Related papers: Extremal Rees Algebras
We study properties of the resolution of almost Gorenstein artinian algebras $R/I,$ i.e. algebras defined by ideals $I$ such that $I=J+(f),$ with $J$ Gorenstein ideal and $f\in R.$ Such algebras generalize the well known almost complete…
We show that for a vertex decomposable simplicial complex $\Delta$, the Rees algebra of $I_{\Delta^{\vee}}$ is a normal Cohen-Macaulay domain. As consequences, we show that any squarefree weakly polymatroidal ideal is normal and we obtain…
We determine sets of elements which, under certain conditions, generate an intersection of ideals up to radical.
In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. In the case of star…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…
There is given a characterization for the Rees algebras of parameters in a Gorenstein local ring to be almost Gorenstein graded rings. A characterization is also given for the Rees algebras of socle ideals of parameters. The latter one…
In this work, we introduce a new class of Leibniz algebras, called quasi-Artinian Leibniz algebras, which generalizes the minimal condition on ideals. Furthermore, we provide some characterizations and give conditions under which a…
We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in…
Finite-dimensional Reedy algebras form a ring-theoretic analogue of Reedy categories and were recently proved to be quasi-hereditary. We identify Reedy algebras with quasi-hereditary algebras admitting a triangular (or…
All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of…
This work is about the structure of the symbolic Rees algebra of the base ideal of a Cremona map. We give sufficient conditions under which this algebra has the "expected form" in some sense. The main theorem in this regard seemingly covers…
One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that…
The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are…
Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.
We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own…
Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which…
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities…
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.
A classical result of Micali asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal…
The properties of the intersection algebra of two principal monomial ideals in a polynomial ring are investigated in detail. Results are obtained regarding the Hilbert series and the canonical ideal of the intersection algebra using methods…