Depth, detection and associated primes in the cohomology of finite groups

For a fixed prime p, let H∗(G) denote the mod p cohomology ring of a finite group G, that is, ExtFpG(Fp,Fp) interpreted algebraically and H ∗(K(G, 1); Fp) interpreted topologically. The cohomology ring is a graded commutative Noetherian ring, or equivalently, a finitely generated graded commutative Fp-algebra. Although the ring is not strictly commutative, the usual concepts from commutative ring theory apply, e.g., Krull dimension and depth. The prime ideal spectrum is well understood due to the work by Quillen. The depth and associated primes appear to be more mysterious. It is well known that the depth of H∗(G) is bounded above by the minimum of the Krull dimensions of H∗(G)/p of all associated primes p of H∗(G). For general finitely generated (graded) commutative Fp-algebras the bound is not tight. In 1995 J. F. Carlson asked if the cohomology rings of finite groups are special in the sense that the bound is always tight. The question is related to a question about detection on certain collections of subgroups. This thesis is an introduction to these fairly unknown questions for which affirmative answers are known as Carlson’s depth conjecture. Abstrakt For et fast primtal p lad H∗(G) betegne mod p kohomologiringen af en endelig gruppe G, dvs. ExtFpG(Fp,Fp) fortolket algebraisk og H ∗(K(G, 1); Fp) fortolket topologisk. Kohomologiringen er en gradueret kommutativ Noethersk ring, eller ækvivalent, en endeligt frembragt gradueret kommutativ Fp-algebra. Selvom ringen ikke er strengt kommutativ, er det muligt at anvende de sædvanlige begreber fra kommutativ ringteori, f.eks. Krull dimension og dybde. Primidealerne er velbeskrevet af Quillen. Dybden og de associerede primidealer forekommer mere mystiske. Det er velkendt, at minimummet af Krull dimensionerne af H∗(G)/p af alle associerede primidealer p i H∗(G) er en øvre grænse for dybden af H∗(G). Der findes endeligt frembragte (graduerede) kommutative Fp-algebraer, hvor dybden er strengt mindre end minimummet. I 1995 stillede J. F. Carlson spørgsm̊alet, om kohomologiringe af endelig grupper er specielle i den forstand, at dybden altid er lig minimummet. Spørgsm̊alet er relateret til et spørgsm̊al om detektion p̊a en bestemt samling af undergrupper. Dette speciale er en introduktion til disse forholdsvis ukendte spørgsm̊al, for hvilke bekræftende svar er kendt som Carlsons dybde-formodning.