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Monthly Archives: February 2015
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
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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
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