Powers of the phantom ideal (Q2809276)

The main goal of this paper is to develop ideal approximation theory further by introducing a new exact structure on the morphism category of an exact category, and then use this theory to consider phantom ideals in module categories and stable module categories.NEWLINENEWLINEExplicitly, given an exact category \((\mathcal{A}, \mathcal{E})\), the authors study its morphism category \(\mathrm{Arr} (\mathcal{A})\), which has a natural exact structure \((\mathrm{Arr}(\mathcal{A}), \mathrm{Arr}(\mathcal{E}))\). They introduce the notion of mono-epi (ME for abbreviation) morphisms of conflations, and prove that \((\mathrm{Arr}(\mathcal{A}), \mathrm{ME})\) is an exact substructure of \((\mathrm{Arr}(\mathcal{A}), \mathrm{Arr}(\mathcal{E}))\). Within this exact substrcture, the authors define extensions and orthogonal relations of ideals, and prove the ideal version of Salce's lemma, Christensen's lemma, and Wakamatsu's lemma.NEWLINENEWLINEAs a main application of the general theory developed above, the authors considered phantom ideals \(\Phi\) in module categories and stable categories, giving sufficient conditions such that powers of phantom ideals vanish or are object ideals of certain special modules such as projective modules or flat modules. For instance, they prove that if the Jacobson ideal \(J\) of a semiprimary ring satisfies \(J^n = 0\), then the \(n\)-th power of the phantom ideal \(\Phi^n\) is the object ideal generated by projective modules (Theorem 9.1). Furthermore, if \(R\) is a quasi-Frobenius ring whose Jacobson ideal \(J\) is nonzero and satisifies \(J^n = 0\), then the \((n-1)\)-th power \(\Phi^{n-1}\) of the phantom ideal vanishes in the stable category (Theorem 9.3). These results have interesting consequences in representation theory of finite groups.