05126146

j

05126146 Pride, Stephen J. Homological finiteness conditions for groups, monoids, and algebras. Commun. Algebra 34, No. 10, 3525-3536 (2006). 2006 Taylor \& Francis, Philadelphia, PA EN graded algebras group rings monoid rings monoids of finite homological type homological finiteness conditions free resolutions

doi:10.1080/00927870600796110

Let $KB$ be a monoid ring over a commutative ring $K$ and a monoid $B$. Consider $K$ as a left $KB$-module with the $KB$-action $k_1b\cdot k_2=k_1k_2$. Then $B$ is said to be of type `left-$FP_n$' (over $K$) if there is a partial free resolution $0\leftarrow{_BK}\leftarrow P_0\leftarrow P_1\leftarrow\cdots\leftarrow P_n$, where $P_i$ are finitely generated free left $KB$-modules. The author calls a semigroup to be of type `weak bi-$FB_n$' if it has this property with respect of partial free resolutions of $(KB,KB)$-bimodules ($K$ is considered then as the $(KB,KB)$-bimodule $_BK_B$). A semigroup $B$ is called of type `bi-$FP_n$' if there exists a partial resolution $0\leftarrow KB\leftarrow F_0\leftarrow F_1\leftarrow\cdots\leftarrow F_n$ of $(KB,KB)$-bimodules where $F_i$ are finitely generated free. It is proved that 1) if a monoid is weak bi-$FB_n$, then it is both left- and right-$FB_n$; 2) If a group or a connected graded algebra is left-$FP_n$ (or, equivalently, right-$FP_n$), then it is bi-$FP_n$ (a graded algebra $A=\bigoplus_{i\geq 0}A_i$ is called `connected' if $A_0$ has basis the identity $1_A$); 3) The properties left-$FP_n$, right-$FP_n$, bi-$FP_n$ and weak bi-$FP_n$ are closed under retractions. Peeter Normak (Tallinn)