The classical category Rep$(S,\mathbb {Z}_{p})$ of representations of a finite poset $S$ over the field $\mathbb {Z}_{p}$ is extended to two categories, Rep$(S,\mathbb {Z}_{p^{m}})$ and uRep$(S,\mathbb {Z}_{p^{m}})$, of representations of $S$ over the ring $\mathbb {Z}_{p^{m}}$. A list of values of $S$ and $m$ for which Rep$(S,\mathbb {Z}_{p^{m}})$ or uRep$(S,\mathbb {Z}_{p^{m}})$ has infinite representation type is given for the case that $S$ is a forest. Applications include a computation of the representation type for certain classes of abelian groups, as the category of sincere representations in (uRep$(S,\mathbb {Z}_{p^{m}})$) Rep$(S,Z_{p^{m}})$ has the same representation type as (homocyclic) $(S,p^{m})$-groups, a class of almost completely decomposable groups of finite rank. On the other hand, numerous known lists of examples of indecomposable $(S,p^{m})$-groups give rise to lists of indecomposable representations.

In 1960, Northcott \cite {DGN} proved that, if $e_0(I)$ and $e_1(I)$ denote the 0th and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that, if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $\rm{br} _0(M)-\rm{br} _1(M)\le \ell (F/M)$, where $\rm{br} _0(M)$ and $\rm{br} _1(M$) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.

In this paper, we characterize the Pr\"ufer $v$-multiplication domain as a class of essential domains verifying an additional property on the closure of some families of prime ideals, with respect to the constructible topology.

We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gr\"obner fan of the defining ideal of our Gorenstein algebra.

We define the Buchberger resolution, which is a graded free resolution of a monomial ideal in a polynomial ring. Its construction uses a generalization of the Buchberger graph and encodes much of the combinatorics of the Buchberger algorithm. The Buchberger resolution is a cellular resolution that, when it is minimal, coincides with the Scarf resolution. The simplicial complex underlying the Buchberger resolution is of interest for its own sake, and its combinatorics is not fully understood. We close with a conjecture on the clique complex of the Buchberger graph.

Let $A$ be a domain finitely generated as an algebra over a field, $k$ of characteristic zero, $ R=A[t_1,\dots ,t_\ell ]\quad \mbox {or}\quad A[[t_1,\dots ,t_\ell ]]$, and $I$ an ideal of $R$. If $A$ has a resolution of singularities, $Y_0$, which is the blowup of $A$ along an ideal of depth at least 2 and is covered by a finite number of open affines with $H^j(Y_0,\mathcal {O}_{Y_0})$ of finite length over $A$ for $j>0$, we prove that $\rm{Ass} _RH^i_I(R) $ is finite for every $i$. In particular, this holds when $A$ is a finite-dimensional normal domain with an isolated singularity which is a finitely generated algebra over a field of characteristic~0.