Index of terminology

The reference numbers indicate the chapter, section and sub-section (or exercise) in that order.

Adèle, restricted, principal restricted adtle : VII.2.4 m-adic filtration: III.2.1 — topology: III.2.5 n-adic integers: III.2.12 Algebra, Azumaya: II.1.7 — finitely generated : III.1.1 — integral, finite, over a ring: V. 1.1 Algebraic closure of a field in an algebra : V. 1.2 Algebraically closed field in an algebra: V. 1.2 — dependent, independent, elements: III.1.1 — free, related, family: III.1.1 Almost all $p \in P(A)$ (property valid for) : VII.4.3 — nilpotent endomorphism: IV. 1.4 Approximation theorem for absolute values : VI.7.3 — theorem for valuations: VI.7.2 Artin-Rees Lemma: III.3.1 Associated (filtered module) with a graded module: III.2.1 — (filtered ring) with a graded ring: III.2.1 — (filtration) with a graduation: III.2.1 — (graded homomorphism) with a homomorphism compatible with filtrations: III.2.4 — (graded module) with a filtered module: III.2.3 — (graded ring) with a filtered ring: III.2.3 — (mapping) with a ring homomorphism: II.4.3 — (prime ideal) with a module: IV. 1.1 — ring, place, valuation: VI.3.3

Canonical decomposition of a place: VI.2.3 — factorization of a valuation: VI.3.2 — homomorphism of the decomposition group of a prime ideal $\mathfrak{p}'$ of $\mathbf{A}'$ into the automorphism group of $\mathbf{A}'/\mathfrak{p}'$: V.2.2 Class, divisor, attached to a finitely generated module : VII.4.7 Classes, divisor (monoid of) : VII.1.2 Closure, algebraic, of a field in an algebra : V.1.2 — integral, of an integral domain : V.1.2 — integral, of a ring in an algebra: V.1.2 Commutative diagram: I.1.2 Compatible (filtration) with a ring structure, module structure: III.2.1 — (homomorphism) with filtrations: III.2.4 Complete system of extensions of a valuation: VI.8.2 Completely integrally closed domain : V.1.4 Component, irreducible (of a topological space) : II.4.1 Conditions, Hensel's: III.4.5 Conductor of a submodule : V.1.5 Content of a polynomial over a pseudo-Bezoutian domain: VII.1.Ex.23 — of a torsion module: VII.4.5 Criterion, Eisenstein’s irreducibility: VII.3.Ex.20

Decomposition, canonical, of a place : VI.2.3 — complete, of a prime ideal: V.2.2 — field of a prime ideal: V.2.2 — group, ring, of a prime ideal: V.2.2 — of an ideal in a Dedekind domain into prime factors: VII.2.3 — primary: IV.2.2 and Ex.20 — reduced primary: IV.2.3 and Ex.20 Decreasing filtration: III.2.1 Dedekind domain: VII.2.1 Defined (topology) by a filtration: III.2.5 Defining ideal: III.3.2 Degree, residue class, of one valuation over another: VI.8.1 Dependence, integral (equation of) : V.1.1 Derived (module filtration) from a ring filtration: III.2.1 Diagram, commutative: I.1.2 — snake : I.1.4 Discrete filtration: III.2.1 — valuation: VI.3.6 Distinguished polynomial : VII.3.8 Divisor, principal divisor: VII.1.1 — determinantal: VII.4.Ex.11 — finitely generated: VII.1.Ex.11

Divisorial fractional ideal : VII. 1.1 Divisors, equivalent: VII. 1.2 Domain, Bezoutian (or Bezout) : VII.1.Ex.20 — completely integrally closed : V. 1.4 — Dedekind: VII.2.1 — factorial: VII.3.1 — integrally closed : V. 1.2 — integrally closed, of finite character: VII.1.Ex.25, 26 and 28 — integrally Noetherian: V.3.Ex.6 — Krull: VII.1.3 — local integral, of dimension 1 : VI.4.Ex.7 — Pruferian (or Prufer): VII.2.Ex.12 — pseudo-Bezoutian: VII. 1.Ex.21 — pseudo-principal: VII. 1.Ex.21 — pseudo-Pruferian: VII.2.Ex.19 — regularly integrally closed: VII. 1.Ex.30 Dominating (local ring) a local ring: VI. 1.1 Dual, algebraic toric, of a module: VI.5.Ex.9 — lattice: VII.4.2 — topological toric: VI.5.Ex.10

Element, topologically nilpotent: III.4.3 Elements, algebraically dependent, independent: III.1.] — strongly relatively prime: III.4.1 Endomorphism, almost nilpotent: IV. 1.4 Equivalent divisors: VII. 1.2 — valuations: VI.3.2 Essential graded ideal: III.1.4 — valuations: VII.1.4 Euclidean ordered field: VI.2.Ex.4 Exhaustive filtration: III.2.1 Extension, quasi-Galois: V.2.2

Factor, invariant: VII.4.Ex. 11 and 14 Factorial domain: VII.3.1 Factorization, canonical, of a valuation : VI.3.2 Faithfully flat module: I.3.1 Family, algebraically free, related : III.1.1 — formally free: III.2.9 Field, algebraically closed in an algebra : V. 1.2 — decomposition: V.2.3 — projective: VI.2.1 — residue, of a local ring: II.3.1

Field, residue, of a place: VI.2.3 — residue, of a valuation: VI.3.2 — value, of a place: VI.2.2 Filtered group, ring, module: III.2.1 Filtration, m-adic: III.2.1 — associated with a graduation: III.2.1 — compatible with a ring structure, module structure: III.2.1 — discrete: III.2.5 — m-good: III.3.1 — increasing, decreasing, separated, exhaustive: III.2.1 — induced, product, quotient: III.2.1 — module, derived from a ring filtration: III.2.1 — trivial: III.2 Finite algebra over a ring : V. 1.1 — (place) at an element: VI.2.2 Finitely generated algebra: III.1.1 Finitely presented: I.2.8 Flat for M, M-flat (module): I.2.2 — module: I.2.3 Formally free family: III.2.9 Fractional ideal: VII.1.1 Function, order: III.2.2

Gaussian integer: V. 1.1 Gauss’s lemma: VII.3.5 Gelfand-Mazur Theorem: VI.6.4. Generated by a subset (multiplicative subset): II.2.1 Generators, formal system of: III.2.9 m-good filtration: III.3.1 Group, decomposition: V.2.2 — filtered: III.2.1 — inertia: V.2.2 — of classes of invertible modules: II.5.7 — of operators, locally finite : V. 1.9 — operating on a ring: V. 1.9 — order, of a valuation: VI.3.2 — ordered, of height $n$, of height $+\infty$: VI.4.4

Height $\leq 1$ (prime ideal of) : VII.1.6 — of an ordered group, of a valuation : VI.4.4 Henselian ring : III.4.Ex.3 Hensel’s conditions: III.4.5 — Theorem: III.4.3

Homomorphism, canonical, of the decomposition group of a prime ideal $p'$ — from $A'$ to the automorphism group of $A'/p'$: V.2.2 — compatible with filtrations: III.2.4 — graded, associated with a homomorphism compatible with filtrations: III.2.4 — local: II.3.1 — pseudo-injective, pseudo-surjective, pseudo-zero, pseudo-bijective: VII.4.4

Ideal, determinantal: VII.4.Ex.10 and 14 — essential graded : III.1.4 — immersed prime: IV.2.3 — integral, fractional ideal : VII.1.1 — invertible fractional: II.5.7 — lying above an ideal : V.2.1 — minimal prime: II.2.6 — of a place: VI.2.3 — of a valuation: VI.3.2 — prime: II.1.1 — prime, associated with a module : IV.1.1 — prime, decomposing completely: V.2.2 — prime, of height $\leq 1$: VII.1.6 — primary, p-primary: IV.2.1 and Ex.20 — unramified: V.2.Ex.18 and 19 Ideally Hausdorff module: III.5.1 Ideals, relatively prime: II.1.2 Identity, Cauchy's: VII.3.Ex.18 Immersed prime ideal: IV.2.3 Improper valuation: VI.3.1 Increasing filtration: III.2.1 Independent valuation rings: VI.7.2 — valuations: VI.7.2 Index, initial, of a subgroup of an ordered group, initial ramification index of a valuation: VI.8.4 — ramification: VI.8.1 Induced filtration: III.2.1 Induction, Noetherian (principle of): II.4.2 Inertia field: V.2.3 — ring, group: V.2.2 Initial ramification index: VI.8.4 Integer, algebraic: V.1.1 — Gaussian: V.1.1 — over a ring: V.1.1

Invertible fractional ideal: II.5.7 — submodule: II.5.6 Irreducible component: II.4.1 — set: II.4.1 — space: II.4.1 Isolated subgroup: VI.4.2

Jacobson ring: V.3.4

Module graded, associated with a filtered module: III.2.3 — ideally Hausdorff 111.5.1 — of fractions defined by a subset of a ring: II.2.2 — projective, of rank $n$: II.5.3 — pseudo-coherent: I.2.Ex.11 — pseudo-zero: VII.4.4 Monoid, divisor class: VII.1.2 Morphism for laws of composition not everywhere defined: VI.2.1 Multiplicative subset: II.2.1

Nilradical of a ring: II.2.6 Noetherian space: II.4.2 Non-degenerate submodule: II.5.5 Normalization lemma: V.3.1 Normed discrete valuation: VI.3.6 Nullstellensatz: V.3.3

Order group of a valuation: VI.3.2 — of an element for a valuation: VI.3.2 — reduced, of a formal power series: VII.3.8 Ordered pair of rings with the linear extension property: I.3.7 Ostrowski’s theorem: VI.6.4

Place, finite at $x$: VI.2.2 — of a field: VI.2.2 — trivial: VI.2.2 Point, generic, of an irreducible space: II.4.Ex.2 Polygon, Newton: VI.4.Ex.11 Polynomial, distinguished: VII.3.8 — minimal: V.1.3 Preparation theorem: VII.3.8 Presentation of a module, — finite: I.2.8 n-presentation: I.2.Ex.6 Presented, finitely (module): I.2.8 Primary decomposition: IV.2.2 and Ex.20 — p-primary, ideal, submodule: IV.2.1 and Ex.20 Prime ideal: II.1.1 — spectrum: II.4.3 Principal divisor: VII.1.1 — restricted adèle: VII.2.4 Principle of Noetherian induction: II.4.2 Product filtration: III.2.1 Projective field: VI.2.1

Pseudo-injective, pseudo-surjective, pseudo-zero, pseudo-bijective (homomorphism): VII.4.4 Pseudo-isomorphism: VII.4.4 Pseudo-zero module: VII.4.4

Quasi-Galois extension: V.2.2 Quotient filtration: III.2.1

Radical of an ideal: II.2.6 Rank at p of a projective module: II.5.3 — of a projective module: II.5.3 — rational, of a commutative group: VI.10.2 — residue: VI.8.5 Rational rank of a commutative group: VI.10.2 Reduced order: VII.3.8 — primary decomposition: IV.2.3 — ring: II.2.6 — series: VII.3.8 Reflexive lattice: VII.4.2 Related local rings: VI.1.Ex.1 Relatively prime ideals: II.1.2 Representative system of extremal elements: VII.3.3 Residue class degree of a valuation: VI.8.1 — field: II.3.1 — rank of a valuation: VI.8.5 Resolution, finite free, of a module: VII.4.7 Restricted adèle: VII.2.4 — formal power series: III.4.2 Ring, absolutely flat: I.2.Ex.17 — coherent (left, right): I.2.Ex.17 — decomposition: V.2.2 — filtered: III.2.1 — filtered, associated with a graded ring: III.2.1 — graded, associated with a filtered ring: III.2.3 — inertia: V.2.2 — integrally closed in an algebra: V.1.2 — Jacobson: V.3.4 — linearly topologized: III.4.2 — local: II.3.1 — local, dominating a local ring: VI.1.1 — local, of A at p, of p (p a prime ideal): II.3.1 — of a place: VI.2.3 — of a valuation: VI.3.2

Ring of fractions defined by a subset of a ring: II.2.1 — reduced: II.2.6 — semi-local: II.3.5 — total, of fractions: II.2.1 — unramified: V.2.Ex.19 — valuation, valuation ring of a field : VI. 1.1 — Zariski: III.3.3 Rings, independent valuation: VI.7.2

Saturation of a submodule with respect to a multiplicative subset (with respect to a prime ideal) : II.2.4 Semi-local ring: II.3.5 Series, reduced: VII.3.8 — restricted formal power: III.4.2 Set, irreducible: II.4.1 — major, in a totally ordered group: VI.3.5 Space, irreducible: II.4.1 — Noetherian: II.4.2 Special topology: II.4.3 Spectrum, prime, of a ring: II.4.3 Strongly Laskerian module: IV.2.Ex.28 — primary submodule: IV.2.Ex.27 — relatively prime elements: III.4.1 Subgroup, isolated, of an ordered group: VI.4.2 Submodule, invertible :II.5.6 — non-degenerate: II.5.5 — primary, p-primary: IV.2.1 Subset, multiplicative, of a ring: II.2.1 — multiplicative, generated by a subset: II.2.1 — saturated multiplicative: II.2.Ex.1 Support of a module: II.4.4 System, complete, of extensions of a valuation: VI.8.2 — formal, of generators: III.2.9 — representative, of extremal elements: VII.3.3

Theorem, approximation, for absolute values: VI.7.3 — approximation, for valuations: VI.7.2 — Gelfand-Mazur: VI.6.4 — Hensel’s: III.4.3 — Hilbert’s zeros (Nullstellensatz): V.3.3 — Krull’s: III.3.1 — Krull-Akizuki: VII.3.5 — Ostrowski’s: VI.6.3

Theorem, preparation: VII.3.8 — Stickelberger's : VI.8.Ex.18 — Zariski's Principal: V.3.Ex.7 Topologically nilpotent element: III.4.3 Topology defined by a filtration: III.2.5 — spectral: II.4.3 — Zariski: II.4.3 Transporter: I.2.10 Trivial filtration: III.2.1 — place: VI.2.2

Ultrametric absolute value: VI.6.1 Uniformizer for a discrete valuation: VI.3.6 Unramified valuation: VI.8.1

Valuation, valuation of an element $x$: VI.3.1 and 2 — discrete, normed discrete valuation: VI.3.6 — essential: VII.1.4 and Ex.26 — improper: VI.3.1 — ring: VI.1.1 — unramified: VI.8.1 Valuations, equivalent: VI.3.2 — independent: VI.7.2 Value field of a place: VI.2.2 — ultrametric absolute: VI.6.1

Weakly associated (prime ideal) with a module: IV.1.Ex.17

Zariski, ring: III.3.3 — topology: II.4.3