Sunil K. Chebolu

Ghosts in modular representation theory

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 composite 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 C2 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.

Freyd’s generating hypothesis with almost split sequences

Freyds 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 Freyds 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 Freyds generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis.

GROUPS WHICH DO NOT ADMIT GHOSTS

A ghost in the stable module category of a group G is a map between representations of G that is invisible to Tate cohomology. We show that the only non-trivial finite p-groups whose stable module categories have no non-trivial ghosts are the cyclic groups C 2 and C 3 . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups G the second power of the Jacobson radical of kG is stably isomorphic to a suspension of k.

Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Let

Finite generation of Tate cohomology

G

COUNTING IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS USING THE INCLUSION-EXCLUSION PRINCIPLE

be a finite group and let

Detecting fast solvability of equations via small powerful Galois groups

k

What Is Special about the Divisors of 12

be a field whose characteristic

Characterizations of Mersenne and 2-rooted primes

p

Galois

divides the order of