  • the maximal ideals of \(A\) are in bijection with the \(k\)-rational points of \(\operatorname{Spec} A\);
  • if \(A\) is a domain, \(\dim A\) (Krull) equals the transcendence degree of \(\operatorname{Frac}(A)\) over \(k\).

Dimension theory

  • Krull dimension, transcendence degree, height of primes.
  • For Noetherian rings: dimension formulas, dimension equals transcendence degree for finitely generated \(k\)-algebras.

Going up / going down

  • Cohen–Seidenberg theorems for integral extensions.
  • Properties of integral / faithfully flat / smooth ring maps.

Smoothness and étale morphisms

For affine group schemes, we need:

  • Smooth ring maps: locally of finite presentation, formally smooth.
  • Étale ring maps: smooth + unramified, equivalently locally-of-finite-presentation + flat + unramified.
  • Faithfully flat: flat + the map is surjective on Spec.

These translate into smooth/étale/faithfully-flat morphisms of schemes under Spec.

Tensor products and base change

For \(R \to R'\) a ring map and \(A\) an \(R\)-algebra, \(A \otimes_R R'\) is the base change of \(A\) along \(R'\). On Spec: \(\operatorname{Spec}(A \otimes_R R') = \operatorname{Spec} A \times_{\operatorname{Spec} R} \operatorname{Spec} R'\).

This is the algebraic shadow of the base change of schemes.

Mathlib

The relevant Mathlib namespaces:

  • Mathlib.Algebra.* — algebra typeclasses (CommRing, CommSemiring, Module, Algebra).
  • Mathlib.RingTheory.* — ring theory (Ideal, Spec, Localization, MvPolynomial).
  • Mathlib.LinearAlgebra.* — modules, tensor products.
  • Mathlib.RingTheory.FiniteType — finite-type algebras (FG of finite-type).

Everything in this DAG that touches commutative algebra reduces to Mathlib API calls; the project does NOT redevelop commutative algebra from scratch.

References

  • Atiyah–Macdonald, Introduction to Commutative Algebra (1969).
  • Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry (GTM 150, 1995).
  • Stacks Project (online) — comprehensive reference.

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