Large self-injective rings and the generating hypothesis (Q2509400)

This is a study of graded commutative rings (commutativity in the graded meaning), large in various senses, which are self-injective. One characterization of a self-injective Noetherian graded commutative ring is that it is Artinian that factors into a finite product of Artinian local rings each of which has one-dimensional socle. After giving a general theory of self-injective rings and criteria for self-injectivity, the authors address the Noetherian case. The topics are as follows: Coherence, Self-injective adjustment, The cube algebra. Pontryagin self-dual rings, The infinite root algebra, The Rado algebra, The 0-algebra, Triangulation. The background motivation comes from Spanier-Whitehead category \(\mathcal F\) of finite spectra. The authors treat the \(p\)-components, for a fixed prime \(p\) by defining Hom sets in a triangulated category named \(\mathcal F_p\) as follows: \( [X, Y] =\mathbb Z_p\otimes{\Hom}_{\mathcal F}(X,Y)\), where \(\mathbb Z_p\) denotes the ring of \(p\)-adic integers. In this context, the authors are interested in Freyd's generating hypothesis in stable homotopy theory, namely whether the functor \(\pi_*:\mathcal F_p\longrightarrow\text{RMod}\) is faithful. The authors construct a number of examples of non-Noetherian graded rings that are injective as modules over themselves, and discuss how their constructs are related to the theory of triangulated categories. In addition to the number of said examples, we list here the following results that shed light on the part of interest of the article: Theorem 6.6. Let \(R\) be a graded-commutative ring such that (a) \(R_k=0\), for \(k<0\), (b) \(R_0=\mathbb Z/2\), (c) \(R_k\) is finite, for all \(k\geq 0\). Then, for every \(N\), there is an injective map \(\phi:R\longrightarrow R'\) of graded rings such that \(R'\) also has properties (a)--(c); \(\phi:R_k\longrightarrow R'_k\) is an isomorphism for \(k<N\) and \(R'\) is self-injective. Proposition 8.2. If \(R\) is a Pontryagin self-dual, then it is self-injective. Corollary 12.8. Neither the infinite exterior algebra nor the cube algebra admits a triangulation structure.