Abstract. For a prime power q = p d and a field F containing a root of unity of order q we show that the Galois cohomology ring H *

Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C 2 or C 3 . We also give various conditions which are equivalent to the generating hypothesis.

Abstract. Freyd's generating hypothesis for the stable module category of a non-trivial finite group G is the statement that a map between finitely generated kG-modules that belongs to the thick subcategory generated by the field k factors through a projective module if the induced map on Tate cohomology is trivial. In this paper we show that Freyd's generating hypothesis fails for kG when the Sylow p-subgroup of G has order at least 4 using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd's generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.

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

C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Using just very basic knowledge of finite fields and the inclusion-exclusion formula, we show how one can see the shape of this formula and its proof almost instantly.Why there are exactly 1 30 (2 30 − 2 15 − 2 10 − 2 6 + 2 5 + 2 3 + 2 2 − 2)irreducible monic polynomials of degree 30 over the field of two elements? In this note we will show how one can see the answer instantly using just very basic knowledge of finite fields and the well-known inclusion-exclusion principle.To set the stage, let F q denote the finite field of q elements. Then in general, the number of monic irreducible polynomials of degree n over the finite field F q is given by Gauss's formula 1 n d|n µ(n/d)q d , where d runs over the set of all positive divisors of n including 1 and n, and µ(r) is the Möbius function. (Recall that µ(1) = 1 and µ(r) evaluated at a product of distinct primes is 1 or -1 according to whether the number of factors is even or odd. For all other natural numbers µ(r) = 0.) This beautiful formula is well-known and was discovered by Gauss [2, p. 602-629] in the case when q is a prime. We will present a proof of this formula that uses only elementary facts about finite fields and the inclusion-exclusion principle. Our approach offers the reader a new insight into this formula because our proof gives a precise field theoretic meaning to each summand in the above formula. The classical proof [3, p. 84] which uses the Möbius' inversion formula does not offer this insight. Therefore we hope that students and users of finite fields may find our approach helpful. It is surprising that our simple argument is not available in textbooks, although it must be known to some specialists.