‎let $i$ be an ideal in a regular local ring $(r,n)$‎, ‎we will find‎ ‎bounds on the first and the last betti numbers of‎ ‎$(a,m)=(r/i,n/i)$‎. ‎if $a$ is an artinian ring of the embedding‎ ‎codimension $h$‎, ‎$i$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $a$ a {it $t-$extended stretched local ring}‎. ‎this class of‎ ‎local rings is a natural generalization of the class of stretched ...

Let ∆ be a triangulated homology ball whose boundary complex is ∂∆. A result of Hochster asserts that the canonical module of the Stanley–Reisner ring of ∆, F[∆], is isomorphic to the Stanley–Reisner module of the pair (∆, ∂∆), F[∆, ∂∆]. This result implies that an Artinian reduction of F[∆, ∂∆] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction ...

First we construct an interesting bijection between the set of h-vectors and the set of socle-vectors of artinian algebras. As a simple consequence, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then we address the main problem of this paper: determining when there is a unique socle-vector associated to a given h-vector. In Sections 3 and 5 we work...