Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Abstract

Abstract. Let G be a finite group, and let k be a field whose characteristic p divides the order of G. Freyd's generating hypothesis for the stable module category of G is the statement that a map between finite-dimensional kG-modules in the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. We show that if G has periodic cohomology, then the generating hypothesis holds if and only if the Sylow p-subgroup of G is C 2 or C 3 . We also give some other conditions that are equivalent to the GH for groups with periodic cohomology.

Ghost Numbers of Group Algebras II

“…This theorem corrects the statement of Lemma 4.2 in [12], which has StMod(kG) in place of Loc G k and stmod(kG) in place of Thick G k . That statement is false whenever B is non-trivial.…”

“…We show that this holds when the Sylow p-subgroup P is a direct factor, in Section 4.1, using a result that shows that there is an equivalence between stmod(kP ) and Thick G k . This last result corrects an error in [12]; see the comments after Theorem 4.1. In Section 4.2, we show that if stmod(B 0 ) = Thick k , then the ghost number of kG is finite.…”

“…In this section, we study the ghost number of certain direct products, making a slight correction to a result in [12]. Let k be a field of characteristic p, and G = A × B with A being a p-group, and the order of B being coprime to p.…”

“…It is known that Thick A 4 k = stmod(kA 4 ) [12]. For convenience, we assume that k contains a third root of unity ζ , i.e.,…”

“…This is motivated by Freyd's generating hypothesis in stable homotopy theory [16], which is still an open question. In a series of papers [5,8,10,12] (with a minor correction made below), it has been shown that the generating hypothesis holds for kG if and only if the Sylow p-subgroup of G is C 2 or C 3 . However, computing the ghost number in cases where it is larger than one has proven to be difficult.…”