By J. Thévenaz
This ebook develops a brand new method of the modular illustration idea of finite teams, introducing the reader to an energetic quarter of study in natural arithmetic. It provides a finished therapy of the speculation of G-algebras and exhibits the way it can be utilized to unravel a couple of difficulties approximately blocks, modules and almost-split sequences. The textual content supplies quick access to a couple complicated contemporary effects, and gives a transparent exposition of the $64000 yet tricky paintings in Puig's idea. This e-book could be of maximum curiosity to postgraduate scholars in algebra.
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Additional info for G-algebras and modular representation theory
If e is an idempotent of A , one can also consider the multiplicity of α in e , namely the number of idempotents in α appearing in a primitive decomposition of e . This number is written mα (e) . 3). 16) PROPOSITION. Let A be an O-algebra and let e and f be two idempotents of A . Then e and f are conjugate if and only if we have mα (e) = mα (f ) for every α ∈ P(A) . Proof. If e and f are conjugate, it is clear that mα (e) = mα (f ) for every α ∈ P(A) . Assume conversely that these equalities hold.
A) Since Q is projective and f surjective, the map g lifts to a homomorphism h : Q → P such that f h = g . The image of h is a submodule of P , which maps surjectively onto M via f because f (Im(h)) = Im(g) = M . Therefore Im(h) = P by definition of a projective cover and so h is surjective. Since P is projective, there exists a homomorphism s : P → Q such that h s = id , that is, h is split. (b) By part (a), there exists a surjective homomorphism h : P → P which is split by a homomorphism s : P → P and such that f h = f .
In that case A ∼ = Endk (V ) ∼ = Mn (k) . Since we shall usually be concerned with algebraically closed fields, we mention the following special case. 7) PROPOSITION. Let k be an algebraically closed field. (a) Any finite dimensional division algebra D over k is isomorphic to k . (b) Any finite dimensional simple k-algebra is split, hence isomorphic to Endk (V ) ∼ = Mn (k) , where V is a k-vector space of dimension n . The previous results contain implicitly Schur’s lemma, which we now state in full.
G-algebras and modular representation theory by J. Thévenaz