FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2070594927> ?p ?o ?g. }

• W2070594927 endingPage "219" @default.
• W2070594927 startingPage "216" @default.
• W2070594927 abstract "The groups Kt(A) of Bass for i < 0 are identified as homotopy groups of the spectrum of algebraic A -theory. The spectrum itself is identified. Applications to Laurent polynomials and to -theory exact sequences are given. Quillen has recently proposed a K-theory for unital rings [12], [13]. He associates to a ring A a space BG1(A) whose homology is that of the group G(A) and whose homotopy groups nt BG1(A) + he defines as Kt(A i ^ 1. The space BG1(A) is known to be an H-space, and indeed an infinite loop space. Hence one is motivated to define Kt(A), for i e Z, as 7Zi(E(A)) where E(A) is the associated Q-spectrum. This note describes E(A) and identifies the groups Kt(A i < 0. In fact, we show that the groups Kt(A) are exactly the groups L~K0(A) discussed in Bass' book [3, p. 664] for i < 0. Recall from the work of Karoubi and Villamayor [10] the cone CA and suspension SA of a ring A. An infinite matrix is called permutant if it is an infinite permutation matrix times a diagonal matrix of finite type. The diagonal matrix is of finite type if its diagonal entries are chosen from a finite subset of the ring. The ring CA is the ring generated by permutant matrices. The cone CA contains the two-sided ideal A = Jn Mn(A) and the quotient ring is called the suspension of A. We can now state our main result. THEOREM A. The space Q (BG1(SA)) has the homotopy type of K0{A) x BGl(A) . COROLLARY. For all ieZ we have Kt(A) = Ki+1(SA). Since Karoubi [9] has already identified X0(<SM) with Bass' groups K-i(A% the Corollary above completes the identification of Bass' groups with the negative homotopy of the spectrum E(A). In proving Theorem A we must first analyze the cone construction. THEOREM B. The space BG1(CA) is contractible. This result generalizes work of Karoubi and Villamayor [11] who show that Ki(CA) = 0 for i ^ 2. To prove Theorem B we observe that it suffices AMS 1970 subject classifications. Primary 18F25, 55B15, 16A54, 13D15, 55F50, 18G30, 55B20, 55D35." @default.
• W2070594927 created "2016-06-24" @default.
• W2070594927 creator A5014327763 @default.
• W2070594927 date "1972-01-01" @default.
• W2070594927 modified "2023-09-24" @default.
• W2070594927 title "On the spectrum of algebraic $K$-theory" @default.
• W2070594927 cites W1976482306 @default.
• W2070594927 cites W2005968475 @default.
• W2070594927 cites W2010595105 @default.
• W2070594927 cites W2598457714 @default.
• W2070594927 doi "https://doi.org/10.1090/s0002-9904-1972-12924-0" @default.
• W2070594927 hasPublicationYear "1972" @default.
• W2070594927 type Work @default.
• W2070594927 sameAs 2070594927 @default.
• W2070594927 citedByCount "48" @default.
• W2070594927 countsByYear W20705949272012 @default.
• W2070594927 countsByYear W20705949272017 @default.
• W2070594927 crossrefType "journal-article" @default.
• W2070594927 hasAuthorship W2070594927A5014327763 @default.
• W2070594927 hasBestOaLocation W20705949271 @default.
• W2070594927 hasConcept C121332964 @default.
• W2070594927 hasConcept C134306372 @default.
• W2070594927 hasConcept C136119220 @default.
• W2070594927 hasConcept C156778621 @default.
• W2070594927 hasConcept C202444582 @default.
• W2070594927 hasConcept C33923547 @default.
• W2070594927 hasConcept C62520636 @default.
• W2070594927 hasConcept C9376300 @default.
• W2070594927 hasConceptScore W2070594927C121332964 @default.
• W2070594927 hasConceptScore W2070594927C134306372 @default.
• W2070594927 hasConceptScore W2070594927C136119220 @default.
• W2070594927 hasConceptScore W2070594927C156778621 @default.
• W2070594927 hasConceptScore W2070594927C202444582 @default.
• W2070594927 hasConceptScore W2070594927C33923547 @default.
• W2070594927 hasConceptScore W2070594927C62520636 @default.
• W2070594927 hasConceptScore W2070594927C9376300 @default.
• W2070594927 hasIssue "2" @default.
• W2070594927 hasLocation W20705949271 @default.
• W2070594927 hasLocation W20705949272 @default.
• W2070594927 hasOpenAccess W2070594927 @default.
• W2070594927 hasPrimaryLocation W20705949271 @default.
• W2070594927 hasRelatedWork W1971227110 @default.
• W2070594927 hasRelatedWork W2038106543 @default.
• W2070594927 hasRelatedWork W2043494397 @default.
• W2070594927 hasRelatedWork W2156697727 @default.
• W2070594927 hasRelatedWork W2159964792 @default.
• W2070594927 hasRelatedWork W2264810569 @default.
• W2070594927 hasRelatedWork W2885964212 @default.
• W2070594927 hasRelatedWork W2892231951 @default.
• W2070594927 hasRelatedWork W3106133691 @default.
• W2070594927 hasRelatedWork W4249580765 @default.
• W2070594927 hasVolume "78" @default.
• W2070594927 isParatext "false" @default.
• W2070594927 isRetracted "false" @default.
• W2070594927 magId "2070594927" @default.
• W2070594927 workType "article" @default.