A basis for the right quantum algebra and the “1 = q” by Foata D., Han G.-N.

By Foata D., Han G.-N.

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10 A. 11. Let G = {Gn } be a simplicial group with face maps δi : Gn → Gn−1 and degeneracies si : Gn → Gn+1 (0 i n). Define πn G = Hn /dn+1 Kn where Hn ⊂ Kn ⊂ Gn are defined by Kn := ker(δ0 ) ∩ · · · ∩ ker(δn−1 ) and Hn = Kn ∩ ker(δn ) . Say that G is acyclic if πn (G) = 0 ∀n. We can regard a simplicial ring as a simplicial group using its additive structure and we say that a simplicial ring is acyclic if πn R = 0 for all n. 12. A simplicial ring R is said to be free if there exists a basis Bn of Rn as a free ring for all n and si (Bn ) ⊂ Bn+1 for all i and all n.

Let Σn denote the n × n permutation matrices. Then Σn can be identified with the n-th symmetric group. Put Σ = lim −→ Σn . Then Σ acts on Γ (a) by conjugation and so, we can form Γ (a) = Γ (a) Σ . One could think of Γ (a) as the group of matrices in GLn (a ⊕ Z) whose image in GLn (Z) is a permutation matrix. Consider the fibration BΓ a → B Γ (a) → B(Σ). Note that B(Σ), B Γ (a) has an associated +-construction which are infinite loop spaces. Define Fa as the homotopy fibre Fa → B Γ (a)+ → BΣ + .

10). In Section 4, we highlight, with copious examples, some fundamental results in higher K-theory, most of which have classical analogues at the zero-dimensional level. 5). We also discuss Waldhausen’s fibration sequence, localization sequence for Waldhausen’s K-theory and a long exact sequence which realizes the cofibre of the Cartan maps as K-theory of a Waldhausen category. Finally, we discuss excision, Mayer–Vietoris sequences and long exact sequence associated to an ideal. In Section 5, we define Galois, étale and motivic cohomologies and discuss their interconnections as well as connections with K-theory.

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