Differential modules are natural generalizations of complexes. In this paper, we study differential modules with complete intersection homology, comparing and contrasting the theory of these differential modules with that of the Koszul complex. We construct a Koszul differential module that directly generalizes the classical Koszul complex and investigate which properties of the Koszul complex can be generalized to this setting.

The aim of this paper is to study conditions on an integral domain $D$ such that any $D$-algebra between the polynomial ring $D[X]$ and the ring of integer-valued polynomials $\text{Int}(D)$ is (locally) free. These results are then extended to several indeterminates.

In this paper, we study the $s$-th symbolic powers of the edge ideals of complete graphs. In particular, we provide a criterion for finding an Eliahou–Kervaire splitting on these ideals, and we use the splitting to provide a description for the graded Betti numbers. We also discuss the symbolic powers and graded Betti numbers of edge ideals of parallelizations of finite simple graphs.

This paper aims to provide several relations between Bass and Betti numbers of a given module and its deficiency modules. Such relations and the tools used throughout allow us to generalize some results of Foxby, characterize Cohen–Macaulay modules in equidimensionality terms, study the Cohen–Macaulay and complete intersection properties of a ring, and furnish a case for the Auslander–Reiten conjecture.

Nagel and Römer introduced the class of weakly vertex decomposable simplicial complexes, which include matroid, shifted, and Gorenstein complexes as well as vertex decomposable complexes. They proved that the Stanley–Reisner ideal of every weakly vertex decomposable simplicial complex is Gorenstein linked to an ideal of indeterminates via a sequence of basic double G-links. In this paper, we explore basic double G-links between squarefree monomial ideals beyond the weakly vertex decomposable setting.

Our first contribution is a structural result about certain basic double G-links which involve an edge ideal. Specifically, suppose $I(G)$ is the edge ideal of a graph $G$. When $I(G)$ is a basic double G-link of a monomial ideal $B$ on an arbitrary homogeneous ideal $A$, we give a generating set for $B$ in terms of $G$ and show that this basic double G-link must be of degree $1$. Our second focus is on examples from the literature of simplicial complexes known to be Cohen–Macaulay but not weakly vertex decomposable. We show that these examples are not basic double G-links of any other squarefree monomial ideals.

We characterize nearly Gorenstein projective monomial curves of codimension 2 and 3.

We investigate log-concavity in the context of level Hilbert functions and pure $O$-sequences, two classes of numerical sequences introduced by Stanley in the late 1970s whose structural properties have since been the object of a remarkable amount of interest in combinatorial commutative algebra. However, a systematic study of the log-concavity of these sequences began only recently, thanks to a paper by Iarrobino.

The goal of this note is to address two general questions left open by Iarrobino’s work: (1) Given the integer pair $(r,t)$, are all level Hilbert functions of codimension $r$ and type $t$ log-concave? (2) What about pure $O$-sequences with the same parameters?

Iarrobino’s main results consisted of a positive answer to (1) for $r=2$ and any $t$, and for $(r,t)=(3,1)$. Further, he proved that the answer to (1) is negative for $(r,t)=(4,1)$.

Our chief contribution to (1) is to provide a negative answer in all remaining cases, with the exception of $(r,t)=(3,2)$, which is still open in any characteristic. We then propose a few detailed conjectures specifically on level Hilbert functions of codimension 3 and type 2.

As for question (2), we show that the answer is positive for all pairs $(r,1)$, negative for $(r,t)=(3,4)$, and negative for any pair $(r,t)$ with $r\ge 4$ and $2\le t\le r+1$. Interestingly, the main case that remains open is again $(r,t)=(3,2)$. Further, we conjecture that, in analogy with the behavior of arbitrary level Hilbert functions, log-concavity fails for pure $O$-sequences of any codimension $r\ge 3$ and type $t$ large enough.