Ghosts in modular representation theory

Abstract

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis---the statement that ghosts between finite-dimensional G-representations factor through a projective---we define the ghost number of kG to be the smallest integer l such that the composition of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C_2 as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.Comment: 15 pages, final version, to appear in Advances in Mathematics. v4 only makes changes to arxiv meta-data, correcting the abstract and adding a do

Ghost Numbers of Group Algebras II

“…We are able to get an upper bound for the ghost number of kA 4 , since the simple modules have bounded ghost lengths. Proof By Theorem 3.2, the simple ghost number of kA 4 is equal to the ghost number of kV , which is known to be 2 (see [11]). Since stmod(kA 4 ) = Thick k and every simple ghost is a ghost, the ghost number of kA 4 is at least 2.…”

“…Thus the simple ghost number of kG, the ghost number of kG and the ghost number of kP are all equal. Since P is a cyclic p-group, its ghost number is known [11]. In particular, this allows us to compute the ghost numbers of the dihedral groups at an odd prime.…”

“…However, computing the ghost number in cases where it is larger than one has proven to be difficult. Some preliminary work was done in [11], where the ghost numbers of cyclic p-groups were computed, and various upper and lower bounds were obtained in other cases. Substantial progress was made in our previous paper [15], where we computed the ghost numbers of k(C 3 × C 3 ) and other algebras of wild representation type, as well as the ghost numbers of dihedral 2-groups, the first non-abelian computations.…”

Finite generation of Tate cohomology

“…The following natural question was raised in [11]: When does the Tate cohomology functor detect trivial maps in the stable module category stmod(kG) of finitely generated kG-modules? A map φ : M → N between finitely generated kG-modules is said to be a ghost if the induced map in Tate cohomology groups …”

The quaternion group has ghost number three

“…Let G be a group and K a field of characteristic p. A map f : M → N in the stable category stmod(KG) of finitely generated KG-modules is called a ghost if it vanishes under Tate cohomology, that is if f * :Ĥ * (G, M) →Ĥ * (G, N) is zero. The ghost maps then form an ideal in stmod(KG), and Chebolu, Christensen and Mináč [3] define the ghost number of KG to be the nilpotency degree of this ideal.…”