For the almost complete intersection ideals (x21, ... , x2n, (x1 + & centerdot; & centerdot; & centerdot; + xn)k), we compute their reduced Gr & ouml;bner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach. 

Consider a standard graded artinian k-algebra B and an extension of B by a new variable, A=B⊗kk[x]/(xd) for some d≥1. We will show how maximal rank properties for powers of a general linear form on A can be determined by maximal rank properties for different powers of general linear forms on B. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties B must have for all extensions B⊗kk[x]/(xd) to have the weak or the strong Lefschetz property.

Motivated by the foundational result that a monomial complete intersection has the strong Lefschetz property (SLP) in characteristic zero, it is natural to ask when monomial almost complete intersections have the SLP. In this paper, using the Hilbert series as a central tool, we investigate the strong Lefschetz property for certain monomial almost complete intersections: those with the non-pure-power generator having support in two variables, and those with symmetric Hilbert series. In the former case, we give a complete classification for when the SLP holds, and in the latter case, we prove that such algebras always have the SLP.

In this licentiate thesis, we consider questions related to the so called weak or strong Lefschetz properties. These are properties of a graded artinian algebra which asks for the existence of a linear form in the algebra such that the multiplication by that linear form, or multiplications by all powers of it, gives a map which always has full rank on the algebra.

In the general introduction, we give background material for understanding the Lefschetz properties, their definitions, and mention some other standard tools used when studying graded artinian algebras. We then introduce a selection of other common methods for proving that an algebra does or does not have the weak or strong Lefschetz property. This includes monomial orders, Macaulay's inverse system, preservation results and more.

Paper I is joint with Samuel Lundqvist and Lisa Nicklasson. It concerns binomial complete intersections of a specific form we call normal form. For a collection of binomials written on normal form, we associate a family of directed labelled graphs that let us determine several properties of such a family of binomials. We give a monomial basis for the associated algebra, its Macaulay dual generator, and a formula for the resultant. 

Paper II gives an answer to the following question. Given a fixed number of variables and fixed number of minimal generators that are possible for a quadratic artinian ideal, does there exist a quadratic artinian monomial ideal with those specifications having the strong Lefschetz property? The main result of this second paper is showing that this always has a positive answer when working over a field of characteristic zero, or large enough characteristic, by giving a concrete construction of such an ideal. Along the way, several interesting facts about the Hilbert series of these ideals are also established.

We consider homogeneous binomial ideals I=(f1,…,fn)𝐼=(𝑓1,…,𝑓𝑛) in K[x1,…,xn]𝐾[𝑥1,…,𝑥𝑛], where fi=aixdii−bimi𝑓𝑖=𝑎𝑖𝑥𝑖𝑑𝑖−𝑏𝑖𝑚𝑖 and ai≠0𝑎𝑖≠0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by xdii𝑥𝑖𝑑𝑖 for i=1,…,n𝑖=1,…,𝑛 form a vector space basis for the corresponding quotient, and we describe the Macaulay dual generator in terms of a directed graph that we associate to I. These two properties can be seen as a natural generalization of well-known properties for monomial complete intersections. Moreover, we give a description of the radical of the resultant of I in terms of the directed graph.

where RLex⁡(x_i⁢x_j) is the ideal generated by all the square-free monomials which are greater than or equal to x_i⁢x_j in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of I. Using a theorem of Lindsey [Proc. Amer. Math. Soc. 139 (2011), no. 1, 79–92], this allows for a short proof that any algebra defined by I has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran [Colloq. Math. 173 (2023), no. 1, 1–8], this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property.