Quillen K-theory

Quillen K-theory
Name	Quillen K-theory
Field	Algebraic topology; Algebraic K-theory
Introduced	1972
Introduced by	Daniel Quillen
Notable for	Definition of higher K-groups via plus-construction and Q-construction

Quillen K-theory Quillen K-theory provides a foundational framework for higher algebraic K-theory by defining K-groups of rings, schemes, and exact categories using homotopy-theoretic constructions. It unifies earlier computations by Alexander Grothendieck, integrates methods from Daniel Quillen's work in homotopical algebra, and connects to classical results associated with John Milnor, Michael Atiyah, and Friedhelm Waldhausen. The theory underpins deep relations among invariants studied by Jean-Pierre Serre, Hyman Bass, Armand Borel, and Jean-Louis Verdier.

Introduction

Quillen introduced two complementary constructions—commonly called the plus-construction and the Q-construction—that produce higher K-groups as homotopy groups of certain spaces, situating algebraic K-theory in the context of homotopy theory, simplicial sets, and the study of classifying spaces of categories and groups. His approach built on precedents in work by Grothendieck, who formulated K0 for vector bundles and coherent sheaves, and by Milnor, who defined low-dimensional K-groups for fields; Quillen extended these notions to an infinite family of groups K_n with robust functoriality properties used by researchers such as Jean Lannes and Gunnar Carlsson.

Definitions and constructions

Quillen's plus-construction defines K-theory for a ring R via the group GL(R), its classifying space BGL(R), and the Quillen plus-construction BGL(R)+ that kills a perfect subgroup while preserving homology; the higher K-groups are K_n(R)=π_n(BGL(R)+). This construction relies on techniques developed in the homotopy-theoretic milieu of J. H. C. Whitehead, Serre spectral sequence methods refined by Jean-Pierre Serre, and categorical foundations from Alexandre Grothendieck's work on exact categories. The Q-construction associates to an exact category E a simplicial category Q(E) whose geometric realization has homotopy groups giving K_n(E); it connects naturally to constructions explored by Friedhelm Waldhausen for categories with cofibrations and weak equivalences and to later model-category frameworks due to Daniel Dugger and Philip S. Hirschhorn. Both constructions satisfy additivity and localization principles influenced by the ideas of Hyman Bass and John Swan.

Theorems and properties

Quillen K-theory satisfies fundamental theorems: homotopy invariance, devissage, localization, and the resolution theorem; these echo structural results obtained earlier by Grothendieck for K0 and by Bass for K1. The localization theorem produces long exact sequences relating K-theory of a closed immersion and its complement, echoing methods from Alexander Grothendieck's cohomological formalism and later refined by Robert Thomason and Thomas Geisser. Quillen’s computations for K-theory of finite fields gave results later interpreted through the work of Armand Borel on stable cohomology and Daniel Fried, and his spectral sequence techniques parallel those introduced by Jean Leray and Jean-Louis Verdier. Moreover, Quillen proved that for regular noetherian schemes the plus-construction and Q-construction coincide, a result used by Gérard Laumon and Luc Illusie in study of arithmetic schemes.

Examples and computations

Quillen computed K_n for finite fields, showing periodicity and explicit values that corroborated conjectures of John Milnor and inspired further work by Andrei Suslin and Vladimir Voevodsky. For R a principal ideal domain such as the ring of integers, computations by Quillen and successors like Hyman Bass and Dennis Sullivan provided K1 and K2 identifications linked to classical objects studied by Emil Artin and Kurt Gödel in algebraic contexts. Calculations for local fields and number rings used input from the conjectures of Markus Rost and results by Fritz Hirzebruch and Armand Borel connecting K-theory with special values of L-functions and regulators explored by Alexander Beilinson.

Applications and relations to other K-theories

Quillen K-theory interfaces with topological K-theory as developed by Michael Atiyah and Friedrich Hirzebruch and with motivic approaches advanced by Vladimir Voevodsky and Spencer Bloch. It feeds into the formulation and partial resolutions of conjectures such as the Beilinson conjectures and the Lichtenbaum conjecture studied by Stephen Lichtenbaum and Henri Gillet. The Q-construction and Waldhausen K-theory interrelate via additive and multiplicative structures investigated by Friedhelm Waldhausen and Bjorn Dundas; these connections are pivotal in applications to manifold theory where tools from William Browder and John Milnor intersect with algebraic K-theory computations. Quillen K-theory also underpins developments in higher category theory and homotopical algebra explored by Jacob Lurie and Charles Rezk.

Historical context and development

Quillen developed his definitions in the early 1970s, publishing groundbreaking papers that reshaped the field and influenced contemporaries including Jean-Pierre Serre, Armand Borel, and Hyman Bass. His methods integrated algebraic, topological, and categorical innovations that stimulated subsequent generations: researchers such as Daniel Grayson, Charles Weibel, Cynthia Weibel, Robert Thomason, and Andrei Suslin expanded computational and theoretical horizons. The interplay with arithmetic and geometric questions attracted contributions from Alexander Beilinson, Spencer Bloch, and Pierre Deligne, and later connections to motivic homotopy theory were driven by Vladimir Voevodsky and Fabien Morel. Quillen's influence extends into modern homotopy theory research by figures like Jacob Lurie, marking the theory as central in contemporary algebraic and arithmetic investigations.

Category:Algebraic K-theory