Mathematics

In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.

We prove that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in P n such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.

Let P be a simple thin polyomino, roughly speaking a polyomino that has no holes and does not contain a square tetromino as a subpolyomino. In this paper, we determine the reduced Hilbert series h(t)/(1−t ) d of K[P] by proving that h(t) is the rook polynomial of P. As an application, we characterize the Gorenstein simple thin polyominoes.

Let A be a Noetherian local ring with the maximal ideal m and I be an m -primary ideal in A . In this paper, we study a boundary condition of an inequality on Hilbert coefficients of an I -admissible filtration I . When A is a Buchsbaum local ring, the above equality forces Buchsbaumness on the associated graded ring of filtration. Our result provides a positive resolution of a question of Corso in a general set up of filtration.

Let L be a general length function for modules over a Noetherian ring R. We use L to define Hilbert series and polynomials for R[X]-modules. The leading term of any such polynomial is an invariant of R[X]-modules, which refines the algebraic entropy.

If I=( f 1 ,?? f r ) is an ideal in S=k[ x 1 ,?? x n ] , and f i are "general" elements of given degrees, there is a conjecture on the Hilbert series of S/I . We are considering the corresponding concepts in bigraded rings.

We prove the existence of HK density function for a pair (R,I) , where R is a N -graded domain of finite type over a perfect field and I⊂R is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair (R,I) , where, in addition R is standard graded. As one of the consequences we show that if G is a finite group scheme acting linearly on a polynomial ring R of dimension d then the HK density function f R G , m G , of the pair ( R G , m G ) , is a piecewise polynomial function of degree d−1 . We also compute the HK density functions for ( R G , m G ) , where G⊂S L 2 (k) is a finite group acting linearly on the ring k[X,Y] .

We give an explicit formula for the Hilbert-Poincaré series of the parity binomial edge ideal of a complete graph K n or equivalently for the ideal generated by all 2×2 -permanents of a 2×n -matrix. It follows that the depth and Castelnuovo-Mumford regularity of these ideals are independent of n .

When studying a graded module M over the Cox ring of a smooth projective toric variety X , there are two standard types of resolutions commonly used to glean information: free resolutions of M and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when X is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety X , we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties.

Let G be a finite simple connected graph on [n] and R=K[ x 1 ,…, x n ] the polynomial ring in n variables over a field K . The edge ideal of G is the ideal I(G) of R which is generated by those monomials x i x j for which {i,j} is an edge of G . In the present paper, the possible tuples (n,depth(R/I(G)),reg(R/I(G)),dimR/I(G),deg h(R/I(G))) , where deg h(R/I(G)) is the degree of the h -polynomial of R/I(G) , arising from Cameron--Walker graphs on [n] will be completely determined.