Algebraic K-Theory by E. M. Friedlander, M. R. Stein

By E. M. Friedlander, M. R. Stein

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Table of Contents

I. basic TOPOLOGY.
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2. Topological areas and non-stop Functions.

three. Connectedness and Compactness.

four. Countability and Separation Axioms.

five. The Tychonoff Theorem.

6. Metrization Theorems and Paracompactness.

7. entire Metric areas and serve as Spaces.

eight. Baire areas and size Theory.

II. ALGEBRAIC TOPOLOGY.
nine. the basic Group.

10. Separation Theorems within the Plane.

11. The Seifert-van Kampen Theorem.

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Example text

Proof. Write xi = r λir vr and yj = Now expand both sides using (4). s µjs ws for some scalars λir and µjs . 23. Let M, N be CG-modules with bases m1 , . . and n1 , . . respectively. Then defining g · (mi ⊗ nj ) = (g · mi ) ⊗ (g · nj ) and extending bilinearly gives M ⊗ N the structure of an CG-module. Extending bilinearly means that if x = nj ∈ M ⊗ N then x · t is defined to be g λg g ∈ CG and t = i,j λij mi ⊗ λg λij g · (mi ⊗ nj ). g,i,j When we do this the two linearity conditions in the definition of a module are guaranteed to hold, and we only need to check that acting gh is the same as acting h then g.

Then for any g ∈ G we have χΩ (g) = | fixΩ (g)|. Version of Sunday 29th March, 2015 at 20:00 40 Proof. Consider the matrix for the action of g with respect to the basis Ω of CΩ. The column of this matrix corresponding to ω ∈ Ω has a 1 at position g · ω and zeros elsewhere, so it contributes 1 to the trace if g · ω = ω and zero otherwise. Thus the trace of this matrix is the number of elements of Ω fixed by g. 4. The character χ of the regular CG-module satisfies χ(g) = 0 if g = e and χ(e) = |G|.

So x = s ∈ SI ∩ V = {0}. This finishes the proof. 2. With the notation of the previous lemma, let f : S1 ⊕ · · · ⊕ Sn → N be a module homomorphism. Then im f is isomorphic to a direct sum of some of the Si . Proof. ker f is a submodule of M , so it has a complement C isomorphic to the direct sum of some of the Si by the previous lemma. Because C is a complement to the kernel of f , the restriction f |C : C → im f defined by f |C (c) = f (c) is an isomorphism of modules. So im f ∼ = C which is isomorphic to a direct sum of some of the Si .

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