Mathematics

In this study, we present the generalization of the concept of r -ideals in commutative rings with nonzero identity. Let R be a commutative ring with 0≠1 and L(R) be the lattice of all ideals of R . Suppose that ϕ:L(R)→L(R)∪{∅} is a function. A proper ideal I of R is called a ϕ−r -ideal of R if whenever ab∈I and Ann(a)=(0) imply that b∈I for each a,b∈R. In addition to giving many properties of ϕ−r -ideal, we also examine the concept of ϕ−r -ideal in trivial ring extension and use them to characterize total quotient rings.

We study generalized F -depth and F -depth, depth-like invariants associated to the canonical Frobenius action on the local cohomology modules of a local ring of prime characteristic. These invariants are naturally associated to the singularity types generalized weakly F -nilpotent and weakly F -nilpotent, which have uniform behavior among the Frobenius closure of all parameter ideals simultaneously. By developing natural lower bounds on these invariants, we are able to provide sufficient conditions which produce broad classes of rings that have these singularity types, including constructions like gluing schemes along a common subscheme, Segre products of graded rings, and Veronese subrings of graded rings. We further analyze upper bounds on Frobenius test exponents of these constructions in terms of the input data.

Let F q be a finite field with q elements, where q is a power of prime p . A polynomial over F q is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of zeroes in the affine torus T=( F ∗ q ) s of any set of r linearly independent square-free polynomials over F q in s variables, under certain conditions on r , s and degree of these polynomials. Applying the results, we partly obtain the generalized Hamming weights of toric codes over hypersimplices and square-free evaluation codes, as defined in \cite{hyper}. Finally, we obtain the dual of these toric codes with respect to the Euclidean scalar product.

Let R be the face ring of a simplicial complex of dimension d−1 and R(n) be the Rees algebra of the maximal homogeneous ideal n of R. We show that the generalized Hilbert-Kunz function HK(s)=ℓ(R(n)/(n,nt ) [s] ) is given by a polynomial for all large s. We calculate it in many examples and also provide a Macaulay2 code for computing HK(s).

In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families of semigroups and we give explicitly their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a given Frobenius number is provided and the realization of positive integers as Frobenius numbers of GSI-semigroups is studied.

The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.

We study Gorenstein ideals of codimension 4 derived from generic doublings of almost complete intersection perfect ideals of codimension 3 . We also investigate spinor coordinates of such Gorenstein ideals with 8 and 9 generators. For an ideal J of commutative ring R , the R/J module J/ J 2 is called conormal module and R/J -dual of J/ J 2 is called normal module. We study properties of conormal and normal modules of almost complete intersection perfect ideals of codimension 3 .

The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional assumptions, such as when the latter is reduced or a domain, the outcome turns out to be stronger. One important application of these considerations is to the specialization of rational maps and of symmetric and Rees powers of a module.

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras.

We study a variety for graded maximal Cohen--Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only a finite number of isomorphism classes of graded maximal Cohen--Macaulay modules with fixed Hilbert series over Cohen--Macaulay algebras of graded countable representation type.