In a recent paper, Jeremy Rickard showed that for any finite-dimensional algebra R over a field, if the R-injectives generate the derived category D(R), then the finitistic dimension of R is finite (recall that the famous finitistic dimension conjecture claims that for any R that is a finite-dimensional algebra over a field, the finitistic dimension of R should be finite). In the same paper, it was noted that there are no known finite-dimensional algebras over fields for which this injective generation property is absent. In this talk, we will consider the case when R is a group algebra (not necessarily a finite-dimensional algebra over a field), and show that for a large class of groups, the finiteness of the finitistic dimension of the group algebra (over any commutative ring of finite global dimension) implies the above injective generation property. We will also show how this question, for group algebras, is very closely connected to some existing conjectures on various cohomological invariants for groups, and that will lead us to a version of the finitistic dimension conjecture for group algebras.
This talk was part of the one-day meeting Triangulated Categories in Representation Theory, which took place online in 2021.
