Combinatorics of Determinantal Ideals

146 Pages·2006·0.614 MB·English

by  Cornel Băețica

COMBINATORICS OF DETERMINANTAL
IDEALS
Cornel B˘ae¸tica
Contents
Introduction iii
1 Preliminaries 1
2 Divisor class group and canonical class of determinantal rings 10
2.1 Integrity and normality of determinantal rings . . . . . . . . . 10
2.2 Divisor class group of determinantal rings . . . . . . . . . . . 17
2.3 The canonical class of determinantal rings . . . . . . . . . . . 20
3 Gr¨obner bases and determinantal ideals 31
3.1 Bitableaux,combinatorialalgorithms,andtheKnuth-Robinson-
Schensted correspondence . . . . . . . . . . . . . . . . . . . . 31
3.2 Schensted and Greene’s theorems . . . . . . . . . . . . . . . . 34
3.3 KRS and generic matrices . . . . . . . . . . . . . . . . . . . . 35
3.4 Gr¨obner bases of determinantal ideals . . . . . . . . . . . . . . 43
4 Powers and products of determinantal ideals 51
4.1 Primarydecompositionofthe(symbolic)powersofdeterminan-
tal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Powers of ideals of maximal minors . . . . . . . . . . . . . . . 59
4.3 Gr¨obner bases of powers of determinantal ideals . . . . . . . . 62
5 Simplicial complexes associated to determinantal ideals 71
5.1 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Multiplicity of determinantal rings . . . . . . . . . . . . . . . 74
5.3 Hilbert series of determinantal rings . . . . . . . . . . . . . . . 88
5.4 The a-invariant of determinantal rings . . . . . . . . . . . . . 92
6 Algebras of minors 97
6.1 Cohen-Macaulayness and normality of algebras R(I ) and A . 97
t t
6.2 Divisor class group of R(I ) and A . . . . . . . . . . . . . . . 103
t t
6.3 Canonical modules of in (R(I )) and in (A ) . . . . . . . . . 110
τ t τ t
i
ii Contents
6.4 Canonical classes of R(I ) and A . . . . . . . . . . . . . . . . 114
t t
7 F-rationality of determinantal rings 121
7.1 F-regularity of determinantal rings . . . . . . . . . . . . . . . 121
7.2 F-rationality of Rees algebras . . . . . . . . . . . . . . . . . . 129
References 132
Introduction
The study of determinantal ideals and of classical determinantal rings is an
old topic of commutative algebra. As in most of the cases, the theory evolved
from algebraic geometry, and soon became an important topic in commutative
algebra. Looking back, one can say that it is the merit of Eagon and North-
cott [49], [50] to be the first who brought to the attention of algebraists the
determinantal ideals and investigated them by the methods of commutative
and homological algebra. Later on, Buchsbaum and Eisenbud, in a series of
articles, went further along the way of homological investigation of determi-
nantal ideals, while Eagon and Hochster [48] studied them using methods of
commutative algebra in order to prove that the classical determinantal rings
are normal Cohen-Macaulay domains. Subsequently, L. Avramov, T. J´ozefiak,
H. Kleppe, R. E. Kutz, D. Laksov, V. Marinov, P. Pragacz et al. have turned
the attention to the study of determinantal (pfaffian) ideals, respectively of
classical determinantal (pfaffian) rings associated to symmetric (alternating)
matrices; see [3], [74], [75], [76], [83], [84], [85]. As shown later by C. De
Concini, D. Eisenbud, and C. Procesi [41], [42], the appropriate framework
including all three types of rings is that of algebras with straightening law,
and the standard monomial theory on which these algebras are based yields
computationally effective results. A coherent treatment of determinantal ide-
als from this point of view was given by Bruns and Vetter in their seminal
book [28].
A new perspective to the study of the determinantal ideals was brought
by Sturmfels’ paper [92], in which he applied the Knuth-Robinson-Schensted
correspondence in order to determine Gr¨obner bases for determinantal ideals.
Later on, this technique was extended by Herzog an Trung [68] to the study
of 1-cogenerated ideals, ladder determinantal ideals and pfaffian ideals.
Our book aims to a thorough treatment of all three types of determinantal
rings in the light of the achievements of the last fifteen years since the publi-
cation of Bruns and Vetter’s book [28]. We implicitly assume that the reader
is familiar with the basics of commutative algebra. However, we include some
of the main notions and results from [28], for the sake of completeness, but
without proofs. We recommend the reader to first look at the book of Bruns
and Vetter in order to get a feel for the flavor of this field. The first two
iii
iv Introduction
chapters of our book follow the line of investigation from Bruns and Vetter
[28], while the rest of the book relies on the combinatorics of Gr¨obner bases,
a method initiated by Sturmfels [92]. As often as possible, we discuss the case
of 1-cogenerated ideals and their corresponding determinantal rings, though
sometimes we restrict ourselves to the classical case of determinantal ideals
generated by minors (pfaffians) of fixed size, either due to lack of results or
difficulties of exposition.
An important and useful feature of the book is that every chapter contains
a “Notes” section, where we present historical remarks, more bibliographical
sources, open questions and further directions for research.
Our book is meant to be a reference text for the current state of research
in the theory of determinantal rings. It was structured in such a way that it
can be used as textbook for a one semester graduate course in advanced topics
in Commutative Algebra, at PhD level.
We now include brief descriptions of the chapters of the book.
In Chapter 1, we introduce the most important notions that will be used
throughout the book. It is a mainly expository chapter, containing few proofs.
We start with the definition of graded algebras with straightening law on a
poset (doset) over an arbitrary commutative ring. In all sections of the book,
the exposition is split into three parts that correspond to the three types of
matrices we are interested in: in part (G) we study generic matrices, in part
(S) generic symmetric matrices, and in part (A) generic alternating matrices.
Chapter 2 deals with the computations of the divisor class group and of the
canonical class of determinantal rings corresponding to 1-cogenerated ideals.
We also determine which of the rings under consideration are Gorenstein rings.
Chapter3isthecoreoftheentirebook. Wefirstdescribethecombinatorial
algorithms INSERT and DELETE, and then present the Knuth-Robinson-
Schensted correspondence, KRS for short, between standard (Young) bita-
bleaux and two-line arrays of positive integers of a certain type. The theorems
of Schensted and Greene are also reviewed. The theorem of Schensted [88]
deals with the determination of the length of the longest increasing (decreas-
ing) subsequence of a given sequence of integers. The length of the longest
increasing (respectively decreasing) subsequence of a given sequence v is the
length of the first row (respectively column) in the tableau INSERT(v). An in-
terpretationoftherestoftheshapeofINSERT(v)isgivenbyGreene’stheorem
[65]. We describe a KRS type correspondence between standard monomials
and ordinary monomials of a set of indeterminates, and at the end of the chap-
ter we use this correspondence to determine Gr¨obner bases for all three types
of determinantal ideals.
In Chapter 4, we first recall some results on the primary decomposition of
the powers (products) of determinantal ideals. It turns out that the primary
decomposition depends on the characteristic of the ground field. We then
Introduction v
determine Gr¨obner bases for the powers of ideals of maximal minors (pfaffians)
without reference to the characteristic. We conclude the chapter by exploiting
the theorems of Schensted and Greene to determine Gr¨obner bases for the
(symbolic) powers of determinantal ideals.
Chapter 5 brings a new perspective to the study of determinantal ideals
based on the principle of deriving properties of ideals and algebras from their
initial counterparts. As a matter of fact, we associate to the initial ideal of
any determinantal ideal a shellable simplicial complex in such a way that the
corresponding residue class ring of the initial ideal is the Stanley-Reisner ring
of the complex. Some of the combinatorial properties of simplicial complexes
are then interpreted in terms of families of lattice paths of a certain type, and
thus the determination of the multiplicity, the Hilbert series, or the a-invariant
of determinantal rings becomes a matter of counting paths.
All these techniques are brought together in Chapter 6 for the investigation
of Rees algebras associated to determinantal (pfaffian) ideals and of algebras
generatedbyminors(pfaffians)offixedsize. WeshowthattheReesalgebrasof
determinantal(pfaffian)idealsandthealgebrasgeneratedbyminors(pfaffians)
are Cohen-Macaulay normal domains in non-exceptional characteristic. The
rest of the chapter is devoted to the description of the divisor class group and
the canonical class of Rees algebras of determinantal (pfaffian) ideals and of
algebrasgeneratedbyminors(pfaffians). Finally, wedetermine theGorenstein
rings among the rings under consideration.
Inchapter7, weprovethat the classicaldeterminantalrings areF-rational,
anextensionofthewell-knownpropertythatdeterminantalringshaverational
singularities in characteristic 0. We also find that the Rees algebras associ-
ated to determinantal (pfaffian) ideals and the algebras generated by minors
(pfaffians) of fixed size are F-rational (in non-exceptional characteristic).
We want to point out that our book is concerned with 1-cogenerated deter-
minantal ideals and determinantal rings associated with them. An important
generalization of the classical determinantal rings is given by the ladder deter-
minantal rings which are defined by the minors of certain subregions, called
ladders, of a generic (symmetric, respectively alternating) matrix. They were
introduced by Abhyankar for the study of the singularities of Schubert vari-
eties of flag manifolds, and share many properties with classical determinantal
rings, such as: ladder determinantal rings are Cohen-Macaulay normal do-
mains, are F-rational, the Gorenstein property can be completely character-
ized in terms of the shape of the ladder, and there are determinantal formulas
for the a-invariant and Hilbert series. As for the determinantal rings most of
these results rely on the combinatorial structure of the Gr¨obner bases of their
defining ideals. For more details on these topics, we refer the reader to Conca
[34], [35], Conca and Herzog [38], De Negri [45], Ghorpade [58], Herzog and
Trung [68], Krattenthaler and Prohaska [80], Krattenthaler and Rubey [81],
vi Introduction
and Narasimhan [86].
Recently, K. R. Goodearl and T. H. Lenagan [61], [62] initiated the study
of quantum determinantal ideals that arise from quantum generic matrices. It
is worth pointing out that a truly interesting question is to determine non-
commutative Gr¨obner bases for these ideals.
Bucharest, Cornel B˘ae¸tica
January 2004
Chapter 1
Preliminaries
The straightening law of Doubilet, Rota and Stein [47], in the approach of De
Concini, Eisenbud and Procesi [41], is an indispensable tool in the study of
determinantal rings. From the algebraic point of view, the determinantal rings
are graded algebras with straightening law, and this yields some of their main
properties.
Definition 1.0.1 Let A be a B-algebra and ∆ (cid:18) A a finite subset with a
partial order (cid:22), called a poset. Then A is a graded algebra with straightening
law (for short ASL) on ∆ over B if the following conditions hold:
⊕
(H ) A = A is a positively graded B-algebra such that A = B, ∆
0 i≥0 i 0
consists of homogeneous elements of positive degree and generates A as
a B-algebra.
(H ) Theproductsδ (cid:1)(cid:1)(cid:1)δ , u 2 N, δ 2 ∆, suchthatδ (cid:22) (cid:1)(cid:1)(cid:1) (cid:22) δ arelinearly
1 1 u i 1 u
independent over B. We call them standard monomials.
(H ) (Straightening law) For all incomparable δ ,δ 2 ∆ the product δ δ has
2 1 2 1 2
a representation
∑
δ δ = λ µ, λ 2 B, λ 6= 0, µ standard monomial ,
1 2 µ µ µ
that satisfies the following condition: every standard monomial µ con-
tains a factor ζ 2 ∆ such that ζ (cid:22) δ , ζ (cid:22) δ (we allow that δ δ = 0,
∑ 1 2 1 2
then the sum λ µ being empty).
µ
InfactthestandardmonomialsformaB-basisofAcalledthestandard basisof
A. The representation of an element of A as a linear combination of standard
monomials is called its standard representation. The relations in (H ) will be
2
referred to as the straightening relations.
1
2 1. Preliminaries
The most interesting examples of ASLs are rings related to matrices and
determinants; see De Concini, Eisenbud and Procesi [42], Bruns and Vetter
[28].
In order to apply the ASL theory to determinantal rings we have to know
that these rings are indeed ASLs. This follows readily from the fact that their
defining ideals have a distinguished system of generators with respect to the
underlying poset. To be precise, let us consider A an ASL on a poset ∆ (over
B), and ∆′ (cid:18) ∆ a poset ideal of ∆, i.e. ∆′ contains all elements x (cid:22) y if y 2 ∆′.
Set I = ∆′A the ideal of A generated by ∆′.
Proposition 1.0.2 The residue class ring A/I is a graded ASL on ∆ n ∆′
over B.
When ∆′ is the poset ideal cogenerated by a subset Ω of ∆, that is ∆′ =
fξ 2 ∆ : ω 6(cid:22) ξ for any ω 2 Ωg, then the ideal I = ∆′A is called the ideal
cogenerated by Ω. In particular, when jΩj = 1 the ideal cogenerated by Ω is
said to be a 1-cogenerated ideal.
An useful generalization of the notion of graded algebra with straightening
law on a poset is that of graded algebra with straightening law on a doset. We
define the underlying notions.
Definition 1.0.3 Let H be a finite poset with order (cid:22). A subset D of H(cid:2)H
is a doset if D satisfies the following conditions:
(a) (α,α) 2 D for all α 2 H;
(b) if (α,β) 2 D, then α (cid:22) β;
(c) if α (cid:22) β (cid:22) γ 2 H, then
(α,γ) 2 D () (α,β) 2 D and (β,γ) 2 D.
⊕
Definition 1.0.4 LetB bearing, andletA = A beapositivelygraded
i≥0 i
B-algebra such that A = B. Let D be a doset of a poset H, and suppose that
0
D (cid:18) A. Then A is a graded algebra with straightening law on doset D over B
(for short DASL) if the following conditions are satisfied:
(H ) D consists of homogeneous elements of positive degree in A.
0
(H ) Theproductsoftheform(α ,α )(cid:1)(cid:1)(cid:1)(α ,α ),k (cid:21) 1,(α ,α ) 2 D,
1 1 2 2k−1 2k 2i−1 2i
such that α (cid:22) α (cid:22) (cid:1)(cid:1)(cid:1) (cid:22) α (cid:22) α form a B-basis of A. We call
1 2 2k−1 2k
them standard monomials.
1. Preliminaries 3
(H ) For (β ,β ) 2 D, i = 1,...,l, let M = (β ,β )(cid:1)(cid:1)(cid:1)(β ,β ). More-
2 2i−1 2i 1 2 2l−1 2l
over, let
∑
M = λ N, λ 2 B, λ 6= 0, N standard monomial
N N N
be the representation of M as a linear combination of standard mono-
mials (standard representation). Let N = (γ ,γ )(cid:1)(cid:1)(cid:1)(γ ,γ ) be any
1 2 2h−1 2h
of the standard monomials appearing on the right side of the equation.
Then for all permutations σ of the set f1,...,2lg we have that the se-
quence (β ,...,β ) is lexicographically greater or equal than the
σ(1) σ(2l)
sequence (γ ,...,γ ).
1 2h
(H ) In the notation of (H ), suppose that there is a permutation σ such that
3 2
β (cid:22) (cid:1)(cid:1)(cid:1) (cid:22) β . The standard monomial
σ(1) σ(2l)
(β ,β )(cid:1)(cid:1)(cid:1)(β ,β )
σ(1) σ(2) σ(2l−1) σ(2l)
must appear with coefficient (cid:6)1 in the standard representation of M.
Let us now consider the most important examples of ASLs related to matrices
and determinants as they were described by De Concini, Eisenbud and Procesi
[42]; see also Bruns and Vetter [28].
(G) Let X = (X ) be an m(cid:2)n matrix of indeterminates over a commutative
ij
ring B, for short a generic matrix, ∆(X) the set of all minors of X, and
B[X] = B[X : 1 (cid:20) i,j (cid:20) n] the polynomial ring over B. Consider the set
ij
∆(X) ordered in the following way:
[a ,...,a j b ,...,b ] (cid:22) [c ,...,c j d ,...,d ] if and only if t (cid:21) s and a (cid:20) c ,
1 t 1 t 1 s 1 s i i
b (cid:20) d for i = 1,...,s.
i i
(As usual, one denotes by [a ,...,a j b ,...,b ] the minor det(X ) .)
1 t 1 t aibj 1≤i,j≤t
Let δ 2 ∆(X), δ = [a ,...,a j b ,...,b ]. We define I(X,δ) as being the ideal
1 t 1 t
generated by all minors which are not greater or equal than δ. Thus the ideal
I(X,δ) is the ideal of B[X] cogenerated by δ. Denote by R(X,δ) the residue
class ring of B[X] with respect to the ideal I(X,δ) and by ∆(X,δ) the set of
all minors which are greater or equal than δ.
In particular, if δ = [1,...,t j 1,...,t] then I(X,δ) is the ideal I (X)
t+1
generated by all the (t+1)-minors of X, and denote by R (X) the analogous
t+1
residue class ring. When B is a field we call R (X) the classical determi-
t+1
nantal ring.
In the investigation of the rings R(X,δ) we rely on the knowledge of their
combinatorial structure enlightened by De Concini, Eisenbud and Procesi [41],
[42], and on the methods developed by Bruns and Vetter in the fundamental