Avhandlingens titel

Around minimal Hilbert series problems for graded algebras

Respondent

Lisa Nicklasson

Opponent

Uwe Nagel (University of Kentucky)

Handledare

Samuel Lundqvist

Abstract

The Hilbert series of a graded algebra is an invariant that encodes the
dimension of the algebra's graded components. 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 $\ell$ such that multiplication by
any power of $\ell$ has maximal rank. Equivalently, the quotient
$A/(\ell^d)$ 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, give
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 homogeneous 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 a set of 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 $I \subset A$ 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.