We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.

This paper applies$K$-homology to solve the index problem for a class of hypoelliptic (but not elliptic) operators on contact manifolds.$K$-homology is the dual theory to$K$-theory. We explicitly calculate the$K$-cycle (i.e., the element in geometric$K$-homology) determined by any hypoelliptic Fredholm operator in the Heisenberg calculus.

In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.