By Aderemi Kuku

ISBN-10: 142001112X

ISBN-13: 9781420011128

ISBN-10: 158488603X

ISBN-13: 9781584886037

Illustration concept and better Algebraic K-Theory is the 1st ebook to offer better algebraic K-theory of orders and staff jewelry in addition to represent greater algebraic K-theory as Mackey functors that bring about equivariant greater algebraic K-theory and their relative generalizations. therefore, this ebook makes computations of upper K-theory of workforce earrings extra available and gives novel concepts for the computations of upper K-theory of finite and a few endless teams. Authored by way of a optimal authority within the box, the ebook starts with a cautious assessment of classical K-theory, together with transparent definitions, examples, and critical classical effects. Emphasizing the sensible worth of the often summary topological structures, the writer systematically discusses larger algebraic K-theory of actual, symmetric monoidal, and Waldhausen different types with purposes to orders and crew jewelry and proves quite a few effects. He additionally defines profinite greater ok- and G-theory of tangible different types, orders, and staff jewelry. delivering new insights into classical effects and establishing avenues for extra purposes, the booklet then makes use of representation-theoretic techniques-especially induction theory-to learn equivariant better algebraic K-theory, their relative generalizations, and equivariant homology theories for discrete workforce activities. the ultimate bankruptcy unifies Farrell and Baum-Connes isomorphism conjectures via Davis-Lück meeting maps.

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**Extra info for Representation Theory and Higher Algebraic K-Theory**

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O. 5 Let S, T, Y be G-sets. Show that there exists a natural isomorphism HomGSet (T × S, Y HomGSet (S, Y T ) where Y T is the set of all set-theoretic maps T → Y made into a G-set by deﬁning gf : T → Y : t → gf (g −1 t) for all g ∈ G, f ∈ Y T , t ∈ T. 6 Let Λ be a ring with identity and P, Q ∈ P(Λ). , if P ⊕ Λn Q ⊕ Λn for some integer n. (b) Any element of K0 (Λ) can be written as [P ] − r[Λ] for some integer r. 3 in the text). 1 in the text, verify that K0 (P(R)G ) is a ring with identity. 8 Let R1 , R2 be two rings with identity.

Through C(X, B)). If (A, ϕ), (A , ϕ ) ∈ C/X (resp. (B, δ), (B , δ ) ∈ X/C), then C/X((A, ϕ), (A , ϕ )) = {ψ ∈ C(A, A )|ϕ = ϕ ψ} resp. , a morphism from (A, ϕ) to (A , ϕ ) (resp. (B, δ) to (B , δ )) is a commutative triangle A@ @@ @@ ϕ @@ ~~ ~~ ~ ~ ~~ ~ δ (resp. B ψ X /A } } }} }} }~ } ϕ XA AA AAδ AA A of C-morphisms) /B ρ (b) If in (a) C = GSet, S ∈ GSet, we have GSet/S (resp. S/GSet). Note that if S ∈ GSet, the category S can be realized as a full subcategory of GSet/S whose objects are all maps G/e → S where s ∈ S is identiﬁed with fs : G/e → S g → gs) and (g, s) is identiﬁed with (ρ(g)) , G/e BB BB BB BB fs S / G/e : x → xg −1 , x ∈ G/e zz zz z z f z} z gs (c) If C = F Set in (i), then we have an equivalence of categories [S, F Set] GSet/S deﬁned as follows.

Also each element u ∈ F ∗ determines a principal fractional ideal Ru. Let PR be the subgroup of IR consisting of all principal fractional ideals. The (ideal) class group of R is deﬁned as IR /PR and denoted by Cl(R). It is well known that if R is the ring of integers in a number ﬁeld, then Cl(R) is ﬁnite (see [39]). 2 Let R be a Dedekind domain with quotient ﬁeld F . An Rlattice is a ﬁnitely generated torsion free R-module. Note that any R-lattice M is embeddable in a ﬁnite-dimensional F -vector space V such that F ⊗R M = V .

### Representation Theory and Higher Algebraic K-Theory by Aderemi Kuku

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