4 - Indecomposable Modules

Abstract

This chapter presents Green's theory on indecomposable modules of group rings over complete discrete valuation rings. Our first goal is the vertex theory of indecomposable modules in Section 3, which is fundamental throughout the theory. The Green correspondence in Section 4 is one of the main themes of this chapter. In this connection, we show a result due to Burry and Carlson (Theorem 4.6) and establish some functorial properties of it (Theorems 4.5 and 5.4). In Section 6 we study the endomorphism rings of induced modules and show a Clifford-type theorem on indecomposable modules (Corollary 6.8). The Green indecomposability theorem in Section 7 is one of the main results of this chapter, and in the proof of it, Theorem 13.27 of Chapter 1 concerning the extension of valuation will play an important role. In Section 8 we discuss Scott modules.