By Tammo tom Dieck

This booklet is a jewel– it explains very important, beneficial and deep subject matters in Algebraic Topology that you just won`t locate somewhere else, conscientiously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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**Extra info for Algebraic Topology and Transformation Groups**

**Sample text**

In the commutative case, the theorem takes the following form, which is often useful. 2) COROLLARY. If A is a commutative O-algebra, then there exists a unique primitive decomposition of 1A . In particular any two primitive idempotents of A are either equal or orthogonal. 15 is the following. 3) THEOREM. Let A be an O-algebra. The set P(A) of points of A is in bijection with both Max(A) and Irr(A) . If α ∈ P(A) , the corresponding maximal ideal mα is characterized by the property e ∈ / mα for some e ∈ α (or equivalently for every e ∈ α ), while the corresponding simple A-module V (α) is characterized by the property e · V (α) = 0 for some e ∈ α (or equivalently for every e ∈ α ).

The following result is another useful fact about decompositions of idempotents. 16) PROPOSITION. Let A be a ring. (a) Let 1A = i∈I i be a decomposition of the unity element. Then A decomposes as the direct sum of left ideals A = ⊕i∈I Ai . (b) Let A = ⊕λ∈Λ Vλ be a finite direct sum decomposition of A into left ideals. Then there exists a decomposition of the unity element 1A = λ∈Λ iλ such that Vλ = Aiλ . (c) An idempotent e of A is primitive if and only if the left ideal Ae is indecomposable. (d) If A is noetherian, there exists a primitive decomposition of the unity element 1A .

Moreover Mλ is indecomposable if and only if eλ is primitive. Since M is finitely generated and A is finitely generated as an O-module, M is finitely generated as an O-module and therefore so is EndO (M ) , as well as its subalgebra EndA (M ) . 16, there exists a primitive decomposition of idM , proving (a). 1, this decomposition is conjugate to the given one by some element ∼ φ ∈ EndA (M )∗ , that is, φeλ φ−1 = eσ(λ) for some bijection σ : Λ → ∆ . Then for every λ ∈ Λ , we have φ(Mλ ) = φ(eλ M ) = φeλ φ−1 M = eσ(λ) M = Mσ(λ) , as required.