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Extra resources for A short course in K-theory

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The 54 A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 relations are generated by elements of the form T (D, {f, g}), where D is a codimension p − 1 subvariety of X, f, g are in k(D)× , {f, g} is the symbol in K2 (k(D)). T is the tame symbol: iZ∗ (−1)ordZ (f )ordZ (g) ( T ({f, g}) = Z f ordZ (g) )|Z , g ordZ (f ) where the sum is over all codimension one subvarieties of the normalization D N of D, and iZ : Z → X is the composition Z ⊂ D N → D ⊂ X. A number of authors, starting with Bloch and Beilinson, have attached analytic invariants to elements of H p (X, Kp+1 ), by associating to i (Zi , fi ) the current ω→ ln(fi )ω + (2πi) i Zi ω, ∆ where ∆ is a 2p-chain (with Q-coefficients) with boundary the (2p − 1)cycle i fi−1 ([0, ∞]).

2) 131 (1990), no. 3, 493–540.

Take g ∈ GL(R). Then the conjugation action by g on BGL(R) extends to an action on BGL(R)+ which is homotopic to the identity. Additionally, one can show that the stable sum ⊕σ extends to an operation ⊕ : BGL(R)+ ×BGL(R)+ → BGL(R)+ which makes BGL(R)+ into an H-group, with the identity matrix as unit. Loday [21] has shown that the tensor product operation on matrices (A, B) → A ⊗ B can be modified to define a product BGL(R)+ ∧ BGL(R)+ → BGL(R)+ which makes the graded group ⊕∞ i=0 Ki (R) into a graded ring.

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