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Graded Betti Numbers and Hilbert Functions of Graded Cohen-Macaulay Modules
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

In this thesis we study graded Cohen-Macaulay modules and their possible Hilbert functions and graded Betti numbers. In most cases the Cohen-Macaulay modules we study are level modules.

In order to use dualization to study Hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences, introduced by Geramita and Lorenzini, to include Hilbert functions of certain level modules. As in the case of level algebras, a level sequence is cancellable, but now by dualization its reverse is also cancellable which gives a new condition on level sequences. We also give a characterization of the cancellable sequences.

We prove that a sequence of positive integers (h0, h1, . . . ,hc) is the Hilbert function of an artinian level module of codimension two if and only if hi−1 − 2hi + hi+1<= 0 for all 0 <= i <= c, where we assume that h−1 = hc+1 = 0. This generalizes a result already known for artinian level algebras.

Zanello gives a lower bound for Hilbert functions of generic level quotients of artinian level algebras. We give a new and more straightforward proof of Zanello’s result.

Conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by a positive rational number are given. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.

The h-vectors and graded Betti numbers of level modules up to multiplication by a rational number are studied. Assuming the conjecture, mentioned above, on the set of possible graded Betti numbers of Cohen-Macaulay modules we get a description of the possible h-vectors of level modules up to multiplication by a rational number. We determine,

again up to multiplication by a rational number, the cancellable h-vectors and the h-vectors of level modules with the weak Lefschetz property. Furthermore, we prove that level modules of codimension three satisfy the upper bound of the Multiplicity conjecture and that the lower bound holds if the module, in addition, has the weak Lefschetz property.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , iii, 13 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 2006:05
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4266ISBN: 978-91-7178-556-5 (print)OAI: oai:DiVA.org:kth-4266DiVA: diva2:11537
Public defence
2007-01-26, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100819Available from: 2007-01-12 Created: 2007-01-12 Last updated: 2010-08-19Bibliographically approved
List of papers
1. Artinian level modules and cancellable sequences
Open this publication in new window or tab >>Artinian level modules and cancellable sequences
2004 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 280, no 2, 610-623 p.Article in journal (Refereed) Published
Abstract [en]

In order to use dualization to study Hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences, introduced by Geramita and Lorenzini, to include Hilbert functions of certain artinian modules. As in the case of algebras a level sequence is cancellable, but now by dualization its reverse is also cancellable which gives a new condition on level sequences. We also give a characterization of the cancellable sequences involving Macaulay representations.

Keyword
Betti numbers; Cancellable sequence; Graded algebra; Graded dual; Graded module; Hilbert function; Level algebra; Level module; Level sequence; Lexicographic ideal; Lexicographic submodule
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6738 (URN)10.1016/j.jalgebra.2004.05.022 (DOI)000224037300012 ()2-s2.0-17344364544 (Scopus ID)
Note
QC 20100819 QC 20110914Available from: 2007-01-12 Created: 2007-01-12 Last updated: 2017-12-14Bibliographically approved
2. Artinian level modules of embedding dimension two
Open this publication in new window or tab >>Artinian level modules of embedding dimension two
2006 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 207, no 2, 417-432 p.Article in journal (Refereed) Published
Abstract [en]

We prove that a sequence of positive integers (h0, h1,...,h(c)) is the Hilbert function of an artinian level module of embedding dimension two if and only if h(i-1) - 2h(i) + h(i+1) <= 0 for all 0 <= i <= c, where we assume that h(-1) = h(c+1) = 0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6739 (URN)10.1016/j.jpaa.2005.10.002 (DOI)000240762000010 ()2-s2.0-33746348626 (Scopus ID)
Note
QC 20100819Available from: 2007-01-12 Created: 2007-01-12 Last updated: 2017-12-14Bibliographically approved
3. On Zanello's lower bound for generic quotients of level algebras
Open this publication in new window or tab >>On Zanello's lower bound for generic quotients of level algebras
2014 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 142, no 12, 4025-4028 p.Article in journal (Refereed) Published
Abstract [en]

We give a shorter and more straightforward proof of a theorem of Zanello on lower bounds for Hilbert functions of generic level quotients of artinian level algebras.

Keyword
Graded modules, level algebras, Hilbert functions
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6740 (URN)000344991600001 ()2-s2.0-84922963060 (Scopus ID)
Note

QC 20100819. QC 20150113. Updated from manuscript to article in journal.

Available from: 2007-01-12 Created: 2007-01-12 Last updated: 2017-12-14Bibliographically approved
4. Graded betti numbers of Cohen-Macaulay modules and the multiplicity conjecture
Open this publication in new window or tab >>Graded betti numbers of Cohen-Macaulay modules and the multiplicity conjecture
2008 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 78, no 1, 85-106 p.Article in journal (Refereed) Published
Abstract [en]

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be nonh negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions is known, that is: for modules of codimension two, for Glorenstein algebras of codimension three and for complete intersections. The motivation for proposing the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.

Keyword
ALGEBRA STRUCTURES; RESOLUTIONS; CODIMENSION; IDEALS; THEOREMS; PURE
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6741 (URN)10.1112/jlms/jdn013 (DOI)000258238800006 ()2-s2.0-47949100888 (Scopus ID)
Note
QC 20100819Available from: 2007-01-12 Created: 2007-01-12 Last updated: 2017-12-14Bibliographically approved
5. Graded Betti numbers and h-vectors of level modules
Open this publication in new window or tab >>Graded Betti numbers and h-vectors of level modules
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study h-vectors and graded Betti numbers of level modules up to multiplication by a rational number. Assuming a conjecture on the possible graded Betti numbers of Cohen-Macaulay modules we get a description of the possible h-vectors of level modules up to multiplication by a rational number. We also determine, again up to multiplication by a rational number, the cancellable h-vectors and the h-vectors of level modules with the weak Lefschetz property. Furthermore, we prove that level modules of codimension three satisfy the upper bound of the Multiplicity conjecture of Herzog, Huneke and Srinivasan, and that the lower bound holds if the module, in addition, has the weak Lefschetz property.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-6742 (URN)
Note
QC 20100819. Uppdaterad från Artikel till Manuskript 20100819.Available from: 2007-01-12 Created: 2007-01-12 Last updated: 2010-08-19Bibliographically approved

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