
Recent Posts
Archives
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
Category Archives: Notes
Modular Representation Theory (IV)
Continuing our discussion of modular representation theory, we will now discuss block theory. Previously, we saw that in any ring R, there is at most one way to write where is a set of orthogonal and centrally primitive idempotents. If such an … Continue reading
Idempotents and Decomposition
Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an Rmodule M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading
Posted in Notes
Tagged blocks, idempotents, indecomposable modules, modules, primitive idempotents
Leave a comment
Modular Representation Theory (III)
Let’s work out some explicit examples of modular characters. First, we have a summary of the main results. Let be the modular characters of the simple k[G]modules; they form a basis of Let be those of the projective indecomposable k[G]modules; they form a basis … Continue reading
Modular Representation Theory (II)
We continue our discussion of modular representations; recall that all modules are finitelygenerated even if we do not explicitly say so. First, we introduce a new notation: for each projective finitelygenerated k[G]module P, we have a unique projective finitelygenerated R[G]module denoted for which … Continue reading
Modular Representation Theory (I)
Let K be a field and G a finite group. We know that when char(K) does not divide G, the group algebra K[G] is semisimple. Conversely we have: Proposition. If char(K) divides G, then K[G] is not semisimple. Proof Let , a twosided … Continue reading
Projective Modules and the Grothendieck Group
This is a continuation of the previous article. Throughout this article, R is an artinian ring (and hence noetherian) and all modules are finitelygenerated. Let K(R) be the Grothendieck group of all finitelygenerated Rmodules; K(R) is the free abelian group generated by [M] for simple … Continue reading
Posted in Notes
Tagged artinian, composition series, grothendieck group, projective modules
Leave a comment
Projective Modules and Artinian Rings
Projective Modules Recall that Hom(M, ) is leftexact: for any module M and exact , we get an exact sequence Definition. A module M is projective if Hom(M, ) is exact, i.e. if for any surjective N→N”, the resulting HomR(M, N) → HomR(M, N”) is … Continue reading
Posted in Notes
Tagged artinian, free modules, leftexact, projective modules, semisimple rings, splitting lemma
2 Comments