Green’s Theory of Indecomposable Modules

This chapter explains one of the cornerstones of the module theoretic approach to modular representation theory. By using the notion of a relatively projective module one obtains a means to measure how far away a module over a group ring of G is from being projective. This leads to the concepts of vertices and sources of such modules. The theory is used to establish Green’s correspondences between indecomposable modules of G and of specific subgroups of G. We prove Green’s theorem that induction from a normal subgroup whose index in G is a power of p preserves indecomposability of modular representations. We also make explicit all concepts and results developed so far for the group $\mathit{SL}_{2}(\mathbb{F}_{p})$ .