Krull

Krull
Title	Krull

Krull

Definition and Overview

Krull denotes a cluster of concepts and results centered on the mathematician whose surname is associated with David Hilbert, Emmy Noether, Richard Dedekind, Emil Artin, and Oscar Zarischi in the development of modern algebraic geometry, commutative algebra, ring theory, module theory, and valuation theory. The term encompasses foundational notions such as Krull dimension, Krull ring, Krull–Schmidt theorem, Krull topology, and valuation-related constructions that tie into the work of Claude Chevalley, Alexander Grothendieck, Jean-Pierre Serre, Hermann Minkowski, and Irving Kaplansky. These notions interconnect with classical objects like Noetherian ring, Dedekind domain, principal ideal domain, unique factorization domain, and localization techniques used by Oscar Zariski and Pierre Samuel.

Historical Development and Etymology

The eponym traces to research contemporaneous with contributions by Emmy Noether, Igor Shafarevich, Richard Brauer, Helmut Hasse, and Krull's peers during the early 20th century in Germany, reflecting a dialogue with Leopold Kronecker-influenced currents and the algebraic program of David Hilbert. Early formulations appeared alongside the evolution of ideal theory by Richard Dedekind and generalizations by Emil Artin and Helmut Hasse, while later refinements connected to the categorical perspective of Samuel Eilenberg and Saunders Mac Lane and to the scheme-theoretic framework advanced by Alexander Grothendieck. The surname became attached to multiple theorems and definitions as subsequent authors such as Oscar Zariski, Pierre Samuel, Jean-Pierre Serre, and Hochster incorporated the concepts into modern curricula.

Krull in Algebra (Krull Dimension and Krull Rings)

Krull dimension formalizes a notion of "height" of chains of prime ideals, interacting with Noetherian ring properties, Jacobson ring criteria, Krull intersection theorem-style statements, and comparisons with transcendence degree in contexts like algebraic variety coordinate rings and function field extensions studied by Oscar Zariski and Pierre Samuel. Krull rings, or Krull domains, generalize Dedekind domain factorization via divisor theories and interplay with Weil divisor concepts used in algebraic geometry by André Weil and Jean-Pierre Serre; they link to divisor class groups, to the work of Emil Artin on class field theory, and to factorization analyses by Paul Erdős and Harold Davenport. In these settings, results often reference Noether normalization lemma, Hilbert's Nullstellensatz, and the behavior of prime spectra under integral closure studied by Oscar Zariski and Pierce.

Krull–Schmidt and Krull's Theorems in Module Theory

The Krull–Schmidt theorem asserts uniqueness of decomposition into indecomposable modules under hypotheses related to Artinian module and Noetherian module conditions, interacting with the structure theory of modules examined by Irving Kaplansky, Emmy Noether, Claude Chevalley, and Nathan Jacobson. Complementary Krull theorems address ascending and descending chain conditions for ideals and modules, connecting to Hopkins–Levitzki theorem, Jordan–Hölder theorem, and decomposition phenomena in categories analyzed by Samuel Eilenberg and Saunders Mac Lane. Applications reach classification problems for representations of finite group algebras studied by Richard Brauer and for linear algebraic groups investigated by Armand Borel and Claude Chevalley.

Krull Topology and Valuation Theory

Krull topology arises in completions and profinite constructions related to Galois group topologies appearing in Évariste Galois-inspired work and to the profinite frameworks used by Jean-Pierre Serre and John Tate; it ties to valuation-theoretic completions like those in p-adic number theory developed by Kurt Hensel and to adèle and idèle formulations central to Andrew Wiles-era arithmetic geometry and John Tate's duality. In valuation theory, Krull valuations generalize discrete valuations of Richard Dedekind and Hermann Minkowski, interfacing with the concept of places employed by Alexander Grothendieck and the structure of valued fields studied by Abraham Robinson and Jonas ​​von Neumann-era model-theoretic contributors. The topology governs continuity properties for modules and rings paralleling constructions in Pontryagin duality and in the study of profinite group completions like those used in Galois cohomology.

Applications and Examples

Examples include classical coordinate rings of affine algebraic varietys where Krull dimension equals geometric dimension, Dedekind and Krull domains appearing in algebraic number theory via ring of integers in number fields studied by Ernst Eduard Kummer and Richard Dedekind, and module-decomposition contexts in representation theory of finite group algebras and quiver representations advanced by Gabriel and Happel. Krull topology features in the construction of profinite Galois groups of field extensions in algebraic number theory applied by Kummer, Kronecker, and Emil Artin; valuation-theoretic examples include non-discrete valuations in function field arithmetic as in the work of Hasse and Weil. Current research threads link Krull-associated ideas to scheme-theoretic methods of Alexander Grothendieck, cohomological techniques of Jean-Pierre Serre, homological algebra of Henri Cartan, and factorization problems revisited by Melvin Hochster and Olaf Zariski.

Category:Mathematics