About: Q-construction

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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 category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space BC.) One puts .

Attributes	Values
rdfs:label	Q-construction (en)
rdfs:comment	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 category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space BC.) One puts . (en)
dcterms:subject	Algebraic K-theory
Wikipage page ID	41403415 (xsd:integer)
Wikipage revision ID	1083543911 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic K-theory Algebraic K-theory Nerve (category theory) Equivariant algebraic K-theory Friedhelm Waldhausen Spectrum (topology) Steinberg group (K-theory) Suspension of a ring Acyclic space Central extension (mathematics) Daniel Quillen Equivariant sheaf Waldhausen category Abelian category Birkhäuser Eilenberg–MacLane space Homotopy fiber Homotopy group with coefficients Module spectrum Grothendieck group Geometric realization Exact category Serre spectral sequence Topological space Ring spectrum Universal central extension S-construction Fibered category Classifying space of a category Quillen's Theorem B Colimit CW-complex Spectral sequence for the covering Homotopy pullback Local coefficient system Loop space functor dbr:Comparison_theorem_for_spectral_sequences dbr:Grayson's_binary_complex
Link from a Wikipage to an external page	http://www.math.uiuc.edu/~dan/Papers/HigherAlgKThyII.pdf http://www.math.rutgers.edu/~weibel/Kbook.html%7Ctitle=
sameAs	Q-construction Q-construction Q-construction
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Reflist dbt:Harvnb dbt:Math_theorem
has abstract	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 category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space BC.) One puts and call it the i-th K-group of C. Similarly, the i-th K-group of C with coefficients in a group G is defined as the homotopy group with coefficients: . The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context. For example, one can define equivariant algebraic K-theory as of of the category of equivariant sheaves on a scheme. Waldhausen's S-construction generalizes the Q-construction in a stable sense; in fact, the former, which uses a more general Waldhausen category, produces a spectrum instead of a space. also gives a construction of algebraic K-theory for exact categories. See also module spectrum#K-theory for a K-theory of a ring spectrum. (en)
prov:wasDerivedFrom	wikipedia-en:Q-construction?oldid=1083543911&ns=0
page length (characters) of wiki page	10377 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Q-construction
is Link from a Wikipage to another Wikipage of	Algebraic K-theory Quillen's theorems A and B Clark Barwick Morita equivalence Equivariant algebraic K-theory Fundamental theorem of algebraic K-theory K-theory K-theory of a category Waldhausen category Quillen's Q-construction
is Wikipage redirect of	Quillen's Q-construction
is foaf:primaryTopic of	wikipedia-en:Q-construction

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