Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen. The precise statements...

College, Oxford. Quillen retired at the end of 2006. He died from complications of Alzheimer's disease on April 30, 2011, aged 70. Quillen's best known contribution...

K-theory for a general variety. The first definition of higher K-theory to be widely accepted was Daniel Quillen's. As part of Quillen's work on the Adams...

This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures...

_{i}(B^{+}{\text{f-gen-Mod}}_{R})} , where B + = Ω B Q {\displaystyle B^{+}=\Omega BQ} is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite...

as a relation between the sphere spectrum and the classifying spaces of the symmetric groups via Quillen's plus construction. The mapping space Map 0...

Kan complexes and CW complexes being given by the geometric realization and the singular functor (Milnor's theorem). The Kan–Quillen model structure...

the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination...

scheme X with action of a linear algebraic group G, via Quillen's Q-construction; thus, by definition, K i G ( X ) = π i ( B + Coh G ⁡ ( X ) ) . {\displaystyle...

spaces, and are the successive quotients of the terms of the Bott periodicity clock. These equivalences immediately yield the Bott periodicity theorems. The...

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially...

concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration...

rings over a field: this is the Quillen–Suslin theorem. Any free module is projective. The converse is true in the following cases: if R is a field or skew...

new proofs and generalizations of the Atiyah-Singer index theorem. Among these generalizations are index theorems based on spectral triples and deformation...

R} , and B {\displaystyle {\cal {B}}} is some localizing subcategory of Mod ⁡ ( R ) {\displaystyle \operatorname {Mod} (R)} . Daniel Quillen's algebraic...

In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact...

In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced...

with action of a linear algebraic group G {\displaystyle G} , via Quillen's Q-construction; thus, by definition, K i G ( X ) = π i ( B + Coh G ⁡ ( X )...

that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category C. The exact functors between Quillen's exact...

(Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids). Brown, R.; Razak, A. (1984), "A van Kampen theorem for unions...

group scheme G. It is a theorem of Quillen that G ≅ Z [ b 1 , b 2 , … ] {\displaystyle G\cong \mathbb {Z} [b_{1},b_{2},\dots ]} and assigns to every ring...

f:x\to y} in a strict 2-category C and g : a → b {\displaystyle g:a\to b} in D, the usual product is given as f × g : ( x , a ) → ( y , b ) {\displaystyle...

spaces of categories, and in particular by Quillen's work of algebraic K-theory. In this work, which earned him a Fields Medal, Quillen developed surprisingly...

seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture. A different proof was...

Quillen's Theorem B as well as Quillen's dévissage argument in the process. Much of his recent work has concerned equivariant algebraic K-theory and equivariant...

(1969), Theorem 2.3.5; Sjödin (1980), Theorem 1. Quillen (1970), section 7. Avramov, Luchezar; Halperin, Stephen (1986), "Through the looking glass: a dictionary...

as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic...

section IV.6. Varadarajan 1984, Theorems 2.7.5 and 3.15.1. Varadarajan 1984, section 2.6. Milnor 2010, Warnings 1.6 and 8.5. Knapp 2001, section III.3...

See local ring, perfect ring and Dedekind ring. Free object free presentation free resolution Quillen–Suslin theorem stably free module generic freeness...

subcomplexes A and B, then the inclusion f: (A,A∩B) → (X,B) induces an isomorphism h i ( A , A ∩ B ) → f ∗ h i ( X , B ) {\displaystyle h_{i}(A,A\cap B){\overset...