By J. Thévenaz

This ebook develops a brand new method of the modular illustration concept of finite teams, introducing the reader to an energetic region of study in natural arithmetic. It supplies a finished remedy of the idea of G-algebras and indicates the way it can be utilized to unravel a few difficulties approximately blocks, modules and almost-split sequences. The textual content supplies easy accessibility to a few complicated fresh effects, and offers a transparent exposition of the $64000 yet tough paintings in Puig's conception. This e-book may be of maximum curiosity to postgraduate scholars in algebra.

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Conversely if f is an idempotent which satisfies f = ef e , then f appears in some decomposition of e , because e = f + (e − f ) is an orthogonal decomposition. These elementary observations will be used repeatedly. Instead of referring to a decomposition of an idempotent e as being a set I , we shall often say abusively that the expression e = i∈I i is a decomposition of e . Recall that two idempotents e and f are called conjugate if there exists u ∈ A∗ such that f = ue . Most of the concepts and constructions which we are going to introduce for idempotents will depend on conjugacy classes of idempotents rather than idempotents themselves.

D) The correspondence in (c) sets up a bijection between the sets P(A) and Max(A) . (e) For every point α of A , there is a unique simple A-module V (up to isomorphism) such that e · V = 0 for some e ∈ α . In fact e · V = 0 for every e ∈ α and V ∼ = Ae . (f) The correspondence in (e) sets up a bijection between the sets P(A) and Irr(A) . (g) Any two primitive decompositions of 1A are conjugate under A∗ . The theorem on lifting idempotents allows us to generalize (c)–(g) to any finite dimensional k-algebra, but we shall consider in Section 3 an even more general situation.

AαA ∩ b) . In particular A = (a) b = α∈P(A) α∈P(A) (b) b ⊆ AαA + J(A) . α∈P(A) α⊆b 28 Chapter 1 . Algebras over a complete local ring Proof. (a) Writing 1A as a sum of primitive idempotents and multiplying (say on the left) by an arbitrary element of b , one obtains immediately b = α∈P(A) (Aα ∩ b) . The result follows from the obvious inclusion Aα ⊆ AαA . (b) It suffices to prove the result for the image of b in A/J(A) . Thus we can assume that A is semi-simple. The result is trivial in that case because an ideal is necessarily a direct sum of some of the simple factors S(α) , and α ⊆ b if and only if S(α) ⊆ b .