Luận án tiến sĩ toán học: Đại số mức không đơn hình mới - Tufts University 2007

SOME NEW NON-UNIMODAL LEVEL ALGEBRAS A dissertation submitted by Arthur Jay Weiss In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics TUFTS UNIVERSITY February, 2007 c 2007, ARTHUR JAY WEISS ° ADVISOR: George McNinch UMI Number: 3244622 UMI Microform 3244622 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.

Box 1346 Ann Arbor, MI 48106-1346 Abstract In 2005, building on his own recent work and that of F. Iar- robino discovered some constructions that, he conjectured, would yield level algebras with non-unimodal Hilbert functions. This thesis provides proofs of non- unimodality for Iarrobino’s level algebras, as well as for other level algebras that the author has constructed along similar lines. The key technical contribution is to extend some results published by Iarrobino in 1984.

Iarrobino’s results provide insight into some naturally arising vector subspaces of the vector space Rd of forms of fixed degree in a polynomial ring in several variables. In this thesis, the problem is approached by combinatorial methods and results similar to Iarrobino’s are proved for a different class of vector subspaces of Rd. The combinatorial methods involve the definition of a new class of matrices called L-Matrices, which have useful properties that are inherited by their subma- trices. A particular class of square L-Matrices, associated with some specialized partially ordered sets having interesting combinatorial properties, is identified.

For this class of L-Matrices, necessary and sufficient conditions are given that they ii be nonsingular. Several larger questions are discussed whose answers are incrementally im- proved by the knowledge that the new non-unimodal level algebras exist. iii Acknowledgements This work was done under the supervision of Professor Anthony Iarrobino of Northeastern University and Professor George McNinch of Tufts University. The author wishes to express his sincere appreciation.

The author wishes to thank Professor Fabrizio Zanello for reviewing an early draft of this work and offering numerous insightful comments and suggestions. The author wishes to thank Professor Juan Migliore for providing a preprint of [GHMS06]. The author wishes to thank Emeritus Professor George Leger and Professor Montserrat Teixidor for their excellent advice throughout the graduate student period. iv Contents Abstract ii Acknowledgements iv Chapter 1.

Polynomials as Differential Operators 10 3. Matlis Duality 12 Chapter 3. Definitions and Preliminaries 17 2. Definitions and Preliminaries 27 2.

PV-Matrices as Parameterized Families 31 Chapter 5. Partially Ordered Sets 34 2. The Partially Ordered Set GQ 40 3. Block L-Matrices Associated to GQ 47 v Chapter 6.

Coefficient Matrices of R j−d ∗ E (C ) 64 2. Coefficient Matrices for Constrained Subspaces of D j 67 3. Intersections of Subspaces of D j 78 Chapter 7. Special Cases for Interesting Choices of Q 82 1.

Absence of Constraints 82 2. Essentially n-fold-constrained 90 Chapter 8. Construction of New Non-Unimodal Level Algebras 102 1. Overview of the Construction of Level Algebras 102 2.

Computations by Computer 107 3. Six Families of Level Algebras, together with Computer-Calculated Hilbert Functions 107 4. Formulas for hE (C) (d) and ∆d 113 5. Formulas for hF (C0 ) (d) and δd 115 6.

Proof of Non-Unimodality 127 8. Computation of Types 131 Chapter 9. For Which Codimensions and Types are Non-Unimodal Level Algebras Possible? 145 vi 2. Minimal Socle Degree 150 Bibliography 153 vii Some New Non-Unimodal Level Algebras 1 CHAPTER 1 Introduction In this section, which is intended to provide an overview, we use some technical terms without stopping to provide definitions.

The definitions can all be found in later sections. For over a century, mathematicians have been investigating Hilbert functions of (standard) graded quotients of polynomial rings, and the subject is still a focus of active study. In particular, among the graded quotients are the Gorenstein Artinian graded algebras, which arise in various contexts. Stanley defined a generalization of this class, the class of level algebras, that is useful for studying Gorenstein Artinian graded algebras, but is also interesting in its own right.

The general question for Hilbert functions of level algebras, that is, what sequences could be their Hilbert functions, is the subject of the recent paper [GHMS06], whose introduction provides an excellent history of work that has been done in this direction to date. Most of that work proceeds in different directions from what is done in this thesis. Here, we focus on a property called unimodality that Hilbert functions of level algebras sometimes have. In studying unimodality of level algebras, it is usual to 2 classify them by codimension and type; and one can ask whether it is possible for a level algebra of some particular codimension and type to be non-unimodal.

The following is a summary of the history so far, for which the author is indebted to A. In codimensions 1 and 2, level algebras of all types are necessarily unimodal. The level algebras of codimension 1 are sufficiently simple that this is easy. The investigations in codimension 2 were performed by F.

Macaulay in [Mac04] and [Mac27], written in the first several decades of the twentieth century. The next step was in showing that Gorenstein Artinian graded algebras in codimension 3 are necessarily unimodal. This was done by R. Stanley in [Sta77], although it was D.

Eisenbud who first determined the actual Hilbert functions in [BE77]. In [Sta78], Stanley also demonstrated a level algebra in codimension 13 that was not unimodal. The next progress was accomplished in [BI92] by D. Iarrobino, who showed that a non-unimodal Gorenstein Artinian algebra could be found in codimension 5 and in any higher codimension.

Meanwhile, groundwork was being laid for further progress. In particular, we note the work of J. Iarrobino in [EI78], which contained some basic concepts underlying the investigation of catalecticants by A. Iarrobino in 3 [I84] and differently by R.

Investigations of non- unimodality in Gorenstein Artinian graded algebras were conducted in [B94] by M. Boij, and by M. Zanello published the first non-unimodal level algebra in codi- mension 3 in [Z06]. Its type is 28.

Later that year, A. Iarrobino used the same general idea to produce a level algebra in codimension 3 of type 5 that, he conjectured, would prove to be non-unimodal, as well as showing how to perform a similar construction for any type higher than 5. Iarrobino also suggested methods for codimension 4 that, he conjectured, would produce non-unimodal level algebras. It is his construction in codimension 3, as well as some constructions in codimensions 3, 4, and 5 that proceed along lines suggested by his work, that are analyzed in this thesis, and shown to be non-unimodal.

1 yes yes yes yes yes yes 2 yes yes yes yes yes yes 3 yes ? ? ? no no 4 ? ? no no no no 5 no no no no no no. no no no no no no TABLE 1. Is a Level Algebra of Codimension r and Type t Necessarily Unimodal? As discussed in a later chapter, with a few additional observations we will be able to summarize the current state of knowledge as follows. Necessarily 4 Unimodal: Codimensions 1 and 2 of all types, codimension 3 of type 1.

Non- unimodals exist: Codimension 3, of types 5 and greater; codimension 4, of types 3 and greater; codimension 5 and greater, of all types. Unknown: Codimension 3, types 2, 3, and 4; codimension 4, types 1 and 2. Among the classes listed as unknown, some useful progress has been made. In particular, we note [IS05].

5 CHAPTER 2 Algebraic Preliminaries 1. Level Algebras We fix k, an algebraically closed field of characteristic 0. Throughout this work, it will be implicitly assumed that all our vector spaces are over the field k. Let R be the polynomial ring over k in r variables: R := k [ X1 ,.

R can L be written as a direct sum R = d≥0 R d , where the subspaces Rd consist of all homogeneous polynomials (forms) in R of degree d. For every d, Rd is a finite- µ ¶ d+r−1 dimensional vector space, of dimension. One basis of Rd consists of r−1 all monomials of degree d. By way of notation, let D := (d1 , ., dr ) be any r-tuple of non-negative integers such that d1 +.

Then D determines a monomial X D := X1d1 · · · Xrdr of degree d, and monomials of degree d are indexed by the r- tuples D. D is sometimes called a multi-index of dimension r and degree d. When considering the monomials X D of R, we sometimes use lexicographic ordering, defined as follows. For two different multi-indexes C := (c1 , ., dr ), we say C comes before D if, in the leftmost co-ordinate for which ci 6= di , ci > di.

In this case, we write C > D. By extension, we place an ordering on the monomials of R: X C > X D ⇔ C > D. In this definition, there is no requirement 6 that X C and X D be monomials of the same degree. When listing all monomials of a fixed degree d, to say that they are listed lexicographically means that the listing is according to lexicographical order.

For example, if R = k [ X1 , X2 , X3 ], we list the monomials of degree 2 lexicographically as follows: X12 , X1 X2 , X1 X3 , X22 , X2 X3 , X32 ., dr ) are two multi-indexes, we define their addition and subtraction co-ordinatewise., cr + dr ), and C − D := (c1 − d1 ,. L The direct-sum decomposition of R = d≥0 R d makes it a graded R-module, graded by total degree, since for non-negative integers d and e, Rd Re ⊆ Rd+e. L Let I = d≥0 Id ⊆ R be a homogeneous ideal of R, where Id consists of all forms in I of degree d. We form the quotient ring A := R/I, which is a k-algebra.

L The direct-sum decomposition A = d≥0 Ad , where Ad : = Rd /Id , makes A both a graded k-algebra and a graded R-module. In each case, the grading is by total degree. In considering R or its graded quotients by homogeneous ideals, the only grading we will ever use is the grading by total degree, which is sometimes called standard. From now on, standard grading will always be implicitly assumed.

7 For any graded quotient A = R/I, we say A is Artinian if it is finite-dimensional L as a vector space. In this case, we write A = 0≤d≤ j Ad , where j is the largest integer for which A j is nonzero. In writing such a direct sum decomposition, we will always assume A j is nonzero unless otherwise stated. L For an Artinian quotient A = 0≤d≤ j A d , we define soc(A), the socle of A, to be the annihilator of the linear part of A: soc(A) := { a ∈ A| aA1 = 0}.

Soc(A) is easily seen to be a homogeneous ideal of A. We remark that A j ⊆ soc(A) since A j A1 ⊆ A j+1 = 0, but equality need not hold. A := k[ X, Y ]/( X 2 , XY, Y 3 ) ' k kX kY kY , where we adopt the usual notation that for any F ∈ R, F denotes the homomorphic 2 L 2 image of F in A = R/I. Then A j = A2 = kY , and soc(A)= kX kY .