arXiv subjects
Publications and source records attributed to Oana Olteanu.
Classes of squarefree monomial ideals were intensively studied being an important connection between two main areas in mathematics: commutative algebra and combinatorics. Many algebraic invariants of the squarefree monomial ideals are given in terms of properties of the combinatorial associated objects (graphs, simplicial complexes etc.).
We consider the closed neighborhood ideal of square of the path graph and study some of its algebraic and homological invariants. We compute the height, the projective dimension and the Castelnuovo-Mumford regularity. We prove that these ideals are sequentially Cohen-Macaulay and characterize when they are Cohen-Macaulay.
Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and algebraic invariants in terms of the combinatorics of the graph. We compute the minimal primary decomposition and characterize the Cohen--Macaulay ideals. Moreover, we provide a formula for computing the Betti numbers, which depends only on the coefficients of the independence polynomial of the graph.
We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path, we prove that they have linear quotients and we characterize the normally torsion-free ideals. We determine a class of non-squarefree ideals, arising from some particular graphs, which are normally torsion-free.
In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.
We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.
Let $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$
In this paper we characterize the componentwise lexsegment ideals which are componentwise linear and the lexsegment ideals generated in one degree which are Gotzmann.