Phantom maps and purity over finite-dimensional self-injective algebras

We investigate phantom maps in the stable module category StModL of a finite-dimensional self-injective algebra Lambda. These are the morphisms in StModL whose restrictions to the finite-dimensional submodules of the source module are null. Phantom maps occur as a manifestation of the fact that a morphism M → N in StModL is not completely determined by its restrictions to the finite-dimensional submodules of M. More generally, if a direct system of morphisms Ma→N a in StModL lifts to a direct system in the ordinary module category ModL , there may be an indeterminacy in the proper definition of a direct limit morphism lim→a Ma&rarrr;N a in StModL . This indeterminacy is captured by phantom maps. Phantom maps are also related to the actual "liftability" of a direct system from StModL to ModL . After the first chapter of preliminaries, we study phantom maps emanating from countable-dimensional modules. Under this restriction, we obtain some concrete results, such as the Milnor short exact sequence and quite explicit examples of phantom maps. The notion of purity from algebra and logic is a necessary means for more general investigations of phantom maps and their composites. We provide a survey of this notion and related concepts, tailored to our needs. The survey ends with an investigation of how ordinary projectivity interacts with purity. We use purity to study the vanishing of composites of phantom maps. We introduce the notions of phantom dimension of a finite-dimensional self-injective algebra, and of phantom index of a finite group. We show how to bound the phantom index for certain finite 2-groups, and carry out explicit calculations for the groups C2m,2n=Z/ 2mxZ/2 n , D2n , Q2n , and SD2n . In particular, we obtain that the quaternion groups Q 2n have phantom index either two or three; so does the Klein four-group V4=Z/2xZ/ 2 . Another interaction between phantom maps and purity occurs when we study the phantom spectral sequence. We obtain an Extended Milnor Sequence applicable with no restrictions on module generation cardinality. The sequence is used to show that there are direct systems in StModL which do not lift to ModL , and that the forgetful functor from pure extensions to ordinary extensions need not be faithful. Finally, we report on some purity and phantom maps related properties of certain Rickard's idempotent modules and the so-called Omega type modules. There are a number of interesting results, for instance that the indecomposable pure-injective Klein modules fall in four categories, three of which are expressible in terms of certain Rickard's modules.