By Steven Dale Cutkosky

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**Extra info for An Introduction to Galois Theory [Lecture notes]**

**Example text**

3. 1. Higher K-theory of exact categories – definitions and examples In [79, Section 3], we discussed K0 of exact categories C, providing copious examples. In this section, we define Kn (C) for all n 0 with the observation that this definition generalizes to higher dimensions the earlier ones at the zero-dimensional level. 1. D EFINITION . e. for any i j exact sequence 0 → C → C → C → 0 in A with C , C in C, we have C ∈ C). In the exact sequence (I) above, we shall refer to i as an inflation or admissible monomorphism, j as a deflation or admissible epimorphism; and to the pair (i, j ) as a conflation.

E. one-connected) and BSt(A)+ is 2-connected. For a comprehensive discussion of Hurewitz maps, see [6]. 4. E XAMPLES /R EMARKS . 2 can be identified respectively with the classical Kn (A). (i) π1 (BGL(A)+ ) = GL(A)/E(A) = K1 (A). (ii) Note that BE(A)+ is the universal covering space of BGL(A)+ and so, we have π2 BGL(A)+ ≈ π2 BE(A)+ ≈ H2 BE(A)+ ∼ = H2 BE(A) ∼ = H2 E(A) ≈ K2 (A). (iii) K3 (A) = H3 (St(A)). For a proof, see [38]. (iv) If A is a finite ring, then Kn (A) is finite (see [73] for a proof).

E XAMPLES /A PPLICATIONS . 5 is satisfied by S = {s i } and a is an S-excision ideal. (ii) Let R be the ring of integers in a number field F , Λ an R-order in a semi-simple F -algebra Σ, Γ a maximal R-order containing Λ. Then there exists an s ∈ Z, s > 0, such that sΓ ⊂ Λ and so a = sΓ is a 2-sided ideal in both Λ and Γ . Since s annihilates Λ/a (also Γ /a), a is an S-excision ideal and so, we have a long exact Mayer–Vietoris sequence → Kn+1 (Γ /a) → Kn (Γ /a) 1 s 1 s → Kn (Λ) 1 s → Kn (Λ/a) 1 1 ⊕ Kn (Γ ) s s →, where we have written A( 1s ) for A ⊗ Z( 1s ) for any Abelian group A.