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This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where…
We extend the notion of type sequence to rings that are not necessarily residually rational. Using this invariant we characterize different types of rings as almost Gorenstein rings and rings of maximal length.
We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are klt if and only if R(-K_X) is finitely generated. We introduce a notion of nefness for…
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.
Some descriptions of linked ideals in a commutative Notherian ring $R$ are provided in terms of the Associated prime ideals of $R$. Then, among other things, we make some characterization of Cohen-Macaulay, Gorenstein and regular local…
We study Gorenstein dimension and grade of a module $M$ over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded…
We study stable trace ideals in one dimensional local Cohen-Macaulay rings and give numerous applications.
We prove a conjecture stated in a previous paper by the author about the existence of canonical filtrations for a family of vertex operator algebras in rational levels.
Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the…
In this paper, we consider whether parameter test ideals, conductors, $F$-ideals, and trace ideals are contained in an ideal whose quotient ring has finite phantom projective dimension (for example, ideals generated by a system of…
We develop the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given…
In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in \'etale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of…
In this paper, we introduce and study the rings of Gorenstein homological dimensions small or equal than 1, which we call Gorenstein (semi)hereditary rings, specially particular cases of these rings, which we call strongly Gorenstein…
We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional $\Delta$-system…
Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection…
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive…
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical…
We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…
With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms…
Even if a ring A is coherent, the polynomial ring A[X] in one variable could fail to be coherent. In this note we show that A[X] is graded coherent with the standard grading deg X=1. More generally, we give a criterion of graded…