We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel. 

We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property, and we give partial results on monomial almost complete intersections. We also study the reverse of the decomposition process – a gluing operation – which gives a way to construct monomial algebras with the Lefschetz properties.

An ideal I subset of k[x(1),..., x(n)] is said to have linear powers if I-k has a linear minimal free resolution, for all integers k > 0. In this paper, we study the Betti numbers of I-k, for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for d = 2,3 or n - 1, and the product of all ideals generated by s variables, for s = n - 1 or n - 2. We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

What can be said about the subalgebras of the polynomial ring, with minimal or maximal Hilbert function? This question was discussed in a recent paper by M. Boij and A. Conca. In this paper we study the subalgebras generated in degree two with minimal Hilbert function. The problem to determine the generators of these algebras transfers into a combinatorial problem on counting maximal north-east lattice paths inside a shifted Ferrers diagram. We conjecture that the subalgebras generated in degree two with minimal Hilbert function are generated by an initial Lex or RevLex segment.

The Hilbert series of a graded algebra is an invariant that encodes the dimension of the algebra's graded compontents. It can be seen as a tool for measuring the size of a graded algebra. This gives rise to the idea of algebras with a "minimal Hilbert series", among the algebras within a certain family.

Let A be a graded algebra defined as the quotient of a polynomial ring by a homogeneous ideal. We say that A has the strong Lefschetz property if there is a linear form L such that multiplication by any power of L has maximal rank. Equivalently, the quotient of A/(L^d) should have the smallest possible Hilbert series, for all d. According to a result by Richard P. Stanley from 1980, every monomial complete intersection in characteristic zero has the strong Lefschetz property. In the first and second paper of this thesis we study the analogue problem for positive characteristic. The main results of the two papers, combined with previous results by David Cook II, gives a complete classification of the monomial complete intersections in positive characteristic with the strong Lefschetz property.

In 1985 Ralf Fröberg conjectured a formula for the minimal Hilbert series of a polynomial ring modulo an ideal generated by homogeneous polynomials, given the number of variables, the number of generators of the ideal and their degrees. The conjecture remains an open problem, although it has been proved in a few cases. The questions studied in the third and fourth paper are inspired by this conjecture. In the third paper we search for the minimal Hilbert series of the quotient of an exterior algebra by a principal ideal. If the principal ideal is generated by an element of even degree, the Hilbert series is known by a result of Guillermo Moreno-Socías and Jan Snellman from 2002. In the third paper we give a lower bound for the series, in the case the generator has odd degree.

Instead of defining our algebra as a quotient, we may consider the subalgebra generated by certain elements. Given positive numbers u and d, which set of u homogeneous polynomials of degree d generates a subalgebra with minimal Hilbert series? This problem was suggested by Mats Boij and Aldo Conca in a paper from 2018. In the fourth paper we focus on the first nontrivial case, which is subalgebras generated by elements of degree two. We conjecture that an algebra with minimal Hilbert series is generated by an initial segment in the lexicographic or reverse lexicographic monomial ordering.

In the fifth paper we shift focus from Hilbert series to another invariant, namely the Betti numbers. The object of study are ideals I with the property that all powers I^k have a linear resolution. Such ideals are said to have linear powers. The main result is that the Betti numbers of A/I^k, if I is an ideal with linear powers, satisfy certain linear relations. When A/I has low Krull dimension, little extra information is needed in order to compute the Betti numbers explicitly.

We give a lower bound on the Hilbert series of the exterior algebra modulo a principal ideal generated by a generic form of odd degree and disprove a conjecture by Moreno-Socias and Snellman. We also show that the lower bound is equal to the minimal Hilbert series in some specific cases.

In this paper we study the Lefschetz properties of monomial complete intersections in positive characteristic. We give a complete classification of the strong Lefschetz property when the number of variables is at least three, which proves a conjecture by Cook II. We also extend earlier results on the weak Lefschetz property by dropping the assumption on the residue field being infinite, and by giving new sufficient criteria.

In this paper we classify the monomial complete intersections, in two variables, and of positive characteristic, which has the strong Lefschetz property. Together with known results, this gives a complete classification of the monomial complete intersections with the strong Lefschetz property.

There is a longstanding conjecture by Fröberg about the Hilbert series of the ring R∕I, where R is a polynomial ring, and I an ideal generated by generic forms. We prove this conjecture true in the case when I is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by m’th powers of generic forms of degree d≥2 gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.

An ideal I of a polynomial ring is said to have linear powers if I^k has a linear minimal free resolution, for all k. In this paper we study the Betti numbers of I^k, for ideals I with linear powers. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square free monomials of degree d, for d=2,3 or n−1, and the product of all ideals generated by s variables, for s=n−1 or n−2. We also study the generators of the Rees ideal, for ideals with linear powers. Especially, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This is related to a conjecture on matroids by White.