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 finitely-generated even if we do not explicitly say so. First, we introduce a new notation: for each projective finitely-generated k[G]-module P, we have a unique projective finitely-generated 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 two-sided … 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 finitely-generated. Let K(R) be the Grothendieck group of all finitely-generated R-modules; K(R) is the free abelian group generated by [M] for simple … Continue reading