Ändra sökning
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf
Subalgebras generated in degree two with minimal Hilbert function
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
Abstract [en]

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.

Nationell ämneskategori
Algebra och logik
Forskningsämne
matematik
Identifikatorer
URN: urn:nbn:se:su:diva-178735OAI: oai:DiVA.org:su-178735DiVA, id: diva2:1391065
Tillgänglig från: 2020-02-03 Skapad: 2020-02-03 Senast uppdaterad: 2020-02-04
Ingår i avhandling
1. Around minimal Hilbert series problems for graded algebras
Öppna denna publikation i ny flik eller fönster >>Around minimal Hilbert series problems for graded algebras
2020 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

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/(Ld) 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 Ik have a linear resolution. Such ideals are said to have linear powers. The main result is that the Betti numbers of A/Ik, 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.

Ort, förlag, år, upplaga, sidor
Stockholm: Department of Mathematics, Stockholm University, 2020. s. 22
Nyckelord
Graded algebras, Hilbert series, Lefschetz properties, Exterior algebra, Linear resolution
Nationell ämneskategori
Algebra och logik
Forskningsämne
matematik
Identifikatorer
urn:nbn:se:su:diva-178763 (URN)978-91-7911-010-9 (ISBN)978-91-7911-011-6 (ISBN)
Disputation
2020-04-07, Stockholm, 15:00 (Engelska)
Opponent
Handledare
Anmärkning

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript.

Tillgänglig från: 2020-02-28 Skapad: 2020-02-04 Senast uppdaterad: 2020-03-27Bibliografiskt granskad

Open Access i DiVA

Fulltext saknas i DiVA

Övriga länkar

arXiv:1911.11038

Sök vidare i DiVA

Av författaren/redaktören
Nicklasson, Lisa
Av organisationen
Matematiska institutionen
Algebra och logik

Sök vidare utanför DiVA

GoogleGoogle Scholar

urn-nbn

Altmetricpoäng

urn-nbn
Totalt: 1 träffar
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf