相关论文: Homological properties of pinched Veronese rings
We studies the nearly Gorenstein property for Veronese subalgebras of (semi-)standard graded algebras. We introduce a condition~$(\natural)$ for Cohen--Macaulay semi-standard graded rings, motivated by the study of Ehrhart rings. We show…
In this paper, we investigate Lefschetz properties of Veronese subalgebras. We show that, for a sufficiently large $r$, the $r$\textsuperscript{th} Veronese subalgebra of a Cohen-Macaulay standard graded $K$-algebra has properties similar…
We determine properties of two-dimensional normal affine semigroup rings, and in particular of weighted Veronese rings, including determinantal presentation, Gr\"obner basis, graded Hilbert series and graded Betti numbers, the structure of…
Let $(R, \frak m)$ be a local ring of prime characteristic $p$ and of dimension $d$ with the embedding dimension $v$, type $s$ and the Frobenius test exponent for parameter ideals $\mathrm{Fte}(R)$. We will give an upper bound for the…
We describe an algorithm for computing parameter-test-ideals in certain local Cohen-Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of…
We study the minimal free resolution of the Veronese modules of the polynomial ring in n variables, by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We characterize when…
Let $A$ be the ring of integers of a number field $K$. Let $G \subseteq GL_3(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y, Z]$ (fixing $A$) and let $S = R^G$ be the ring of invariants. Assume the Veronese subring $S^{<m>}$ of…
We provide algorithmic methods to check the Cohen--Macaulayness, Buchsbaumness and/or Gorensteiness of some families of semigroup rings that are constructed from the dilation of bounded convex polyhedrons of $\R^3_{\geq}$. Some families of…
Let $F^n(-)$ be the Frobenius functor of Peskine and Szpiro. In this note, we show that the maximal Cohen-Macaulayness of $F^n(M)$ forces $M$ to be free, provided $M$ has a rank. We apply this result to obtain several Frobenius related…
We determine when graded rings have F-rational or F-regular Veronese subrings, and develop techniques of constructing F-rational rings which are not F-regular.
In this article we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen-Macaulay singularities. This is achieved by introducing…
We study ideals which are generated by monomials of degree $d$ in the polynomial ring in $n$ variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese…
We address explicit constructions of new variants of $F$-nilpotent singularities. In particular, we explore how (generalized) weakly $F$-nilpotent singularities behave under gluing, Segre products, Veronese subrings, and the formation of…
Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have…
In this paper we present a condition on a local Cohen-Macaulay F-injective ring of positive characteristic $p > 2$ which implies that its top local cohomology module with support in the maximal ideal has finitely many Frobenius compatible…
We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such…
We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul…
The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under…
In this paper we adress the question of I. Smirnov and K. Tucker on the dual $F$-signature of the Veronese subrings of polynomial rings in $n$ variables using methods of commutative algebra.
The purpose of this note is to relate certain ring-theoretic properties of rings in mixed and positive characteristics that are related to each other by a tilting operation used in perfectoid geometry. To this aim, we exploit the…