Lefschetz properties and Jordan types of Artinian
algebras
NASRIN ALTAFI
Doctoral Thesis in Mathematics
Stockholm, Sweden 2020
ISBN 978-91-7873-589-1 SWEDEN Akademisk avhandling som med tillst˚and av Kungl Tekniska h¨ogskolan framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen i matema-tik fredagen den 18 september 2020 klockan 14.00 i Kollegiesalen, Brinellv¨agen 8, Kungl. Tekniska h¨ogskolan, Stockholm.
© Nasrin Altafi, 2020
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Abstract
This thesis contains six papers concerned with studying the Lefschetz properties and Jordan types of linear forms for graded Artinian algebras. Lefschetz properties and Jordan types carry information about the ranks of multiplication maps by linear forms on graded Artinian algebras. A graded Artinian algebra A is said to have the weak Lefschetz property (WLP) if multiplication map by some linear form ` ∈ A1 on A has maximal rank in
all degrees. If it holds for all powers of ` the algebra is said to have the strong Lefschetz property (SLP). Jordan type P`,Ais a partition determining
the Jordan block decomposition for the multiplication map by ` on A. It is a finer invariant than the WLP and SLP, which determines the ranks of multiplication maps by all powers of ` on A.
Papers A and B concern the study of the Lefschetz properties of Artinian algebras quotient of a polynomial ring in n ≥ 3 variables over a field of characteristic zero by a monomial ideal I generated in a single degree d ≥ 2. Paper A studies the connection between the Lefschetz properties of such algebras and their minimal free resolutions, namely the number of linear steps in their resolutions. Paper B consists of two parts. The first part provides sharp lower bounds for the Hilbert function in degree d of such algebras failing the WLP. The second part of Paper B deals algebras quotients of polynomial rings by ideals generated by forms of the same degree and invariant under action of a cyclic group. The main result of this part classifies such algebras satisfying the WLP in terms of the representation of the action.
Papers C and D both deal with determining the Jordan type partitions of linear forms for graded Artinian algebras with codimension two. Paper C concerns the problem for complete intersection Artinian algebras over an algebraically closed field of characteristic zero or large enough. The results of Paper C classify partitions of an integer n that occur as Jordan type parti-tions for Artinian complete intersection algebras and some linear forms. Such classifications are provided in terms of the numerical conditions of the par-titions. Also for a given Hilbert function of such algebras, the Jordan type partitions are completely determined by which higher Hessians vanish at the point corresponding to the linear form. Some combinatorial invariants of such partitions, namely branch label or hook code, have been studied in this paper as well.
Paper D concerns the generalization of the results of Paper C. The family GT of graded Artinian quotients A = S/I of S = k[x, y], having arbitrary
Hilbert function hA = T has been studied. The cell V(EP) corresponding
to a partition P having diagonal lengths T is comprised of all ideals I in S whose initial ideal is the monomial ideal EP determined by P . These cells
give a decomposition of the variety GT into affine spaces. The main result
of Paper D determines the generic number κ(P ) of generators for the ideals in each cell V(EP); generalizing a result of Paper C. In particular, partitions
having the generic number of generators for an ideal defining an algebra A in GT are determined.
Paper E concerns the SLP of Artinian Gorenstein algebras via studying the higher Hessians of dual generators. The main result of this paper
charac-terizes the Hilbert functions of Artinian Gorenstein algebras having arbitrary codimension satisfying the SLP. It proves that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the SLP if and only if h is an SI-sequence. This is done by studying the higher Hessians of the dual generator of an Artinian Gorenstein quotient of the co-ordinate ring of a set of points in the projective space. Using this approach, we provide families of Artinian Gorenstein algebras obtained by points in the projective plane satisfying the SLP.
Paper F concerns the study of Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. This paper in-troduces rank matrices of linear forms for such algebras. There is a 1-1 corre-spondence between rank matrices and Jordan degree types. The main result of this paper classifies all matrices that occur as the rank matrix for some Artinian Gorenstein algebra A of codimension three and a linear form ` such that `3 = 0. As a consequence, we prove that Jordan types with parts of
v
Sammanfattning
Denna avhandling best˚ar av sex artiklar om Lefschetzegenskaper och Jor-dantyper av linj¨araformer f¨or graderade artinska algebror. Lefschetzegenska-per och JordantyLefschetzegenska-per inneh˚aller information om avbildningarnas rang f¨or linj¨ara former p˚a graderade artinska algebror. En graderad artinsk algebra A s¨ags ha den svaga Lefschetzegenskapen (WLP) om multiplikationen med n˚agon linj¨arform ` ∈ A1 p˚a A har maximal rang i alla grader. Om detta g¨aller f¨or
alla potenser av ` s¨ags algebran ha den starka Lefschetzegenskapen (SLP). Jordantyp P`,A ¨ar en partition som best¨ammer Jordanblockuppdelningen f¨or
multiplikation av ` p˚a A. Det ¨ar en finare invariant ¨an WLP och SLP, som best¨ammer rangen f¨or multiplikation med alla potenser av ` p˚a A.
Artiklarna A och B behandlar studiet av Lefschetzegenskaper hos artins-ka algebror som ¨ar kvoter av en polynomring i n ≥ 3 variabler ¨over ett kropp med karakteristik noll med ett monomalideal I som genereras av ett enda grad d ≥ 2. Artikel A studerar kopplingen mellan Lefschetzegenskaper f¨or s˚adana algebror och deras minimala fria uppl¨osningar, n¨amligen antalet linj¨ara steg i deras uppl¨osningar. Artikel B best˚ar av tv˚a delar. Den f¨orsta delen tillhandah˚aller skarpa undre gr¨anser f¨or Hilbertfunktionen i grad d f¨or s˚adana algebror som inte har WLP. Den andra delen av Artikel B handlar om artinska algebror som ¨ar kvoter av en polynomring med ett ideal I som genereras av former och invariant under verkan av en cyklisk grupp. Huvud-resultatet av denna del klassificerar s˚adana algebror som har WLP i termer av representationen av gruppen.
Artiklarna C och D behandlar att best¨amma Jordantyppartitionen f¨or linj¨ara former i graderade artinska algebror av kodimension tv˚a. Artikel C avser problemet f¨or artinska algebror som ¨ar fullst¨andiga sk¨arningar ¨over ett algebraisk sluten kropp av karakteristisk noll eller tillr¨ackligt stor. Resultaten i Artikel C klassificerar partitioner av ett heltal n som f¨orekommer som parti-tioner av Jordantyp f¨or artinska algebror som ¨ar fullst¨andiga sk¨arningar och n˚agra linj¨araformer. S˚adana klassificeringar tillhandah˚alls med avseende p˚a de numeriska villkoren f¨or partitionerna. ¨Aven f¨or en fix Hilbertfunktion av s˚adana algebror best¨ams Jordantypen fullst¨andigt f¨or vilka h¨ogre Hessianerna ¨
ar noll vid den punkt som motsvarar den linj¨arformen. Vissa kombinatoriska invarianter av s˚adana partitioner, n¨amligen grenetiketter eller krokcode, har studerats i denna artikel.
Artikel D behandlar generalisering av resultaten fr˚an Artikel C. Familjen GTf¨or graderade artiniska algebror som ¨ar kvotienter A = S/I av S = k[x, y],
med arbitr¨ar Hilbertfunktion hA = T har studerats. Cellen V(EP)
motsva-rande en partition P med diagonala l¨angder T best˚ar av alla ideal I i S vars begynnelseideal ¨ar det monomidealet EP best¨amt av P . Dessa celler ger en
uppdelning av varietet GT i affina rummet. Huvudresultatet f¨or Artikel D
best¨ammer det generiska talet κ(P ) f¨or generatorer f¨or idealen i varje cell V(EP) vilket generaliserar resultat av Artikel C. I synnerhet best¨ams
parti-tioner som har det generiska antalet generatorer f¨or ett ideal som definierar en algebra A i GT.
Artikel E avser SLP f¨or artinska Gorenstein algebror genom att studera de h¨ogre Hessianerna av duala generatorer. Huvudresultatet av denna artikel
klassificerar Hilbertfunktionerna i artinska Gorensteinalgebror som har god-tycklig kodimension och SLP. Det visar att en f¨oljd h av ickenegativa heltal ¨
ar Hilbertfunktionen f¨or n˚agon artinsk Gorensteinalgebra med SLP om och endast om h ¨ar en SI-f¨oljd. Detta g¨ors genom att studera de h¨ogre Hessianer-na av dualgeneratorn av en artinsk GorensteiHessianer-nalgebra som ¨ar koordinatringen av n˚agra punkter i det projektiva rummet. Med hj¨alp av denna metod till-handah˚aller vi familjer av artinska Gorensteinalgebror som f˚as fr˚an genom punkter i det projektiva planet och med SLP.
Artikel F avser studie av Jordantyper av linj¨arformer f¨or graderade artins-ka Gorensteinalgebror som har godtycklig kodimension. Detta artikeln intro-ducerar rangmatriser av linj¨arformer f¨or s˚adana algebror. Det finns en 1-1 korrespondens mellan rangmatriser och Jordangradtyper. Huvudresultatet av denna artikel klassificerar alla matriser som f¨orekommer som rangmatris f¨or n˚agon artinsk Gorensteinalgebra A av kodimension tre och en linj¨arform ` s˚a att `3= 0. Som en konsekvens av detta bevisar vi att Jordantyper vars delar
Contents
Contents vii
Acknowledgments ix
I
Introduction and summary
1
1 Introduction 3
2 Background and previous results 7
1 Hilbert functions and minimal free resolutions . . . 7
2 Artinian Gorenstein algebras and their h-vectors . . . 10
3 Macaulay inverse systems . . . 11
4 The Lefschetz properties . . . 13
5 Higher Hessians and Lefschetz properties . . . 14
6 Jordan types of graded Artinian algebras . . . 17
3 Summary of results 21
References 41
II Scientific papers
Paper A
Lefschetz properties of equigenerated monomial algebras with almost linear resolution.
(joint with N. Nemati)
Communications in Algebra, 48:4, 1499-1509.
Paper B
The weak Lefschetz property of equigenerated monomial ideals. (joint with M. Boij)
Journal of Algebra, Volume 556, 2020, Pages 136-168.
Paper C
Complete intersection Jordan types in height two. (joint with A. Iarrobino and L. Khatami)
Journal of Algebra, Volume 557, 2020, Pages 224-277.
Paper D
Jordan types for graded Artinian algebras in height two. (joint with A. Iarrobino, L. Khatami and J. Yam´eogo) Preprint: http://arxiv.org/abs/2006.11794.
Paper E
Hilbert functions of Artinian Gorenstein algebras with the strong Lefschetz property.
Preprint: http://arxiv.org/abs/2007.10684.
Paper F
Jordan types with small parts for Artinian Gorenstein algebras of codimension three.
Acknowledgments
First and foremost, I wish to express my deepest gratitude toward my advisor, Mats Boij, for all his support, patience, and guidance. His invaluable knowledge and insights made our meetings always very inspiring and exciting. I would like to thank my co-adviser, Sandra Di Rocco, for providing me with help when needed.
I am using this opportunity to thank all members of the algebra and geometry group and problem solving seminar group for all the discussions during weekly seminars and fikas. I would like to express my gratitude to all organizers and participants of the Lefschetz workshops in Stockholm, Levico-Terme, and Luminy for inviting me and giving me the opportunity to work with such knowledgeable researchers in the field.
In the work leading up to this thesis, I have also had the pleasure of collaborating with Anthony Iarrobino, Leila Khatami, and Joachim Yam´eogo. I am very grateful to them for generously sharing their knowledge and experience.
I wish to thank Afshin, Eric, and Samuel for all the interesting discussions we have had. In the same spirit, I would like to thank my other friends and colleagues who made my time much more enjoyable: David, Ehsan, Fatemeh, Gerard, Jeroen, Julian, Lenna, Lisa, Martina, Milo, Oliver, Parikshit, Thomas, and Tomas. With special thanks to my best friends Maryam and Mahtab, who made this journey much easier.
I wish to express my deep and sincere thankfulness to Pary, Masood, and Mahsa for all their support and inspiration. Thank you for always believing in me and trusting me in all the decisions I have ever made.
Finally, thank you Navid for all your support and patience, especially during the preparation of this thesis. Thank you for always being there for me. You made my life much more meaningful and fun.
Part I
Introduction and summary
1
Introduction
Starting with the broad topic, this thesis is in the area of commutative algebra and algebraic geometry. Commutative algebra can be seen as the algebraic approach to geometrical problems, that sometimes can be better understood in the algebraic terms. Techniques from commutative algebra can be applied to tackle problems in algebraic geometry and vice-versa. Among the most important objects in commu-tative algebra are polynomial rings over a field, and their quotients. The coordinate rings of varieties are quotients of polynomial rings by ideals. The geometric study of algebraic varieties can be transferred into algebra via this connection. Much of the modern development of the commutative algebra emphasizes graded rings, which correspond to projective varieties.
In the nineteenth century, D. Hilbert solved the problem of existence of finite systems of generators for rings of invariants in a wide range of cases. This problem was one of the most concerned problems in his era. Along with solving this problem, Hilbert proved three breakthrough results in [25] and [26]. The basis theorem, syzygy theorem and that the Hilbert function is determined by only finitely many of its values. Hilbert’s basis theorem directly implies the problem of invariant theory. This result also provides a construction for a free resolution of any finitely generated graded module over a Noetherian ring. The existence of a finite free resolution is guaranteed by Hilbert’s syzygy theorem. His motivation to prove both theorems was to compute the Hilbert function of a module via comparing it with free modules. These results have played an enormous role in determining the shape of commutative algebra.
Any graded module is a direct sum of vector spaces corresponding to the graded pieces of the module; the Hilbert function gives the dimensions of these vector spaces. F. S. Macualay’s celebrated result from 1927 [36] provides a complete classification of the Hilbert functions of standard graded algebras. A very broad and fascinating problem is to describe the algebraic, geometric, and homological consequences forced on the algebras by conditions on the Hilbert function. This thesis is centered on the ubiquitous concern: what conditions the structure of the algebra imposes on the Hilbert function, and conversely what we can deduce about the algebra from knowledge of the Hilbert function. Specifically, this thesis deals with the study of Lefschetz properties and Jordan types of linear forms for graded
Artinian (0-dimensional) algebras.
The study of Lefschetz properties is of importance in algebra, combinatorics, ge-ometry and topology; and it has been crucial in the investigation of graded Artinian algebras. The study of Lefschetz properties has its origin in the hard Lefschetz the-orem. The hard Lefschetz theorem [34] was a breakthrough in algebraic topology and geometry. For a smooth complex projective variety, the cup product with pow-ers of the hyperplane class gives an isomorphism between its cohomology classes. In 1971, P. McMullen [38] posed a conjecture that completely characterizes the f -vectors (vector giving the number of i-dimensional faces) of simplicial polytopes. The f -vector and the h-vector (the Hilbert function of Artinian algebra) contain equivalent combinatorial information about the simplicial complex, and they are related by linear relations. The hard Lefschetz theorem imposes strong constraints on the Hilbert function; this plays a significant role in R. Stanley’s proof of the necessity part of the McMullen’s conjecture [44]. The sufficiency conditions of Mc-Mullen’s conjecture was proved by L. Billera and C. Lee [5]. The g-conjecture is a natural generalization of this theorem to all simplicial spheres and it is one of the main open problems in the theory of f - vectors, or equivalently h-vectors.
The investigation of the Lefschetz properties of Artinian algebras in general was suggested in mid 1980’s. A standard graded Artinian algebra A is said to satisfy the weak Lefschetz property (WLP) if the multiplication map by a generic linear form ` on each graded piece of A has maximal rank. Furthermore, A is said to satisfy the strong Lefschetz property (SLP) if the same holds for all powers of a generic linear form `. The hard Lefschetz Theorem implies that the cohomology ring of a smooth complex projective variety has the SLP.
Over the last years, this topic has attracted increasing attention from math-ematicians of different areas, such as commutative algebra, algebraic geometry, combinatorics, algebraic topology and representation theory. One of the main fea-tures of the WLP and the SLP is their ubiquity and the quite surprising and still not completely understood relations with other themes, including linear configu-rations, interpolation problems, vector bundle theory, plane partitions, differential geometry, coding theory among others. As one of the more recent applications of Lefschetz properties is combinatorics and algebraic geometry, one can mention the work of K. Adiprasito, J. Huh, and E. Katz [1]; they prove that the hard Lefschetz theorem holds for a commutative ring associated to arbitrary matroid.
The study of the Lefschetz properties of Artinian algebras is closely related to one of the most intriguing conjectures, the Fr¨oberg’s conjecture. In 1985, R. Fr¨oberg [15] conjectured that an Artinian quotient of a polynomial ring by an ideal generated by generic forms has the least possible Hilbert function in every degree. It is proved by J. Migliore, R. Mir´o-Roig and U. Nagel [40] that if the Fr¨oberg’s conjecture holds for Artinian quotients of polynomial rings with n variables, then all Artinian quotients of polynomial rings with n + 1 variables have the WLP.
For linear forms of Artinian algebras that fail to be Lefschetz elements, it is natural to ask how far they are from being the weak or Strong Lefschetz element. This motivates the present thesis to concern the study of Jordan types of Artinian
5
algebras. The Jordan type of a linear form ` for a graded Artinian algebra A is the partition determining the Jordan block decomposition of the multiplication by ` on A. The weak and the strong Lefschetz properties of algebra A can be determined from the Jordan type of a generic linear form of A.
The hard Lefschetz Theorem implies that the Hilbert function of the cohomology rings of compact K¨ahler manifolds are unimodal and symmetric. The cohomology rings of compact K¨ahler manifolds are Poincar´e duality algebras. It is known [16] that commutative Poincar´e duality algebras are exactly Gorenstein algebras. It is natural to investigate the study the Hilbert functions and the Lefschetz properties of Gorenstein algebras. They are geometrically interesting as well; the coordinate rings of all plane curves and even more generally all complete intersections are Goren-stein. The h-vectors of Gorenstein algebras with embedding dimension less than four are completely characterized [43] and they are exactly SI-sequences (Stanley-Iarrobino sequences). A. (Stanley-Iarrobino and R. Stanley independently conjectured [42] that a vector of non-negative integers is a h-vector of some Gorenstein algebra if it is unimodal, symmetric and its first difference is a h-vector (by Macualay’s charac-terization). There are examples of Gorenstein algebras with embedding dimensions at least five and non-unimodal h-vectors and hence non-SI h-vectors. It is still open if the h-vectors of Gorenstein algebras with embedding dimension four are SI-sequences. One of the main contributions of this thesis to the field of commutative algebra pertains determining the Lefschetz properties and Jordan types of Artinian Gorenstein algebras.
2
Background and previous results
In this section we summarize background information about the topic studied in this thesis. We outline the historical developments in the literature that led to the current topic of research.
We first fix our setting which will be used throughout this section. How-ever, some parts of this introduction make sense in more general settings. Let S = k[x1, . . . , xn] be a polynomial ring where k is a field. We assume k has
charac-teristic zero unless stated otherwise. We consider S as a standard graded ring with deg(xi) = 1 for all i. A polynomial f is homogeneous of degree d if f ∈ Sd, that is,
all monomial terms of f have the same degree d. An ideal I is homogeneous ( or graded) if it has a system of homogeneous generators. In this thesis I stands for a graded ideal in S. We denote the quotient S/I by A which inherits the grading by Ai= Si/Ii for all i.
1
Hilbert functions and minimal free resolutions
A famous and important numerical invariant of a finitely generated graded module over S is the Hilbert function. It gives the dimension of the module as a k-vector space in various degrees. More precise definition is the following.
Definition 2.1. Let M = ⊕d∈ZMd be a finitely generated graded S-module. The
Hilbert function of M is the function Z → Z defined by hM(i) := dimk(Mi). The
Hilbert series of M is the generating function HM(t) =
X
i∈Z
hM(i)ti.
In fact, hM is determined by finitely many of its value. D. Hilbert proved that
for large values of i this function is a polynomial function, [25]. This polynomial is called the Hilbert polynomial of M .
It is worth noticing that A = S/I is a coordinate ring of a projective algebraic set X = Z(I) ⊂ Pn−1k = ProjS. The homogeneous part Ii is a linear subspace of
the n+i−1n−1 -dimensional k-vector space Si. The dimension of Ii is the number of
independent hypersurfaces of degree i containing X. The Hilbert function hA(i) is
equal to the number of independent functions of degree i defined on the variety X. Example 2.2. For an algebra A = S/I where S = C[x1, x2, x3] and I = (x31, x22, x23),
we have that
dimC(A0) = 1, dimC(A1) = 3, dimC(A2) = 4, dimC(A3) = 3, dimC(A4) = 1,
and dimC(Ai) = 0, for all i ≥ 5. So the Hilbert series of A is equal to
HA(t) = 1 + 3t + 4t2+ 3t3+ t4. (2.1)
The study of Hilbert functions and of resolutions are closely related. For in-stance, the Hilbert series can be computed from its graded free resolution.
The study of free resolutions is an important and central topic in algebra. They are useful tools for studying modules over finitely generated graded k-algebras.
For a graded S-module M and integer a ∈ Z, the shifted module M (a) is the graded module with M (a)b = Ma+b. If ϕ : M −→ N is homogeneous of degree
b ∈ Z, then the induced map ˜ϕ : M (−b) −→ N is homogeneous of degree 0. Given homogeneous elements mi ∈ M of degree ai that generate M as an
S-module, we may define a map from the graded free module F0= ⊕iS(−ai) onto M
by sending the i-th generator to mi (which is a homogeneous map). Let M1⊂ F0
be the kernel of this map F0 −→ M . By the Hilbert Basis Theorem [26], M1 is
also a finitely generated module. The elements of M1 are called syzygies of M .
Choosing finitely many homogeneous syzygies that generate M1, we may define a
map from a graded free module F1 to F0 with image M1. Continuing in this way
we may construct a sequence of maps of graded free modules, called a graded free resolution of M and denoted by F .
F : · · · −→Fi ϕi
−→Fi−1−→ · · · −→F1 ϕ1
−→F0.
It is an exact sequence of degree zero maps between graded free modules such that the cokernel of ϕ1 is M .
Theorem 2.3 (Hilbert’s Syzygy Theorem). Any finitely generated graded S-module M has a finite graded free resolution
F : 0−→Fm ϕm
−→Fm−1−→ · · · −→F1 ϕ1
−→F0.
Moreover, we may take m ≤ n, the number of variables in S.
For a module M the free resolution F is said to be a minimal free resolution if for each i the map ϕimaps a basis of Fito a minimal set of generators of the image
of ϕi. Now suppose that F : · · · → F1 → F0 is the minimal free resolution of an
S-module M , where Fi = ⊕jS(−j)βi,j, that is Fi requires βi,j minimal generators
1. HILBERT FUNCTIONS AND MINIMAL FREE RESOLUTIONS 9
sometimes written βi,j(M ). The Hilbert series and graded Betti numbers of M are
related in the following way HM(t) = Pn i=0 P j∈N(−1) iβ i,jtj (1 − t)n . (2.2)
There are positive integers (h0, h1, . . . , hs) such that
HM(t) =
h0+ h1t + · · · + hsts
(1 − t)d ,
where d is the Krull dimension of M . The vector hM = (h0, h1, . . . , hs) is called
the h-vector of M . So the Hilbert function of M is determined by it h-vector and its Krull dimension.
An algebra A = S/I is Artinian if its Krull dimension is zero. So for Artinian algebra the Hilbert function is identified by its h-vector. Artinian rings are of great interest. In fact, many problems of local rings often reduce to problems of artinian rings. This thesis is concerned with the study of properties of graded Artinian algebras.
Example 2.4. Let A = S/I be the graded algebra of Example 2.2 where S = C[x1, x2, x3] and I = (x31, x
2 2, x
2
3). The minimal free resolution of A is equal to
0−→S(−7)−→S(−4) ⊕ S(−5)2−→S(−2)2⊕ S(−3)−→S.
So β0,0 = 1, β1,2 = 2, β1,3 = 1, β2,4 = 1, β2,5 = 2 and β3,7 = 1 Using (2.2) we get
that
HA(t) =
1 − 2t2− t3+ t4+ 2t5− t7
(1 − t)3 = 1 + 3t + 4t
2+ 3t3+ t4.
Notice that A is Artinian therefore the h-vector of A is equal to hA= (1, 3, 4, 3, 1)
which is equal to what we obtained in Example 2.2.
A famous theorem of Macaulay [36] provides bound on the growth of Hilbert function. To state Macaulay’s theorem we need following definitions and notations. Let h and i be positive integers. Then h can be written uniquely in the following form h =mi i +mi−1 i − 1 + · · · +mj j , (2.3)
where mi> mi−1> · · · > mj ≥ j ≥ 1. This expression for h is called the i-binomial
expansion of h. Also define hhii=mi+ 1 i + 1 +mi−1+ 1 i + · · · +mj+ 1 j + 1 , (2.4)
Theorem 2.5 (Macaulay [36]). There exists a standard graded algebra A such that hA= (h0, h1, . . . ) if and only if h0= 1 and hi+1≤ hhiii for all i ≥ 1.
Such a sequence of integers satisfying the conditions in Macaulay’s theorem is called an O-sequence.
The problem of characterizing the Hilbert functions of different classes of alge-bras is a very interesting problem in the field. One of the classes of algealge-bras that have been studied a lot in this manner is Gorenstein algebras.
2
Artinian Gorenstein algebras and their h-vectors
One of the very interesting and important classes of algebras are Gorenstein alge-bras. They have lots of important properties and there exist useful tools to study them although they are far from being completely understood. Gorenstein algebras are geometrically very interesting. The coordinate rings of all plane curves and even more generally all complete intersections are Gorenstein. The Stanley-Reisner rings of homology spheres are Gorenstein. The study of Gorenstein algebras could be approached by first trying to understand Artinian Gorenstein algebras. In fact, given any Gorenstein algebra of arbitrary dimension one can always mod out by ideal generated by a system of parameters and obtain an Artinian Gorenstein alge-bra. In the paper by Bass [3], a Gorenstein local ring is defined to be a local ring that has finite self injective dimension.
For a graded algebra A = S/I the socle of A is defined to be SocA = 0 : m, where m is the maximal ideal of S. For an Artinian algebra A, the socle degree is said to be the largest integer d such that md6= 0; i.e. md
* I. The type of A is the dimension of its socle, that is dimkSocA.
Definition 2.6. An Artinian algebra A = Ld
i=0Ai is called a Poincar´e duality
algebra if dimkAd = 1 and the map
Ai× Ad−i→ Ad
(α, β) → αβ
defined by the multiplication is a perfect paring for every i = 0, . . . , d.
Graded Artinian Gorenstein algebras are the same as commutative Poincar´e duality algebras.
Proposition 2.7. [37, Proposition 2.1] A graded Artinian k-algebra A is Goren-stein if and only if is a commutative Poincar´e duality algebra.
Let hA= (1, h1, . . . , hd) be the h-vector of a graded Artinian Gorenstein algebra.
Then by the above result we conclude that hd= 1 and it is symmetric about d2; i.e.
hi= hd−ifor every i = 0, . . . , d.
Recall the relation between Hilbert functions and graded Betti numbers. For an Artinian Gorenstein algebra the Betti numbers are also symmetric and the last
3. MACAULAY INVERSE SYSTEMS 11
Betti number is equal to one. Moreover, the minimal free resolution of such an algebra is self dual.
One of the very interesting open problems in the field is to characterize the Hilbert functions of Artinian Gorenstein algebras. Symmetry is a necessary but not sufficient condition for the Hilbert function of such algebras. R. Stanley and A. Iarrobino independently posed a conjecture which characterizes the Hilbert func-tions of Artinian Gorenstein algebras, see [42].
Conjecture 2.8. There exists a Gorenstein algebra with Hilbert function h = (h0, h1, . . . , hd) if and only if h satisfies the following three conditions
(i) h is unimodal; h0≤ h1≤ · · · ≤ hi≥ hi+1≥ · · · ≥ hd,
(ii) h is symmetric,
(iii) ∆h = (h0, h1− h0, . . . , ht− ht−1) is an O-sequence, where t = min{i | hi ≥
hi+1}.
A sequence satisfying the three conditions in the above conjecture is called an SI-sequence. R. Stanley [43] showed that the conjecture is true for Gorenstein algebras with embedding dimension at most three, h1≤ 3.
In general there are vectors which occur as the h-vectors of Gorenstein algebras that are not SI-sequences. R. Stanley [43] gave the first example with h-vector h = (1, 13, 12, 13, 1). Later D. Bernstein and A. Iarrobino [4] and M. Boij and D. Laksov [7] provided examples of non-unimodal (and therefore non SI) Gorenstein sequences with h1 = 5. It is not known if the conjecture holds for Gorenstein
algebras with h1 = 4. However, SI-sequences provide a broad class of Hilbert
functions that occur for Gorenstein algebras which we will discuss it in the next sections.
3
Macaulay inverse systems
The aim of this section is to investigate the theory of Inverse System. In 1916 Macaulay established a one-to-one correspondence between Artinian Gorenstein algebras and suitable polynomials. This correspondence has been deeply studied in the homogeneous case, among other authors, by A. Iarrobino in a series of papers. For more details and list of references we refer to [29] and [17].
We first introduce the Macaulay inverse system in a more general setting for Artinian algebras (not necessarily Gorenstein algebras).
Let S = k[x1, . . . , xn] be a polynomial ring and k is a field of characteristic zero
and R = k[X1, . . . , Xn] be a new polynomial ring. We make R into an S-module by
defining the action of S on R by partial differentiation; xj◦ F = ∂F/∂Xj, for any
spaces:
h, i : S × R −→ k hf, gi → (f ◦ g)(0).
For homogeneous ideal I ⊂ S we define the inverse system I−1⊂ R as I−1:= {g ∈ R | f ◦ g = 0, for all f ∈ I}.
Let deg Xi= 1, for all i, and consider R = ⊕i≥0R−isuch that R−iis the k-vector
space spanned by forms in R with degree i. This defines a grading on R as S-module and so the inverse system of a homogeneous ideal is a graded S-S-module. By denoting by Riinstead of R−i, we are abusing notation. In general I−1is not an
ideal. We observe that when I is a monomial ideal the inverse system module (I−1)d
is generated by the monomials in Rdcorresponding to the monomials in Sdbut not
in Id. The Hilbert function of Artinian algebra A = S/I can be computed by its
Macaulay dual module; hA(i) = dimk(I−1)i. There is a one-to-one correspondence
between graded Artinian algebras and finitely generated graded S-submodules M of R.
For polynomial rings over a field of arbitrary characteristic, the Macaulay inverse system is defined by contraction action. This thesis considers field of characteristic zero unless stated otherwise.
Theorem 2.9 (Macaulay duality). Let S = k[x1, . . . , xn] be a polynomial ring
with n variables over a field k of characteristic zero. There is an order-reversing bijection between the set of finitely generated S-submodules of R = k[X1, . . . , Xn]
and the set of Artinian ideals of S given by: if M is an S-submodule of R then AnnS(M ) = (0 :S M ), and I−1= (0 :RI) for an ideal I ⊂ S.
For Artinian Gorenstein algebras the correspondence is given in the following theorem.
Theorem 2.10. [37, Theorem 2.1] Let A = S/I be a graded Artinian algebra. Then A is Gorenstein if and only if there exists a homogeneous polynomial F ∈ R = k[X1, . . . , Xn] such that I = AnnS(F ).
The polynomial F in the above theorem is called the Macaulay dual generator, for short dual generator, for graded Artinian Gorenstein algebra A.
Example 2.11. The Artinian algebra A = S/I given in Example 2.2 is Gorenstein and even complete intersection algebra with socle degree 4. Using Theorem 2.10 there is a polynomial of degree 4 in the dual ring R = C[X1, X2, X3] such that
I = AnnS(F ). One can easily check that for F = X12X2X3 the dual generator of
A; we have I = hx31, x22, x23i = AnnS(F ).
As an example of Artinian Gorenstein algebra that is not a complete intersec-tion, consider the polynomial G = X14+ X24+ X34. Then
AnnS(G) = hx1x2, x1x3, x2x3, x41− x 4 2, x 4 1− x 4 3, x 4 2− x 4 3i,
4. THE LEFSCHETZ PROPERTIES 13
and its Hilbert function is equal to (1, 3, 3, 3, 1).
4
The Lefschetz properties
The study of Lefschetz properties has its origin in the Hard Lefschetz theorem for smooth complex projective varieties [35]. The investigation of the Lefschetz proper-ties of Artinian local rings was first suggested in 1985 in a seminar on combinatorics and commutative algebra. We first give the formal definitions.
Definition 2.12. Let A = S/I be a graded Artinian ideal. We say that A has the weak Lefschetz property (WLP) if there is a linear form ` ∈ A1 such that, for all
integers i, the multiplication map
×` : Ai−→ Ai+1
has maximal rank, i.e. it is injective or surjective. In this case the linear form ` is called a weak Lefschetz element of A. Algebra A is said to satisfy the strong Lefschetz property (SLP) if there is a linear form ` ∈ A1 such that, for all integers
i and j the multiplication map
×`j : A
i−→ Ai+j
has maximal rank, i.e. it is injective or surjective. In this case the linear form ` is called a strong Lefschetz element.
Lefschetz properties are open conditions in a sense that Lefschetz elements of Artinian algebra A form a Zariski open subsets (possibly empty) of projective space ProjS = Pn−1. Therefore, algebra A has the WLP (SLP) if and only if a generic
linear form ` is the weak Lefschetz element (strong Lefschetz element).
Using the Macaulay inverse system of Artinian algebras we may define the Lefschetz properties by differentiation maps of modules in the Macaulay dual ring instead. More precisely, the rank of the multiplication map ×`j : A
i → Ai+j is
equal to the rank of the differentiation map ◦`j : (I−1)
i+j → (I−1)i, for every
i, j ≥ 0.
The study of SLP of quotient algebra A = S/I where I is generated by generic forms is related to the well known Fr¨oberg’s conjecture [15].
Conjecture 2.13. Fix positive integers a1, . . . , asfor some s > 1. Let F1, . . . , Fs⊂
S be generic forms of degrees a1, . . . , as respectively and let I = (F1, . . . , Fs). Then
for each 2 ≤ i ≤ s, and for all t, the multiplication by Fi on S/(F1, . . . , Fi−1) has
maximal rank, from degree t − ai to degree t. As a result, the Hilbert function of
S/I can be computed inductively.
It is known that all Artinian algebras of embedding dimension two has the SLP (and hence the WLP), see [15,23,28] and therefore the conjecture is true in two variables. D. Anick [2] proved the conjecture for algebras over polynomial rings
with three variables. J. Migliore, R. Mir´o-Roig and U. Nagel in [40] explain the connection of Fr¨oberg’s conjecture and the weak Lefschetz property. They show that if Fr¨oberg’s conjecture is true for all ideals generated by general forms in n variables, then all ideals generated by general forms in n + 1 variables have the WLP.
One another interesting problem is to classify the Hilbert functions of Artinian algebras in terms of the Lefschetz properties. More precisely, one may ask which Hilbert functions forces the Artinian algebras to satisfy or fail the WLP or SLP. It is known that the unimodality of the Hilbert function is a necessary condition for algebras satisfying the WLP or SLP [23]. As we mentioned earlier there are examples of Artinian Gorenstein algebras with non-unimodal Hilbert functions and hence failing the WLP. These examples are for algebras of codimension 5 and higher. T. Harima, J. Migliore, U. Nagel and J. Watanabe proved that over a field of characteristic zero all complete intersection algebras of codimension three satisfy the WLP [23]. Their proof translates the problem into a problem on rank two vector bundles on P2and uses Geauert-M¨ulich theorem. There have been studies in positive characteristics as well, [8,10,11]. It is not known that complete intersection algebras in higher codimensions have the WLP or SLP. It is a very interesting problem whether Artinian Gorenstein algebras with codimension 3 have the WLP or SLP. Besides the partial results this problem is open. In Paper E of this thesis we partially give an affirmative answer to this question.
T. Harima [21] classified the Hilbert functions of Artinian Gorenstein algebras satisfying the WLP.
Theorem 2.14. [21, Theorem 1.2] Let h = (h0, h1, . . . , hc) be a sequence of
posi-tive integers. Then h is the Hilbert function of some Artinian Gorenstein algebras satisfying the WLP if and only if h is an SI-sequence.
This thesis studies the Hilbert functions of Artinian Gorenstein algebras satis-fying the SLP. We will see in this thesis that the h-vectors of Artinian Gorenstein algebras satisfying the SLP are exactly SI-sequences.
5
Higher Hessians and Lefschetz properties
We have seen that Artinian Gorenstein algebras are precisely Poincar´e duality al-gebras. Cohomology rings of compact K¨ahler manifolds also satisfy the Poincar´e duality. So Artinian Gorenstein algebras are the natural objects in commutative algebra which correspond to K¨ahler manifolds. T. Maeno and J. Watanabe [37] provided a criterion for Artinian Gorenstein algebras having the SLP of WLP us-ing higher Hessians of the dual generators of such algebras. Recall from Theorem 2.10 that an Artinian Gorenstein algebra has a single dual generator.
Definition 2.15. [37, Definition 3.1] Let F be a polynomial in R and A = S/ AnnS(F ) be its associated Gorenstein algebra. Let Bj = {α
(j)
5. HIGHER HESSIANS AND LEFSCHETZ PROPERTIES 15
be a k-basis of Aj. The entries of the j-th Hessian matrix of F with respect to Bj
are defined as
(Hessj(F ))u,v= (α(j)u α (j) v ◦ F ).
Note that when j = 1 the form Hess1(F ) coincides with the usual Hessian. Up to non-zero constant multiple, det Hessj(F ) is independent of the choice of basis Bj.
So we denote Bj= {α (j)
i }i. Hessians were first studied by O. Hesse [24]. He studied
the forms that their Hessians have vanishing determinants. He claimed that if the determinant of the Hessian of a form vanishes then there is a change of variables where a variable is eliminated. Later P. Gordan and M. N¨other [20] proved that Hesse’s claim holds for forms of degree at least 3 in polynomial rings with at most three variables. They also showed that for forms in polynomial rings with at least 4 variables Hesse’s claim is not true. See also [45] for more discussion and detailed proof of P. Gordan and M. N¨other.
For a linear form ` = a1x1+ · · · + anxn in S we correspond to ` a point in the
projective space and denote it by P = (a1: · · · : an). We denote by Hess j `(F ) the
j-th Hessian of F evaluated at the point P . Also denote hessj`(F ) = det Hessj`(F ). The rank of certain multiplication maps could be obtained from Hessian matrices. Lemma 2.16. [37] Let A = S/ AnnS(F ) be an Artinian Gorenstein quotient of
S with socle degree d. Let ` be a linear form and consider the multiplication map ×`d−2j : A
j → Ad−j. Pick bases Bj and Bd−j for Aj and Ad−j respectively, then
there is a non-zero constant cj,A such that
det(×`d−2j) = cj,Ahess j `(F ).
The following result by T. Maeno and J. Watanabe provides a criterion for Artinian Gorenstein algebras satisfying the SLP.
Theorem 2.17. [37, Theorem 3.1] Let A = S/ AnnS(F ) be an Artinian
Goren-stein quotient of S with socle degree d. Let ` be a linear form and consider the mul-tiplication map ×`d−2j : Aj → Ad−j. Pick any bases Bj for Aj for j = 0, . . . , bd2c.
Then linear form ` is a strong Lefschetz element for A if and only if hessj`(F ) 6= 0,
for every j = 0, . . . , bd2c.
Example 2.18. Let A = S/I be an Artinian Gorenstein (complete intersection) algebra where S = C[x1, x2, x3] and I = (x31, x22, x23). The dual generator of A is
F = X2
1X2X3. The 0-th Hessian of F is trivially a square matrix of size one with
F as its entry. The first Hessian of F with respect to B1= {x1, x2, x3} is equal to
Hess1(F ) = 2X2X3 2X1X3 2X1X2 2X1X3 0 X12 2X1X2 X12 0 , (2.5)
so hess1(F ) = det Hess1(F ) = 6X4
1X2X3. The second Hessian of F with respect to
basis B2= {x21, x1x2, x1x3, x2x3} of A2 is equal to the following scalar matrix
Hess2(F ) = 0 0 0 2 0 0 2 0 0 2 0 0 2 0 0 0 , (2.6)
so hess2(F ) = 8. The Hessian matrices evaluated at linear form ` = x1+ x2+ x3;
i.e. P = (1, 1, 1) are non-zero,
hess0`(F ) = 1, hess1`(F ) = 6, hess2`(F ) = 8. So Theorem 2.17 implies that A has the SLP.
In order to determine that an Artinian Gorenstein algebra has the WLP it is enough to determine the rank of a single multiplication map by a generic linear form `.
Proposition 2.19. [41, Proposition 2.1] Let A be a graded Artinian Gorenstein algebra with socle degree d and ` be a linear form. Then ` is the weak Lefschetz element if and only if ×` : Abd
2c→ Ab d
2c+1 has maximal rank.
For an Artinian Gorenstein algebra for which all higher Hessians have non-vanishing determinants, Theorem 2.17 shows that all the multiplication maps have maximal rank. It is natural to ask: if an Artinian Gorenstein algebra A has at least one Hessian with vanishing determinant which multiplication maps have maximal rank and which ones do not. R. Gondim and G. Zappal`a [19] introduced mixed Hessians generalizing higher Hessians.
Definition 2.20. [19, Definition 2.1] Let F be a polynomial in R and A = S/ AnnS(F ) be its associated Gorenstein algebra. Let Bj = {α
(j)
i }i and Bk =
{βi(k)}i be ordered k-bases of Aj and Ak respectively. The entries of the mixed
Hessian matrix of order (j, k) of F with respect to Bj and Bk are given by
(Hess(j,k)(F ))u,v= (α(j)u β (k) v ◦ F ).
This generalizes the definition of higher Hessians and we have Hessj(F ) = Hess(j,j)(F ).
Theorem 2.21. [19, Theorem 2.4] Let A = S/ AnnS(F ) be an Artinian
Goren-stein quotient of S with socle degree d. Pick bases Bj and Bk for Aj and Ak
respec-tively. Let ` be a linear form and M be the matrix associated to the multiplication map ×`j−k : A
k → Aj for k ≤ j with respect to Bj and Bk. Then
6. JORDAN TYPES OF GRADED ARTINIAN ALGEBRAS 17
6
Jordan types of graded Artinian algebras
For a graded Artinian algebra A and linear form ` the Jordan type of A and ` is denoted by P`,A= P`and is a partition determining the Jordan block decomposition
for the (nilpotent) multiplication map by ` on A. Jordan types of Artinian algebras have been studied recently also in more general settings than we consider in this thesis, for example it has been studied for non-graded Artinian algebras, see [30–32]. For a generic linear form `; i.e. ` is in an open Zariski subset of Pn−1, the Jordan type P` is called the generic Jordan type. The Jordan type of a linear
form ` for Artinian algebra A determines whether or not ` is the weak or strong Lefschetz element for A. A linear form ` is a weak Lefschetz element for A if the number of parts of P`,A is the Sperner number of A, the maximum value of the
Hilbert function hA[22, Proposition 3.5]. It is also proved in [31, Proposition 2.9]
and [22, Proposition 3.64] that ` is a strong Lefschetz element for A if P`= h∨A; the
conjugate partition of Hilbert function hA. So Jordan type is a finer invariant of
an Artinian algebra than the Lefschetz properties. For an Artinian algebra failing the SLP, the Jordan type determines how far the algebra is from having the SLP.
If P`,A = (p1, . . . , pt) is the Jordan type for ` of A, then there exist elements
z1, . . . zt∈ A, which depend on `, such that {`izk | 1 ≤ k ≤ t, 0 ≤ i ≤ pk− 1} is
a k-basis for A. The Jordan blocks of the multiplication ×` is determined by the strings Sk = {zk, `zk, . . . , `pk−1zk}, and A is the direct sum A = hS1i ⊕ · · · ⊕ hSti.
Denote by dk the degree of zk. Then the Jordan degree type, is defined to be the
indexed partition S`,A= (p1d1, . . . , ptdt), see [31, Definition 2.28].
Example 2.22. Let A = S/I where S = C[x1, x2, x3] and I = (x31, x22, x23). The
computations in Example 2.18 shows that for a generic linear form `, the Jordan type partition A and ` is equal to
P`,A= (5, 3, 3, 1) = h∨A= (1, 3, 4, 3, 1)∨.
One can see that the Jordan degree type of ` for A is equal to S`,A= (50, 31, 31, 12).
Let ` = x1 and note that `3 = 0. We trivially have rk(×`0) = dimkA = 12. One
can easily check that rk(×`) = dimk(S/(I, `)) = 8, rk(×`2) = dimk(S/(I, `2)) = 4
and rk(×`i) = 0, for all i ≥ 3. The Jordan type of A and ` is given by the sizes of the dual partition to the following partition
(rk(×`0) − rk(×`), rk(×`) − rk(×`2), rk(×`2) − rk(×`3)) = (4, 4, 4). So we have that
P`,A= (3, 3, 3, 3),
and
S`,A= (30, 31, 31, 32).
We observe that the linear from ` = x1is a weak Lefschetz element for A but not
One another natural question, which has been treated in this thesis for some Artinian algebras, is to ask for non-generic Jordan types even for algebras having the SLP. For Artinian graded algebras of codimension two the generic Jordan type is known, as we mentioned previously that they all satisfy the SLP.
Given a vector of non-negative integers
T = (1, 2, . . . , d, td, td+1, . . . , tj, 0) where d ≥ td ≥ td+1≥ · · · ≥ tj> 0,
there have been studies in [13,14,33] on the family of Artinian graded Gorenstein algebras with the Hilbert function equal to T . The family GT of Artinian graded
quotients A = S/I where I is an ideal of S = k[x, y], for which the Hilbert function hA= T is a smooth projective variety, that is locally affine space of known
dimen-sion, see [27,33]. The variety GT has a cellular decomposition into the cells V(EP)
where P runs through the set P(T ) of partitions P having diagonal lengths T and EP is a monomial ideal determined by P .
Given a partition P = (p1, p2, . . . , pt) ∈ P(T ) of n = P pi where p1 ≥ p2 ≥
· · · ≥ pt, we let CP be the set of n monomials that fill the Ferrers diagram of P as
follows: for i ∈ [1, t] the, i-th row counting from the top is filled by the monomials yi−1, yi−1x, . . . , yi−1xpi−1. We let E
P be the complementary set of monomials to
C(P ) and denote by (EP) the ideal they generate.
Definition 2.23. A hook of a partition P is a subset of CP consisting of a corner
monomial c, an arm (c, xc, . . . , ν = xu−1c) and a leg (c, yc, . . . , µ = yv−1c), such
that xν ∈ EP and yµ ∈ EP (Figure 2.1). The arm length is u and the leg length is
v; the hook has arm-leg difference u − v. The monomial ν ∈ CP is said to be the
hand, and the monomial µ ∈ CP is said to be the foot of the hook.
c h
f
Figure 2.1: Difference-one hook with hand h, foot f , corner c.
Example 2.24. Let P = (4, 4, 1). The hook with corner x in the Ferrers diagram CP has arm length 3, foot length 2, hand x3, foot yx, so has (arm − leg) difference
one (Figure 2.2). Here T (P ) = (1, 2, 3, 2, 1), ∆(P ) = ∆3and the degree-3-diagonal
of CP has the two spaces corresponding to the monomials y2x and y3.
Theorem 2.25. [33,§3-B,Theorem 3.12, and §3-F] The cell V(EP,`) is an affine
space of dimension the total number of difference-one hooks in CP.
Fix ` ∈ S1. The variety GT parametrizing all graded quotients A = S/I of
6. JORDAN TYPES OF GRADED ARTINIAN ALGEBRAS 19
1 x x2 x3
y yx yx2 yx3
y2
Figure 2.2: Difference-one hook in the Ferrers diagram of partition (4, 4, 1).
cells,
GT =
[
P ∈P(T )
V(EP,`). (2.7)
In the next part we discuss how this machinery is used to determine non-generic Jordan types for graded Artinian algebras of codimension two.
3
Summary of results
The results of this thesis are distributed in the following way.
Papers A and B concern the study of Lefschetz properties of graded Artinian algebras A = S/I where S is a polynomial with n ≥ 3 variables and I is a monomial ideal generated in a single degree d ≥ 2. The results of Paper A are in terms of the minimal free resolution of A. One of the main results of Paper B provides a sharp lower bound for the Hilbert function of A in degree d when A fails the WLP. The other main result of Paper B characterizes Artinian algebras A = S/I where I is an ideal fixed by an action of the cyclic group Z/dZ. Papers A and B were also included in the author’s licentiate thesis.
Papers C and D provide possible Jordan type partitions for graded Artinian algebras A = S/I of S = k[x, y]. Paper C provides all possible Jordan types that occur for Artinian complete intersection algebras and some linear form. This has been done in terms of combinatorial invariants of partitions - such as branch labels and hook codes - as well as algebraic invariants of such algebras, namely higher Hessians of the dual forms. Paper D generalizes this result to arbitrary Artinian algebra of codimension two. The main result of Paper D concerns determining the least number of generators of a homogeneous ideal I for a given vector T such that the Hilbert function of A = S/I is equal to T .
Papers E and F concern the study of Lefschetz properties and Jordan types of Artinian Gorenstein algebras and putting more focus on codimension three. Paper E concerns the study of SLP of such algebras, namely the ones that are obtained from the ideal of points in the projective space via studying the higher Hessians of dual forms. Paper E introduces an approach to determine the Jordan types of such algebras as well as computing the Jordan types with small parts for such algebras of codimension three.
Summary of Paper A
Lefschetz properties of equigenerated monomial algebras with almost linear resolu-tion.
(joint with N. Nemati)
Communications in Algebra (2020).
Paper A is dealing with the Lefschetz properties of Artinian quotients A = S/I where I is a monomial ideal generated in a same degree d ≥ 2 in polynomial ring S = k[x1, . . . , xn] over a filed k of characteristic zero, denote the maximal ideal of
S by m = (x1, . . . , xn). For a monomial ideal I, the Macaulay dual module I−1 in
degree d is generated by monomials of degree d in the dual ring R = k[X1, . . . , Xn]
which are not among the generators of I. This paper concerns the study of Lef-schetz properties of such algebras under certain conditions on their minimal free resolutions. This work was motivated by the following conjecture of Eisenbud, Huneke and Ulrich.
Conjecture 3.1. [12, Conjecture 5.4] Suppose I ⊂ S is artinian ideal generated in degree d and its minimal free resolution is linear for p − 1 steps then
md⊆ I + (lp, lp+1, . . . , ln)2
for sufficiently general linear forms lp, . . . , ln.
In Paper A, we give an affirmative answer to the above conjecture for Artinian monomial algebras where their minimal free resolutions have linear syzygies for n − 1 steps. More generally, we prove the following.
Theorem 3.2. Suppose that there exist integers i, j, a and b satisfying 1 ≤ i < j ≤ n and a, b ≥ 0 and that xa
ixbj|m for every monomial m ∈ (S/I)d. Then the
multiplication map ×(xi+ xj)a+b : (S/I)k−a−b → (S/I)k has maximal rank for
every a + b ≤ k ≤ d.
Moreover, this paper provides classes of Artinian monomial algebras satisfying the WLP and SLP.
Theorem 3.3. Suppose that for every monomial m ∈ Sd we have that xixj|m if
and only if m /∈ I, for some 1 ≤ i < j ≤ n. Then S/I enjoys the SLP.
Theorem 3.4. Suppose that md+1⊂ I and that the minimal free resolution of S/I is linear for n − 2 steps. Then S/I satisfies the WLP.
Contribution to the paper
This paper was developed through regular discussions and my contributions can be found in all aspects of the work.
23
Summary of Paper B
The weak Lefschetz property of equigenerated monomial ideals. (joint with M. Boij)
Journal of Algebra (2020).
Paper B is split into two parts. Both dealing with the weak Lefschetz property of Artinian algebras that are quotients of polynomial ring in n variables by monomial ideals generated in degree d. In the first part, we provide a sharp lower bound for the Hilbert function of such algebras that fail the WLP.
Let S = k[x1, . . . , xn] be a polynomial ring where k is a field of characteristic
zero. Let A = S/I be an Artinian algebra where I is an ideal generated by mono-mials in degree d ≥ 2. Let R = k[X1, . . . , Xn] be the Macaulay dual ring to S.
Macaulay inverse system module of I in degree d, (I−1)d, is generated by the dual
elements in Rdto the monomials in Sd\ Id. Because of the action of the torus (k∗)n
on monomial algebras, the linear form ` = x1+· · ·+xnis a canonical linear form for
the weak and strong Lefschetz element, see [41, Proposition 2.2]. To provide a bound on hA(d) where A fails the WLP we study the multiplication map ×` : Ad−1→ Ad
or equivalently the differentiation map ◦` : (I−1)d→ (I−1)d−1. In order to provide
a lower bound on hA(d) for A failing the WLP, it is enough to study the surjectivity
of the multiplication map by ` on Ad−1, i.e. when hA(d) ≤ hA(d − 1).
The following result of Paper B provides a bound on the number of monomials with non-zero coefficients in a homogeneous polynomial F ∈ Rd in the kernel of the
differentiation map by arbitrary power of ` on (I−1)d.
Theorem 3.5. Suppose that `d−a◦F = 0 for some 0 ≤ a ≤ d−1, then | Supp(F )| ≥
a + 2.
Recall that any Artinian algebra over a polynomial ring with less than three variables satisfies the SLP and so the WLP. The lower bounds on hA(d) for an
algebra A that fails the WLP is different for n = 3 and n ≥ 4.
Theorem 3.6. Let A = S/I be an Artinian algebra where S = k[x1, x2, x3] and I
is a monomial ideal generated in degree d ≥ 2. Assume that A fails the WLP, then hA(d) ≥
3d − 3 if d is odd 3d − 2 if d is even. Moreover, the bounds are sharp.
Theorem 3.7. Let A = S/I be an Artinian algebra where S = k[x1, . . . , xn] for
n ≥ 4 and I is a monomial ideal generated in degree d ≥ 2. Assume that A fails the WLP, then
hA(d) ≥ 2d
In the second part of Paper B, we let k be an algebraically closed field of char-acteristic zero, we assume k = C. Let I be a monomial ideal in Sd generated by
monomials fixed by action of the cyclic group Z/dZ. Let d ≥ 2 and ξ = e2πi/d be
the primitive d-th root of unity. Consider the diagonal matrix
Ma1,...,an= ξa1 0 · · · 0 0 ξa2 · · · 0 .. . ... ... 0 0 · · · ξan
representing the cyclic group Z/dZ, where a1, a2, . . . , an are integers and the action
is defined by [x1, . . . , xn] 7→ [ξa1x1, . . . , ξanxn]. Since ξd= 1, we may assume that
0 ≤ ai ≤ d − 1, for every 1 ≤ i ≤ n. Let I ⊂ S be the ideal generated by all the
forms of degree d fixed by the action of Ma1,...,an. It is shown in this paper that
these ideals are Artinian and generated by monomials of degree d, the proof for n = 3 was given in [39]. Since A = S/I is a monomial Artinian algebra the linear form ` = x1+· · ·+xnis the canonical weak (and strong) Lefschetz element for A. E.
Mezzetti and R. M. Mir´o-Roig [39] proved that for n = 3 and gcd(a1, a2, a3, d) = 1
such ideals define Artinian algebras that fail the WLP by failing injectivity of the multiplication map by ` = x1+ x2+ x3 in degree d − 1. In paper B we provide the
number of generators of such ideals when n = 3.
Theorem 3.8. For integers a1, a2, a3 and d ≥ 2, the number of monomials in
S = k[x1, x2, x3] of degree d fixed by the action of Ma1,a2,a3 is equal to
1+gcd(a2− a1, a3− a1, d) · d + gcd(a2− a1, d) + gcd(a3− a1, d) + gcd(a3− a2, d)
2 .
We then conclude that the number of generators for I depends on the integers a1, a2and a3. If gcd(a1, a2, a3, d) 6= 1 the multiplication map by ×` : Ad−1→ Adis
not necessarily the assertion of injectivity. In the main result of this part we extend the result in [39].
Theorem 3.9. For integers d ≥ 2, n ≥ 3 and 0 ≤ a1, . . . , an ≤ d − 1, let Ma1,...,an
be a representation of the cyclic group Z/dZ and I ⊂ S = k[x1, . . . , xn] be the ideal
generated by all forms of degree d fixed by the action of Ma1,...,an. Then, I satisfies
the WLP if and only if at least n − 1 of the integers ai are equal.
Sketch of the proof. If all integers ai’s are equal then I is the d-th power of the
maximal ideal and clearly A satisfies the WLP. Also if n − 1 of the integers ai’s
are equal then every monomial in (I−1)d is divisible by one of the variables and
therefore A has the WLP.
In the other cases we provide an element in the kernel of the differentiation map ◦` : (I−1)
d→ (I−1)d−1which implies that ×` : Ad−1→ Ad is not surjective. Also
using similar argument as in [39] and [9] we conclude that ×` : Ad−1→ Ad is not
injective.
25
Lemma 3.10. For integer d ≥ 2 and distinct integers 0 ≤ a1, a2, a3 ≤ d − 1, let
Ma1,a2,a3 be a representation of Z/dZ. Define the linear form
L = l X j=0 ξjx1+ l+k+1 X j=l+1 ξjx2+ 2d−1 X j=l+k+2 ξjx3,
where l and k are the residues of a2− a3− 1 and a3− a1− 1 modulo d. Then
the support of the form F = Ld − Ld are exactly the monomials of degree d in k[x1, x2, x3] which are not invariant under the action of Ma1,a2,a3, where L is the
conjugate of L and ξ is a primitive d-th root of unity.
Contribution to the paper
The results and proofs in this paper grew out of joint discussions. My contributions lie in all aspects of the work.
Summary of Paper C
Complete intersection Jordan types in height two. (joint with A. Iarrobino and L. Khatami)
Journal of Algebra (2020).
In this paper, we determine all partitions of n that occur as Jordan type par-titions for some Artinian Complete intersection algebra A of codimension two and linear form ` ∈ A1. The Hilbert function of a graded Artinian algebra A = S/I
where S = k[x, y] with socle degree j is symmetric about b2jc and is in the following form
hA= T = (1, 2, . . . , d − 1, dk, d − 1, . . . , 2, 1), (3.1)
with Sperner number d. For a fixed T satisfying (3.1), recall P(T ) is the set of partitions with diagonal lengths T . We determine the ones that occur as Jordan type partitions for some Artinian complete intersection algebra A with hA= T and
some linear form ` ∈ A1. We call them CIJT partitions. We first provide numerical
conditions of any partition to be a CIJT partition. The main result of this paper in this regard is split into the following theorems.
Theorem 3.11 (CIJT partitions, k > 1). Assume that T satisfies (3.1) with d ≥ 2 and k > 1. A partition P can occur as the Jordan type of a linear form for some complete intersection algebra of Hilbert function T satisfying (3.1) if and only if
there exists an integer n ∈ [0, d] and an ordered partition n = n1+ · · · + nc (empty
partition when n = 0) such that P = pn1 1 , . . . , p nc c , (d − n) d−n+k−1, (3.2) where pi= k − 1 + 2d − ni− 2 X j<i nj, for 1 ≤ i ≤ c.
Theorem 3.12 (CIJT partitions, k = 1). Assume that T satisfies (3.1) with d ≥ 2. A partition P can occur as the Jordan type of a linear form for some complete intersection algebra of Hilbert function T if and only if there exists an integer n ∈ [0, d − 1] and an ordered partition n = n1+ · · · + nc (empty partition when n = 0)
such that P has d parts of the form P = pn1 1 , . . . , p nc c , (d − n) d−n, (3.3) where pi= 2d − ni− 2 X j<i nj, for 1 ≤ i ≤ c.
We conclude that a partitions P with diagonal lengths T satisfying (3.1) is a CIJT partition if and only the number of parts in P is either d or d + k − 1. We also give a simple combinatorial characterization for CIJT partitions.
Corollary 3.13. A partition P is a CIJT partition if and only if P = (pn1
1 , . . . , p nt
t )
such that for each i ∈ [2, t],
pi−1= ni−1+ ni+ pi. (3.4)
Above theorems allow us to count the CIJT partitions.
Corollary 3.14. Let T satisfy Equation (3.1). We have that P(T ) = 2·3d−1, when k > 1 and P(T ) = 3d−1 when k = 1. Moreover, among those partitions, there are 2d CIJT partitions having diagonal lengths T if k > 1, and 2d−1if k = 1.
We also study CIJT partitions with diagonal lengths T in terms of the algebraic invariants of Artinian complete intersection algebras with Hilbert function T . For an Artinian complete intersection algebra with dual generator F , A = S/ Ann(F ), and Hilbert function hA = T we study higher Hessians of F on a linear form `
and their connection with the CIJT partitions P` = P . The following key result
characterizes the vanishing of properties of higher Hessian in terms of numerical properties of a CIJT partition.
Theorem 3.15 (When is a Hessian non-zero?). Let P` = (p1, p2, . . . , pd, . . . ) be
the Jordan type partition for a linear form ` of an Artinian complete intersection algebra A = S/ Ann(F ) of Hilbert function hA = (1, 2, . . . , dk, . . . , 2, 1), for an
integer k ≥ 1. Then for each i ∈ [0, d − 1] we have
hessi`(F ) 6= 0 ⇐⇒ i+1
X
j=1
27
In particular, P` has d parts unless k ≥ 2 and hessd−1` = 0.
Using the above theorem we are able to prove that there is a one-to-one corre-spondence between the set of higher Hessians that vanish and the CIJT partition P .
Theorem 3.16 (Hessians and partitions). Assume that T satisfies Equation (3.1) for d ≥ 2 and k ≥ 2 (k = 1, respectively). Then there is a one-to-one correspondence between the CIJT partitions P` of diagonal lengths T , and the 2d (when k > 1), or
2d−1 (when k = 1) subsets of the active Hessians for T that vanish at ` in S 1.
In particular, let P be a partition of diagonal lengths T . The following are equivalent.
i. P = P`,Afor a linear form ` ∈ R and an Artinian complete intersection algebra
A = S/ Ann(F ), and there is an ordered partition n = n1+ · · · + nc of an
integer n satisfying 0 ≤ n ≤ d (or 0 ≤ n ≤ d − 1, respectively) such that hessn1+···+ni−1
` (F ) 6= 0, for each i ∈ [1, c], and the remaining Hessians are
zero; ii. P satisfies P = pn1 1 , . . . , p nc c , (d − n) d−n+k−1, (3.6)
where pi = k − 1 + 2d − ni− 2(n1+ · · · + ni−1), for 1 ≤ i ≤ c.
An immediate consequence of Theorem 3.16 is that for an Artinian complete intersection algebra with a given Hilbert function and a linear form for which a set of higher Hessians vanish there is exactly one CIJT partition. This implies that there is only one collection of ranks of higher and mixed Hessians for vanishing Hessians that occurs for a pair (A, `) where A is any Artinian complete intersection algebra, but the Jordan type P` is fixed.
Proposition 3.17 (Ranks of mixed and higher Hessians). Assume that A = S/ Ann(F ) is an Artinian complete intersection algebra with the Hilbert function T satisfying (3.1), for d ≥ 2 and k ≥ 1. Denote the socle degree of A by j = 2d+k −3. Assume further that for a linear form ` ∈ S1, and a non-negative integer m, we
have
hessm` (F ) = hessm+1` (F ) = · · · = hessm+n` (F ) = 0, and hessm−1` (F ) 6= 0 for m 6= 0. Then
(i) If m + n ≤ d − 2 and hessm+n+1` (F ) 6= 0, (recall hessd−1` 6= 0), then for every i ∈ [0, n] we have rk Hessm+i,s` (F ) = max{j + i − (n + s), m} if s ∈ [j − (m + n), j − (m + i)] , m + i + 1 if s ∈ [d, j − (m + n + 1)] . In particular,
rk Hessm+i` (F ) = m, rk Hessm+n−i` (F ) = m + n − 2i, for every i ∈h0, n 2
i . (3.7)
(ii) If k ≥ 2, and m + n = d − 1, then for every i ∈0, n + k
2 − 1 and every
s ∈ [d, j − (m + i)] we have
rk Hessm+i,s` (F ) = max{2m + n + i + 1 − s, m}.
For two partitions of the integer n, P = (p1, . . . , pt), p1 ≥ · · · ≥ pt and Q =
(q1, q2, . . . , qt0),
q1≥ · · · ≥ qt0 define the dominance partial order by
Q ≤ P ⇔ i X j=0 qj≤ i X j=0
pj for all i ≤ min{t, t0}. (3.8)
For an Artinian complete intersection algebra A = S/ Ann(F ) and linear form ` the set of integers i such that hessi`(F ) 6= 0 corresponds uniquely to a CIJT partition
P . Denote by HP,`= HP,`(F ) the set of integers i where hessi`(F ) 6= 0.
Theorem 3.18. Let T satisfy Equation (3.1) and asuume that P = P`= (p0, p1, . . .)
and Q = Q`0 = (q0, q1, . . .) are CIJT partitions having diagonal lengths T . Then
Q ≤ P in the dominance order if and only if HQ,`0 ⊆ HP,`.
Recall that for a Hilbert function T that occurs for an Artinian quotient of S = k[x, y] the projective variety GT parametrizes graded algebra quotients A = S/I of
S having Hilbert function T : it is smooth of dimension 2(d − 1) + 1 when k ≥ 2 and 2(d − 1) when k = 1.
Proposition 3.19. [Proper intersection of CIJT cells of GT] Let T satisfy
Equa-tion (3.1), and let P, Q be CIJT partiEqua-tions of diagonal lengths T , and P ∩ Q their intersection in the poset of partitions. Then, fixing `, we have
V(EP) ∩ V(EQ) = V(EP ∩Q) and
V(EP) =
[
P0≤P
V(EP0). (3.9)
Furthermore, the codimension of the cell V(EP,`) in GT is the number of Hessians
that vanish at p`. The cells V(EP) and V (EQ) intersect properly.
As a consequence of the above result and the correspondence of CIJT parti-tions with vanishing higher Hessians we conclude the frontier property of the cells V(EP). Denote by CIT the open dense subvariety of GT parametrizing complete
intersections.
Theorem 3.20. [Closure of V(EP)] Assume that T satisfies Equation(3.1). The
Zariski closure in CIT of the locus V(EP) of Artinian algebras whose Jordan type
is a CIJT partition P ∈ P(T ), is equal to V(EP) =
[
P0≥P
V(EP0)
29
A combinatorial invariant that is assigned to a partition P is the difference-one hook code, Definition 2.23, that provides the dimension of the cell V(EP). As the
last main result of this paper we provide a characterization of CIJT partitions with a given diagonal lengths T in terms of the difference-one hook code of P . We do so via the characterization of CIJT partitions and subsets of vanishing higher Hessians Theorem 3.16.
Proposition 3.21. Assume that P` is the Jordan type partition of a linear form
` of an Artinian CI algebra A = S/ Ann(F ) of Hilbert function T satisfying (3.1). Let h` be the difference-one hook code of P`. We have the following,
(i) If k ≥ 2, then
hessd−i` (F ) = 0 ⇔ (h`)i<
1 if i = 1, 2 if i ∈ [2, d]. (ii) If k = 1, then for each i ∈ [1, d − 1], we have that
hessd−1−i` (F ) = 0 ⇔ (h`)i< 2.
Contribution to the paper
While I was involved in discussions for all parts of the paper, I was the main contributor in the parts related to higher Hessians.
Summary of Paper D
Jordan types for graded Artinian algebras in height two. (joint with A. Iarrobino, L. Khatami and J. Yam´eogo) Preprint: arXiv:2006.11794.
In this paper we consider the family GT of graded Artinian algebras A = S/I
where S = k[x, y], having arbitrary Hilbert function which is in the following form hA= T = (1, 2, . . . , d, td, td+1, . . . , tj, 0) where d ≥ td≥ td+1≥ · · · ≥ tj> 0,
(3.10) where ti = dimkAi, where j is the socle degree of T , d is the lowest degree of the
elements of any I such that hS/I = T and n = |T | =P ti= dimkA.
In this paper we generalize the main result of Paper C for arbitrary Hilbert function T satisfying (3.10). We do so as the following. We associate to a partition P a monomial ideal EP, determined by the Ferrers graph of P ; the diagonal lengths
