# Download Algebraic Topology and Transformation Groups by Tammo tom Dieck PDF

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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Transformation Groups

This publication is a jewel– it explains very important, worthy and deep subject matters in Algebraic Topology that you simply won`t locate in different places, conscientiously and intimately. """" Prof. Günter M. Ziegler, TU Berlin

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.