Representation Theory Seminar

ABSTRACT: For a long time, this speaker has been calling for a study of stable categories. Most often this term refers to the category of modules modulo projectives. Its objects are modules, but the morphisms are quotients of the usual homomorphisms by the subgroup of homomorphisms factoring through projectives. This tool has numerous uses in diverse areas of representation theory, group cohomology, algebraic number theory, algebraic geometry, commutative algebra, and even topology (in fact, this concept originated in the work of Eckmann and Hilton on duality in homotopy theory). But stable categories don't seem to have been studied for their own sake. The first attempt at a phenomenological study of such categories was recently undertaken in joint work with Dali Zangurashvili (J. Pure Appl. Algebra 219 (2015), no. 9, 4061-4089). It became immediately clear that there were surprisingly tight and unexpected connections between the properties of the ring and the properties of its projectively stable category.
    In the last few weeks it has transpired that additive functors defined on stable categories, also known as stable functors, bring tremendous additional power to the study of rings and modules. This talk will concentrate on two new applications of such functors. The first one is a definition of the torsion submodule of a module, the second is a definition of the cotorsion quotient module of a module. This will be done in utmost generality: for any module over any ring. The new definitions are remarkably simple but, for a person not used to working with functors, may seem highly counterintuitive. One of the goals of this talk is to demystify these definitions and convince the audience that the language of functors is simple, convenient, and natural. The obtained results are new even in the classical setting of abelian groups.
    Time permitting, we shall see that the Auslander-Gruson-Jensen functor sends the cotorsion functor to the torsion functor (of opposite chirality). If the injective envelope of the ring is finitely presented, then the right adjoint of the AGJ functor sends the torsion functor back to the cotorsion functor. In particular, this correspondence establishes a duality between the requisite functors on the categories of all modules over an artin algebra.
    This will be an expository talk, no prior familiarity with functor categories is assumed. For the part on the AGJ functor, notes on Jeremy Russell's talk last week are helpful, but not essential. Graduate students looking for research opportunities are especially welcome.
    This is joint work with Jeremy Russell.