Homological Finiteness Conditions for Groups, Monoids, and Algebras

Abstract

Recently, Alonso and Hermiller (2003) introduced a homological finiteness condition bi − FP _n (here called “weak bi-FP _n ”) for monoid rings, and Kobayashi and Otto (2003) introduced a different property, also called bi − FP _n (we adhere to their terminology). From these and other articles we know that: bi − FP _n  ⇒ left and right FP _n  ⇒ weak bi − FP _n ; the first implication is not reversible in general; the second implication is reversible for group rings. We show that the second implication is reversible in general, even for arbitrary associative algebras (Theorem 1′), and we show that the first implication is reversible for group rings (Theorem 2). We also show that the all four properties are equivalent for connected graded algebras (Theorem 4).

Key Words:

• Graded algebra
• Group ring
• Monoid ring
• Type-FP _n

Mathematics Subject Classification:

• 20M05
• 16E40
• 20J05

ACKNOWLEDGMENT

I thank Peter Kropholler and Alexandro Olivares for helpful discussions.

Notes

Communicated by V. Gould.