I. The Pure Semisimple Conjecture. The conjecture states that if every left R-module over a ring R is a direct sum of indecomposable modules, then so is every right R-module. The following two papers contain a procedure that shows how to construct, starting with a putative counterexample, one with two simple modules and Jacobson radical squared zero:
A test for finite representation type
Finitely presented modules over a pure semisimple ring
II. The Bruer-Thrall Conjectures. This paper contains a proof of Crawley-Boevey’s result that if an artin algebra has infinitely many indecomposable modules of bounded length, then it has a generic module:
The Ziegler spectrum of a locally coherent Grothendieck category
III. Krull-Gabriel dimension of an artin algebra. This paper contains a proof of the conjecture that the Krull-Gabriel dimension of the free abelian category over an artin algebra cannot be 1:
The endomorphism ring of a localized coherent functor
IV. The Phantom number of a finite group. This paper with X.H. Fu contains a proof of the conjecture of Benson and Gnacadja that if G is a finite group, then the nilpotency index of the phantom ideal in the stable category of the group ring kG is finite: