The Frobenius complexity of Hibi rings

We study the Frobenius complexity of Hibi rings over fields of characteristic $p>0$. In particular, for a certain class of Hibi rings (which we call ${\omega }^{(?1)}$-level), we compute the limit of the Frobenius complexity as $p\to \infty$.     

The Frobenius exponent of Cartier subalgebras

Let R be a standard graded finitely generated algebra over an F-finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that the Frobenius complexity of R is finite. Moreover, we extend this result to Cartier subalgebras of R.     

Families of artinian and low dimensional determinantal rings

Let $\mathrm{GradAlg}(H)$ be the scheme parameterizing graded quotients of $R=k[x_0,\dots ,x_n]$ with Hilbert function H (it is a subscheme of the Hilbert scheme of ${\mathbb{P}}^n$ if we restrict to quotients of positive dimension, see definition below). A graded quotient $A=R/I$ of codimension c is called standard determinantal if the ideal I can be generated by the $t\times t$ minors of a homogeneous $t\times (t+c?1)$ matrix $(f_{ij})$. Given integers $a_0\le a_1\le ...\le a_{t+c?2}$ and $b_1\le ...\le b_t$, we denote by $W_s(\underset{\_}{b};\underset{\_}{a})?\mathrm{GradAlg}(H)$ the stratum of determinantal rings where $f_{ij}\in R$ are homogeneous of degrees $a_j?b_i$.In this paper we extend previous results on the dimension and codimension of $W_s(\underset{\_}{b};\underset{\_}{a})$ in $\mathrm{GradAlg}(H)$ to artinian determinantal rings, and we show that $\mathrm{GradAlg}(H)$ is generically smooth along $W_s(\underset{\_}{b};\underset{\_}{a})$ under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of ${\mathbb{P}}^n$ is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of $\mathrm{GradAlg}(H)$.     

On the kernels of some higher derivations in polynomial rings

Let A=R[x₁,…,x_n] be the polynomial ring in n variables over an integral domain R with unit, let D be a rational higher R-derivation on A and let be the extension of D to the quotient field of A. We prove that, if the transcendental degree of the kernel of D over R is not less than n−1, then the quotient field of the kernel of D equals the kernel of . Moreover, when n=2, we give a necessary and sufficient condition for an R-subalgebra of A to be expressed as the kernel of a rational higher R-derivation on A.     

On equations defining Veronese rings

For a standard graded noetherian algebra S that is of weakly linear type, the defining equations of the Veronesian subrings S^(d) are described explicitly, for d sufficiently large. Received: 19 January 2005     

Tight closure of parameter ideals in local rings F-rational on the punctured spectrum

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring of characteristic $p>0$. The aim of this paper is to show that ${?}_R({\mathfrak{q}}^{?}/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided R is an F-injective local ring that is F-rational on the punctured spectrum.     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown-McCoy radical, it can happen that the polynomial ring R[X] in an infinite set X of commuting indeterminates over a ring R is Behrens radical while the polynomial ring R〈X〉 in an infinite set Y of non-commuting indeterminates over R is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.     

Intermediary rings in normal pairs

We establish in this paper a result that gives the number of intermediary rings between R and S where (R,S) is a normal pair of rings. This result answers in particular a question which was left open in [A. Jaballah, Finiteness of the set of intermediary rings in a normal pair, Saintama Math. J. 17 (1999) 59-61]. Further applications are also given.     

Initial complex associated to a jet scheme of a determinantal variety

We show in this paper that the principal component of the first-order jet scheme over the classical determinantal variety of m×n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.     

3-standardness of the maximal ideal

We study a notion called n-standardness (defined by M.E. Rossi (2000) in [10] and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is 3-standard, first proving results for when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic 0 case. As an application, we extend a result due to T. Puthenpurakal (2005) [9] and show that a certain length associated with a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.     

On iteration of Cox rings

We characterize all varieties with a torus action of complexity one that admit iteration of Cox rings.     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].