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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 3, March 2009, Pages 1371–1395 S 0002-9947(08)04636-9 Article electronically published on October 17, 2008 MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS ALBERTO CORSO AND UWE NAGEL Abstract. Each partition λ =(λ 1 2 ,...,λ n ) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying sev- eral features of toric rings of Ferrers graphs. In particular we recover/establish formulæ for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory. 1. Introduction A Ferrers graph is a bipartite graph on two distinct vertex sets X = {x 1 ,...,x n } and Y = {y 1 ,...,y m } such that if (x i ,y j ) is an edge of G, then so is (x p ,y q ) for 1 p i and 1 q j . In addition, (x 1 ,y m ) and (x n ,y 1 ) are required to be edges of G. For any Ferrers graph G there is an associated sequence of non-negative integers λ =(λ 1 2 ,...,λ n ), where λ i is the degree of the vertex x i . Notice that the defining properties of a Ferrers graph imply that λ 1 = m λ 2 ≥···≥ λ n 1; thus λ is a partition. Alternatively, we can associate to a Ferrers graph a diagram T λ , dubbed a Ferrers tableau, consisting of an array of n rows of cells with λ i adjacent cells, left justified, in the i-th row. Ferrers graphs/tableaux have a prominent place in the literature as they have been studied in relation to chromatic polynomials [2, 18], Schubert varieties [16, 15], hypergeometric series [29], permutation statistics [9, 18], quantum mechanical operators [50], and inverse rook problems [23, 16, 15, 42]. More generally, algebraic and combinatorial aspects of bipartite graphs have been studied in depth (see, e.g., [46, 30] and the comprehensive monograph [51]). In this paper, which will be followed by [13], we are interested in the algebraic properties of the edge ideal Received by the editors January 22, 2007. 2000 Mathematics Subject Classification. Primary 05A15, 13D02, 13D40, 14M25; Secondary 05C75, 13C40, 13H10, 14M12, 52B05. The second author gratefully acknowledges partial support from the NSA under grant H98230- 07-1-0065. ©2008 American Mathematical Society Reverts to public domain 28 years from publication 1371 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Transcript of American Mathematical Society · 2018-11-16 · TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY...

Page 1: American Mathematical Society · 2018-11-16 · TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 3, March 2009, Pages 1371–1395 S 0002-9947(08)04636-9 …

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 361, Number 3, March 2009, Pages 1371–1395S 0002-9947(08)04636-9Article electronically published on October 17, 2008

MONOMIAL AND TORIC IDEALSASSOCIATED TO FERRERS GRAPHS

ALBERTO CORSO AND UWE NAGEL

Abstract. Each partition λ = (λ1, λ2, . . . , λn) determines a so-called Ferrerstableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed aFerrers ideal, is a squarefree monomial ideal that is generated by quadrics. Weshow that such an ideal has a 2-linear minimal free resolution; i.e. it definesa small subscheme. In fact, we prove that this property characterizes Ferrersgraphs among bipartite graphs. Furthermore, using a method of Bayer andSturmfels, we provide an explicit description of the maps in its minimal freeresolution. This is obtained by associating a suitable polyhedral cell complexto the ideal/graph. Along the way, we also determine the irredundant primarydecomposition of any Ferrers ideal. We conclude our analysis by studying sev-eral features of toric rings of Ferrers graphs. In particular we recover/establishformulæ for the Hilbert series, the Castelnuovo-Mumford regularity, and themultiplicity of these rings. While most of the previous works in this highlyinvestigated area of research involve path counting arguments, we offer here

a new and self-contained approach based on results from Gorenstein liaisontheory.

1. Introduction

A Ferrers graph is a bipartite graph on two distinct vertex sets X = {x1, . . . , xn}and Y = {y1, . . . , ym} such that if (xi, yj) is an edge of G, then so is (xp, yq) for1 ≤ p ≤ i and 1 ≤ q ≤ j. In addition, (x1, ym) and (xn, y1) are required to beedges of G. For any Ferrers graph G there is an associated sequence of non-negativeintegers λ = (λ1, λ2, . . . , λn), where λi is the degree of the vertex xi. Notice thatthe defining properties of a Ferrers graph imply that λ1 = m ≥ λ2 ≥ · · · ≥ λn ≥ 1;thus λ is a partition. Alternatively, we can associate to a Ferrers graph a diagramTλ, dubbed a Ferrers tableau, consisting of an array of n rows of cells with λi

adjacent cells, left justified, in the i-th row.Ferrers graphs/tableaux have a prominent place in the literature as they have

been studied in relation to chromatic polynomials [2, 18], Schubert varieties [16,15], hypergeometric series [29], permutation statistics [9, 18], quantum mechanicaloperators [50], and inverse rook problems [23, 16, 15, 42]. More generally, algebraicand combinatorial aspects of bipartite graphs have been studied in depth (see,e.g., [46, 30] and the comprehensive monograph [51]). In this paper, which willbe followed by [13], we are interested in the algebraic properties of the edge ideal

Received by the editors January 22, 2007.2000 Mathematics Subject Classification. Primary 05A15, 13D02, 13D40, 14M25; Secondary

05C75, 13C40, 13H10, 14M12, 52B05.The second author gratefully acknowledges partial support from the NSA under grant H98230-

07-1-0065.

©2008 American Mathematical SocietyReverts to public domain 28 years from publication

1371

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1372 A. CORSO AND U. NAGEL

I = I(G) and the toric ring K[G] associated to a Ferrers graph G. The edge idealis the monomial ideal of the polynomial ring R = K[x1, . . . , xn, y1, . . . , ym] over thefield K that is generated by the monomials of the form xiyj , whenever the pair(xi, yj) is an edge of G. K[G] is instead the monomial subalgebra generated by theelements xiyj . An example is illustrated in Figure 1.

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5 y6

• • • • •

• • • • • •

Ferrers graph

x1

x2

x3

x4

x5

y1 y2 y3 y4 y5 y6

Ferrers tableau with partitionλ = (6, 4, 4, 2, 1)

I = (x1y1, x1y2, x1y3, x1y4, x1y5, x1y6, x2y1, x2y2, x2y3, x2y4,

x3y1, x3y2, x3y3, x3y4, x4y1, x4y2, x5y1)

Figure 1. Ferrers graph, tableau and ideal

Throughout this article λ = (λ1, . . . , λn) will always denote a fixed partitionassociated to a Ferrers graph Gλ with corresponding Ferrers ideal Iλ. In Section2 we describe several fine numerical invariants attached to the ideal Iλ. In Theo-rem 2.1 we show that each Ferrers ideal defines a small subscheme in the sense ofEisenbud, Green, Hulek, and Popescu [20]; i.e. the free resolution of Iλ is 2-linear(see also Remark 3.3). More precisely, we give an explicit — but at the same timesurprisingly simple — formula for the Betti numbers of the ideal Iλ; namely, weshow that

βi(R/Iλ) =(

λ1

i

)+

(λ2 + 1

i

)+

(λ3 + 2

i

)+ . . . +

(λn + n − 1

i

)−

(n

i + 1

)for 1 ≤ i ≤ max{λi + i − 1}. Furthermore, the Hilbert series is

∑k≥0

dimK [R/Iλ]k · tk =1

(1 − t)m+

t

(1 − t)m+n+1·

n∑j=1

(1 − t)λj+j .

Notice that the formula for the Betti numbers involves a minus sign. This is quitean unusual phenomenon for Betti numbers, as they tend, in general, to have anenumerative interpretation. In order to determine the Betti numbers it is essentialto find a (not necessarily irredundant) primary decomposition of Iλ. We refinethis decomposition into an irredundant one in Corollary 2.6, where we observe, inparticular, that the number of prime components is related to the outer corners ofthe Ferrers tableau. For instance, in the case of the ideal Iλ described in Figure 1we have that it is the intersection of 5 (= 4 outer corners +1) components:

Iλ = (y1, . . . , y6) ∩ (x1, y1, y2, y3, y4)∩ (x1, x2, x3, y1, y2) ∩ (x1, x2, x3, x4, y1) ∩ (x1, . . . , x5).

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1373

We conclude Section 2 by identifying, in terms of the shape of the tableau, the un-mixed (Corollary 2.7) and Cohen-Macaulay (Corollary 2.8) members in the familyof Ferrers ideals. The latter result also follows from recent work of Herzog and Hibi[30].

There are relatively few general classes of ideals for which an explicit minimalfree resolution is known. The most noteworthy such families include the Koszulcomplex, the Eagon-Northcott complex [17], and the resolution of generic mono-mial ideals [3] (see also [4]). In Section 3 we analyze even further the minimal freeresolution of a Ferrers ideal Iλ and obtain a surprisingly elegant description of thedifferentials in the resolution in Theorem 3.2. In some sense, this is a prototypicalresult as it provides the minimal free resolution of several classes of ideals obtainedfrom Ferrers ideals by appropriate specializations of the variables (see [13] for fur-ther details). Our description of the free resolution of a Ferrers ideal relies on thetheory of cellular resolutions as developed by Bayer and Sturmfels in [3] (see also[41]). More precisely, let ∆n−1 × ∆m−1 denote the product of two simplices ofdimensions n− 1 and m− 1, respectively. Given a Ferrers ideal Iλ, we associate toit the polyhedral cell complex Xλ consisting of the faces of ∆n−1 × ∆m−1 whosevertices are labeled by generators of Iλ (see Definition 3.1). By the theory of Bayerand Sturmfels, Xλ determines a complex of free modules. Using an inductive argu-ment we show in Theorem 3.2 that this complex is in fact the multigraded minimalfree resolution of the ideal Iλ. While leaving the details to the main body of thepaper, we illustrate the situation in the case of the partition λ = (4, 3, 2, 1), whichis the largest we can draw. In this case the polyhedral cell complex Xλ can actuallybe identified with the subdivision of the simplex ∆3 pictured below (see [13] foradditional details):

x1

x2

x3

x4

y1 y2 y3 y4

x1y1

x2y1

x3y1

x4y1

x1y4

x1y3

x1y2x2y3

x3y2

x2y2

Figure 2. Ferrers tableau and associated polyhedral cell complex

In particular, we observe that Xλ has four 3-dimensional cells: two of them areisomorphic to ∆3 whereas the remaining two are isomorphic to either ∆1 × ∆2 or

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1374 A. CORSO AND U. NAGEL

∆2 × ∆1. A grey shading in the picture above also indicates how the polyhedralcell complex corresponding to the partition (3, 2, 1) sits inside Xλ.

In Section 4 we prove the converse of Theorem 2.1. Namely, we show that anyedge ideal of a bipartite graph with a 2-linear resolution necessarily arises froma Ferrers graph (see Theorem 4.2). One of the ingredients of the proof is a well-known characterization of edge ideals of graphs with a 2-linear resolution in termsof complementary graphs, due to Froberg [21] (see also [19]).

The starting point of Section 5 is the observation that the toric ring of aFerrers graph can be identified with a special ladder determinantal ring. Wethen proceed to recover/establish formulæ for the Hilbert series and other in-variants associated with these rings. We remark that this is a highly investi-gated part of mathematics that has been the subject of the work of many re-searchers. Among the extensive, impressive and relevant literature we single out[1, 8, 10, 11, 12, 26, 32, 34, 35, 36, 37, 38, 43, 45, 52]. While most of these worksinvolve — to a different extent — path counting arguments, we offer here a new andself-contained approach that yields easy proofs of explicit formulæ for the Hilbertseries, the Castelnuovo-Mumford regularity, and the multiplicity of the toric ringsof Ferrers graphs. This method, which is based on results from Gorenstein liai-son theory (see [39] for a comprehensive introduction), has been pioneered in [33],where it was proved that every standard determinantal ideal is glicci; i.e. it is in theGorenstein liaison class of a complete intersection (see also [40]). Recently, Gorla[24] has considerably refined these arguments to show that all ladder determinantalideals are glicci. This result can be used to establish first a simple recursive formulafor the Hilbert series, which we then turn into an explicit formula that involves onlypositive summands. We end by saying that the numerical data we explicitly de-termine also allow us to compute the reduction number of Ferrers ideals, which isa key ingredient in determining the structure of the core of ideals. Indeed, someclasses of monomial ideals possess a surprising interpretation for their core (see[44]). Preliminary investigations also show that the core of a Ferrers ideal has aremarkable structure related to the shape of the corresponding Ferrers tableau.

2. Betti numbers and primary decompositions of Ferrers ideals

The main result of this section, Theorem 2.1, provides a (not necessarily irredun-dant) primary decomposition of a Ferrers ideal Iλ as well as the Betti numbers ofits minimal free resolution. A particularly relevant situation is given by a maximalbipartite graph, in which case the Ferrers tableau has a rectangular shape of sizen × m and Iλ = (x1, . . . , xn)(y1, . . . , ym); using a variety of techniques, includingdeterminantal ideals, residual intersections and Grobner basis techniques, [14] and[6] describe additional features of this and other related subideals in connectionwith the so-called Dedekind-Mertens Lemma. A hook-shaped tableau is the otherextremal case; in this situation, Iλ = x1(y1, . . . , ym) + y1(x1, . . . , xn).

For the sake of simplicity we will denote the partition associated to a Ferrersgraph by λ = (λ1, λ2, . . . , λs, 1, . . . , 1) with λ1 ≥ λ2 ≥ · · · ≥ λs ≥ 2. Furthermore,we denote the dual partition by λ∗ = (λ∗

1, λ∗2, . . . , λ

∗m), where λ∗

j is the degree of thevertex yj . Observe that λ1 = m, λ∗

1 = n, and 1 ≤ s = λ∗2 ≤ n. We also recall that

the Hilbert series of the graded K-algebra R/Iλ is

P (R/Iλ, t) :=∑k≥0

hR/Iλ(k) · tk :=

∑k≥0

dimK [R/Iλ]k · tk,

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1375

where hR/Iλis the Hilbert function of R/Iλ. It is well known that this series is a

rational function.

Theorem 2.1. Let G be a Ferrers graph with associated partition λ = (λ1, λ2, . . . ,λs, 1, . . . , 1) and let Iλ be the edge ideal in K[x1, . . . , xn, y1, . . . , ym] associated toG. Then a (not necessarily irredundant) primary decomposition of Iλ is

(y1, . . . , yλ1) ∩ (x1, y1, . . . , yλ2) ∩ (x1, x2, y1, . . . , yλ3) ∩ . . .

∩ (x1, . . . , xs−1, y1, . . . , yλs) ∩ (x1, . . . , xs, y1) ∩ (x1, . . . , xn),

and the minimal Z-graded free resolution of R/Iλ is 2-linear with i-th Betti numbergiven by

βi(R/Iλ) =(

λ1

i

)+

(λ2 + 1

i

)+

(λ3 + 2

i

)+ . . . +

(λn + n − 1

i

)−

(n

i + 1

)

for 1 ≤ i ≤ maxj{λj + j − 1}. Furthermore, the Hilbert series is

P (R/Iλ, t) =1

(1 − t)m+

t

(1 − t)m+n+1·

n∑j=1

(1 − t)λj+j .

Proof. We proceed by induction on λ1 + · · · + λn > 0. If this sum is one, thenIλ = (x1y1) and all the claims are easily verified. Otherwise, we use inductionon n. If n = 1, then λ = (λ1) = (m), s = 1 and Iλ = x1(y1, . . . , ym) = (x1) ∩(y1, . . . , ym) = (y1, . . . , ym) ∩ (x1, y1) ∩ (x1). Moreover, the resolution of I is givenby (a shifted) Koszul complex on m generators. Hence the i-th Betti number is

βi(R/Iλ) =(

m

i

)=

(m

i

)−

(1

i + 1

),

as the latter term is zero. Furthermore, the Hilbert series is

P (R/Iλ, t) =1

(1 − t)m+

t

(1 − t),

as claimed.Suppose now n ≥ 2. We distinguish two main cases: λn = 1 and λn ≥ 2. We

first deal with the case λn = 1. In addition to the partition λ we consider thepartition λ′ = (λ1, . . . , λn−1). Notice that the index s is the same for both λ andλ′. By the induction hypothesis we have that a primary decomposition of Iλ′ is

(y1, . . . , yλ1) ∩ (x1, y1, . . . , yλ2) ∩ (x1, x2, y1, . . . , yλ3) ∩ . . .

∩ (x1, . . . , xs−1, y1, . . . , yλs) ∩ (x1, . . . , xs, y1) ∩ (x1, . . . , xn−1).

Let J denote the intersection of all the components in the above primary decom-position that contain xny1. Thus, Iλ′ = J ∩ (x1, . . . , xn−1). Now observe thatIλ = Iλ′ + (xny1); thus using the above primary decomposition for Iλ′ , we get

Iλ = J ∩ (x1, . . . , xn−1, xny1)= J ∩ (x1, . . . , xn−1, y1) ∩ (x1, . . . , xn−1, xn)= J ∩ (x1, . . . , xn),

which is, after some inspection, exactly the asserted primary decomposition of Iλ.

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1376 A. CORSO AND U. NAGEL

We now turn to the Betti numbers of Iλ. From the given primary decomposition,it follows that Iλ′ : xny1 = (x1, . . . , xn−1). Hence we have the following short exactsequence:

(2.1) 0 → R/(x1, . . . , xn−1)[−2]·xny1−→ R/Iλ′ −→ R/Iλ → 0.

Using a mapping cone construction we obtain that

βi(R/Iλ) = βi(R/Iλ′) + βi−1(R/(x1, . . . , xn−1)) = βi(R/Iλ′) +(

n − 1i − 1

).

Hence, using our inductive assumption, we have that βi(R/Iλ) is given by(λ1

i

)+

(λ2 + 1

i

)+

(λ3 + 2

i

)+ . . . +

(λn−1 + n − 2

i

)−

(n − 1i + 1

)+

(n − 1i − 1

).

However, one can easily check the identity(n−1i−1

)−

(n−1i+1

)=

(ni

)−

(n

i+1

), which

provides the expected form of βi(R/Iλ).Moreover, since the resolution of R/(x1, . . . , xn−1) is linear, the mapping cone

procedure applied to sequence (2.1) also shows that the resolution of R/Iλ is 2-linear. Furthermore, the same exact sequence provides for the Hilbert series

P (R/Iλ, t) = P (R/Iλ′ , t) − t2

(1 − t)m+1.

Using the induction hypothesis, an easy computation provides the claim for theHilbert series of R/Iλ.

Suppose now λn ≥ 2 and consider the partition λ′ = (λ1, λ2, . . . , λn−1, λn − 1).If in addition we assume that λn ≥ 3, then the index s of the partition λ′ alsoequals n. The inductive hypothesis provides that a primary decomposition of Iλ′ is

(y1, . . . , yλ1) ∩ (x1, y1, . . . , yλ2) ∩ (x1, x2, y1, . . . , yλ3) ∩ . . .

∩ (x1, . . . , xn−1, y1, . . . , yλn−1) ∩ (x1, . . . , xn, y1) ∩ (x1, . . . , xn).

Let J denote the intersection of all the components in the above primary decom-position that do contain xnyλn

. Thus, Iλ′ = J ∩ (x1, . . . , xn−1, y1, . . . , yλn−1). Nowobserve that Iλ = Iλ′ + (xnyλn

); thus using the above primary decomposition forIλ′ , we get

Iλ = J ∩ (x1, . . . , xn−1, y1, . . . , yλn−1, xnyλn)

= J ∩ (x1, . . . , xn−1, y1, . . . , yλn−1, yλn) ∩ (x1, . . . , xn−1, y1, . . . , yλn−1, xn)

= J ∩ (x1, . . . , xn−1, y1, . . . , yλn),

which is, after some inspection, exactly the asserted primary decomposition ofIλ. Turning to the Betti numbers of Iλ, the given primary decomposition impliesthat Iλ′ : xnyλn

= (x1, . . . , xn−1, y1, . . . , yλn−1). Hence, by a similar mapping coneargument as above, we obtain that

βi(R/Iλ) = βi(R/Iλ′) + βi−1(R/(x1, . . . , xn−1, y1, . . . , yλn−1))

= βi(R/Iλ′) +(

λn − 1 + n − 1i − 1

)and that the resolution of R/Iλ is 2-linear. Therefore, using our inductive assump-tion we get that βi(R/Iλ) is given by(

λ1

i

)+ . . . +

(λn−1 + n − 2

i

)+

(λn − 1 + n − 1

i

)−

(n

i + 1

)+

(λn − 1 + n − 1

i − 1

),

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1377

which can be rewritten in the form(λ1

i

)+ . . . +

(λn−1 + n − 2

i

)+

(λn + n − 1

i

)−

(n

i + 1

),

as claimed.To finish our proof, let us assume that λn = 2 and consider the partition λ′ =

(λ1, . . . , λn−1, 1), whose index s is n − 1. Moreover, by the inductive assumptionwe have that a primary decomposition of Iλ′ is

(y1, . . . , yλ1) ∩ (x1, y1, . . . , yλ2) ∩ (x1, x2, y1, . . . , yλ3) ∩ . . .

∩ (x1, . . . , xn−2, y1, . . . , yλn−1) ∩ (x1, . . . , xn−1, y1) ∩ (x1, . . . , xn).

Let J denote the intersection of all the components in the above primary decom-position that contain xny2. Thus, Iλ′ = J ∩ (x1, . . . , xn−1, y1). Now observe thatIλ = Iλ′ + (xny2); thus using the above primary decomposition for Iλ′ , we get

Iλ = J ∩ (x1, . . . , xn−1, y1, xny2)= J ∩ (x1, . . . , xn−1, y1, y2) ∩ (x1, . . . , xn−1, y1, xn)= J ∩ (x1, . . . , xn−1, y1, y2),

which is, after some inspection, exactly the asserted primary decomposition of Iλ.The given primary decomposition of Iλ′ provides that Iλ′ : xny2 =(x1, . . . , xn−1, y1).Hence, a mapping cone argument implies that

βi(R/Iλ) = βi(R/Iλ′) + βi−1(R/(x1, . . . , xn−1, y1)) = βi(R/Iλ′) +(

n

i − 1

).

Therefore, using our inductive assumption, we see that βi(Iλ) is given by(λ1

i

)+ . . . +

(λn−1 + n − 2

i

)+

(n

i

)−

(n

i + 1

)+

(n

i − 1

),

which can be rewritten in the form(λ1

i

)+ . . . +

(λn−1 + n − 2

i

)+

(n + 1

i

)−

(n

i + 1

),

which gives the asserted formula as(n+1

i

)=

(2+(n−1)

i

).

The claims about the 2-linearity of the resolution and about the Hilbert seriesfollow similarly as in the case λn = 1. We omit the details. �

Theorem 2.1 allows us to compute further invariants of Ferrers ideals:

Corollary 2.2. Adopt the notation of Theorem 2.1. Then the height of the edgeideal Iλ of a Ferrers graph G is min{n, min

1≤j≤n{λj + j − 1}}, the projective dimen-

sion of the factor ring R/Iλ is max1≤j≤n

{λj + j − 1} and the Castelnuovo-Mumford

regularity reg(Iλ) is equal to 2.

Proof. It suffices to discuss the claim concerning the projective dimension. Seti = max

1≤j≤n{λj + j − 1} and observe that i ≥ n. Hence the formula in Theorem 2.1

implies βi(R/Iλ) ≥ 1. �

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1378 A. CORSO AND U. NAGEL

Example 2.3. Consider the Ferrers ideal Iλ with λ = (6, 4, 4, 2, 1) described inFigure 1. Using our result we get that the projective dimension of R/Iλ is 6, theheight of Iλ is 5, and the minimal free resolution of R/Iλ is of the form

0 → R2(−7) −→ R15(−6) −→ R44(−5)

−→ R65(−4) −→ R50(−3) −→ R17(−2) −→ R −→ R/Iλ → 0.

Remark 2.4. By a result of Herzog-Hibi-Zheng [31, Theorem 3.2], all the powers Ikλ

have a linear resolution so that the Castelnuovo-Mumford regularity of Ikλ , for any

integer k ≥ 1, is reg(Ikλ) = 2k.

Example 2.5. The edge ideal Iλ of the complete bipartite graph on n and mvertices, respectively, is Iλ = (x1, . . . , xn)(y1, . . . , ym) = (x1, . . . , xn) ∩ (y1, . . . , ym)and has i-th Betti number:

βi(R/Iλ) =(

m + n

i + 1

)−

(m

i + 1

)−

(n

i + 1

)for 1 ≤ i ≤ n+m− 1. In fact, the formula for the Betti numbers follows from The-orem 2.1 and [25, Formula (1.48)], or simply by using the Mayer-Vietoris sequence.

The primary decomposition of the Ferrers ideal Iλ described in Theorem 2.1 canbe refined into an irredundant one by using the shape of the Ferrers tableau Tλ.More precisely, define recursively indices j0, . . . , jt by setting j0 = 0 and, for i ≥ 0,

ji+1 = max{k |λk = λji+1}.Note that λj1 = m, jt = n, and that the pairs (ji, λji

), i = 1, . . . , t, are thecoordinates of the outer corners of the Ferrers tableau Tλ. In addition, set λjt+1 = 0and, accordingly, (x1, . . . , xj0) = (0) = (y1, . . . , yλjt+1

). With this notation, westate next our refinement of the primary decomposition described in Theorem 2.1.

Corollary 2.6. The irredundant primary decomposition of the Ferrers ideal Iλ is

Iλ =t+1⋂i=1

(x1, . . . , xji−1 , y1, . . . , yλji),

where the pairs (ji, λji), i = 1, . . . , t, correspond to the t outer corners of the Ferrers

tableau of Iλ. In particular, Iλ is the intersection of t + 1 prime ideals.

Proof. This follows by inspecting the decomposition given in Theorem 2.1. �

Corollary 2.7. Adopt the notation of Theorem 2.1 as well as the one establishedabove. Then the following conditions are equivalent:

(a) Iλ is unmixed ;(b) n = m and the inside corners (ji−1, λji

), for i = 2, . . . , t, of the Ferrerstableau Tλ of Iλ lie on the main anti-diagonal of Tλ, i.e. on {(p, q) | p+q =m}.

Proof. The equivalence of the conditions follows immediately from Corollary 2.6.�

Ferrers ideals are rarely Cohen-Macaulay. In fact, we get:

Corollary 2.8. The following conditions are equivalent:(a) Iλ is unmixed and connected in codimension one;

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1379

(b) n = m and λ = (n, n − 1, n − 2, . . . , 3, 2, 1);(c) Iλ is a Cohen-Macaulay ideal.

In particular, in this case the i-th Betti number is:

βi(R/Iλ) = i

(n + 1i + 1

)for 1 ≤ i ≤ n (= m). Moreover, the Cohen-Macaulay type is n and the Hilbertseries P (R/Iλ, t) is given by

P (R/Iλ, t) =1 + nt

(1 − t)n.

Proof. Corollary 2.2 shows that (b) implies (c). Condition (a) is always a conse-quence of (c). Using Corollary 2.6, we see that (a) implies (b).

Concerning the Betti numbers of Iλ, Theorem 2.1 and the shape of the partitionλ provide that

βi(R/Iλ) = n

(n

i

)−

(n

i + 1

)for 1 ≤ i ≤ n. On the other hand, an easy calculation shows that the latterexpression equals i

(n+1i+1

), as claimed. The statement concerning the Hilbert series

follows immediately from Theorem 2.1. �

We note that the equivalence of conditions (b) and (c) above can also be deducedfrom a recent result of Herzog and Hibi [30, Theorem 3.4]. In general, the conditionthat a projective subscheme Z ⊂ Pn is equidimensional and connected in codimen-sion one is only a necessary condition for Z being arithmetically Cohen-Macaulay.However, if Z is defined by a monomial ideal that has a combinatorial interpreta-tion, then it seems often the case that this condition is also sufficient. Note howeverthat it is not always true. For example, if Z ⊂ P4 is defined by (xy, xzw, ywt, zwt),then Z is connected in codimension 1, but Z is not arithmetically Cohen-Macaulay.

3. Minimal free resolutions of Ferrers ideals

In this section we explicitly describe the minimal free resolution of every Ferrersideal. For our construction we use cellular resolutions and polyhedral cell complexesas introduced by Bayer and Sturmfels in [3]. First, we briefly recall some basicnotions, but we refer to [3] (or [41]) for a more detailed introduction to the topic.A polyhedral cell complex X is a finite collection of convex polytopes (in some RN )called faces (or cells) of X such that:

(1) if P ∈ X and F is a face of P , then F ∈ X;(2) if P, Q ∈ X, then P ∩ Q is a face of both P and Q.

Let Fk(X) be the set of k-dimensional faces. Each cell complex admits an incidencefunction ε on X where ε(Q, P ) ∈ {1,−1} if Q is a facet of P ∈ X. X is calleda labeled cell complex if each vertex i has a vector ai ∈ NN (or the monomialzai , where zai denotes a monomial in the variables z1, . . . , zN ) as label. The labelof an arbitrary face Q of X is the exponent aQ, where zaQ := lcm (zai | i ∈ Q).Each labeled cell complex determines a complex of free R-modules, where R =K[z1, . . . , zN ]. The cellular complex FX supported on X is the complex of freeZ

N -graded R-modules

FX : 0 → SFd(X) ∂d−→ SFd−1(X) ∂d−1−→ · · · ∂2−→ SF1(X) ∂1−→ SF0(X) ∂0−→ S → 0,

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1380 A. CORSO AND U. NAGEL

where d = dim X and SFk(X) :=⊕

P∈Fk(X)

R[−aP ]. The map ∂k is defined by

∂k(eP ) :=∑

Q facet of P

ε(P, Q) · zaP −aQ · eQ,

where {eP |P ∈ Fk(X)} is a basis of SFk(X) and e∅ := 1. If FX is acyclic, thenit provides a free Z

N -graded resolution of the image I of ∂0, that is, the idealgenerated by the labels of the vertices of X. In this case, FX is called a cellularresolution of I.

We are ready to describe a cellular minimal free resolution for each Ferrers ideal.First, let us consider the complete bipartite graph Kn,m that corresponds to the edgeideal (x1, . . . , xn)(y1, . . . , ym) (or the partition λ, where λi = m for i = 1, . . . , n).To this graph, we associate the polyhedral cell complex Xn,m given by the facecomplex of the polytope ∆n−1 × ∆m−1 obtained by taking the cartesian productof the (n− 1)-simplex ∆n−1 and the (m− 1)-simplex ∆m−1. Labeling the verticesof ∆n−1 by x1, . . . , xn and the ones of ∆m−1 by y1, . . . , ym, the vertices of the cellcomplex Xn,m are naturally labeled by the monomials xiyj with 1 ≤ i ≤ n and1 ≤ j ≤ m. An easy example is illustrated in Figure 3.

x1

x2

y1 y2 y3

••

••

x1y1

x1y2

x1y3

x2y1

x2y2

x2y3

Figure 3. Topological viewpoint, ∆1 × ∆2

To simplify notation, we denote the monomial that labels the face P ∈ Xn,m by mP .Recall that mp is the least common multiple of the monomial labels of the verticesof P . In general, we observe that deg mP = dimP + 2 for each face P ∈ Xn,m.

We are now in a position to define the cell complex that will support the cellularresolution of a given Ferrers ideal. As above, we fix a partition λ = (λ1, λ2, . . . , λn)with λ1 = m, corresponding to a Ferrers graph Gλ and a Ferrers ideal Iλ ⊂ R =K[x1, . . . , xn, y1, . . . , ym].

Definition 3.1. The polyhedral cell complex Xλ associated to the partition λ isthe labeled subcomplex of Xn,m consisting of the faces of Xn,m whose vertices arelabeled by all the monomials generating the Ferrers ideal Iλ.

Using the Ferrers tableau Tλ we get a more explicit, yet simple, description ofthe cell complex Xλ. In fact, it is easy to see that the facets of Xλ are in one-to-one correspondence with the outer corners of Tλ. More precisely, if (i, λi) isan outer corner of Tλ, then the product of the polytopes (simplices) with vertices{x1, . . . , xi} and {y1, . . . , yλi

} is a facet of Xλ. Each facet of Xλ determines a

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1381

rectangular region in the Ferrers tableau Tλ. The intersection of the regions corre-sponding to two facets is again a rectangle that corresponds to a product of smallersimplices. This product polytope is the intersection of the two facets of Xλ. Anexample is illustrated in Figure 4.

x1

x2

x3

y1 y2 y3

x1y1

x2y1

x3y1

x1y3

x1y2x2y3

x3y2

x2y2

Figure 4. Faces of the polyhedral cell complex Xλ

The main result of this section is:

Theorem 3.2. The complex FXλprovides the minimal free Zm+n-graded resolution

of Iλ.

It is clear that FXλalso gives a Z-graded minimal free resolution of Iλ. In fact,

since the label of each k-dimensional face of Xλ has degree k + 2 as noted above,we get RFk(Xλ) ∼= Rfk(−k − 2), where fk := |Fk(Xλ)|. Hence, we find again (asseen in Theorem 2.1) that the Z-graded minimal free resolution of Iλ is 2-linear.

Proof. For fixed m = λ1, we will induct on |λ| = λ1 + . . . + λn ≥ m. If |λ| = m,then Xλ = X1,m and the claim follows from the discussion of complete bipartitegraphs above Definition 3.1. Let |λ| > m. We divide the argument into five steps:

(I) For each k ≤ n + λn − 2, let Gk−1 ⊂ RFk(Xλ) denote the free R-modulegenerated by the k-dimensional faces of Xλ involving the vertex xnyλn

. Its rank is

rank Gk−1 =(

n + λn − 2k

).

Indeed, each such face corresponds to the boxes lying on a suitable grid of a rowsand b columns indexed by i1 < i2 < · · · < ia = n and j1 < j2 < · · · < jb = λn,where a + b − 2 = k. In this way, we see that the number of such k-dimensionalfaces is

rank Gk−1 =k+1∑a=1

(n − 1a − 1

)(λn − 1b − 1

)=

k+1∑a=1

(n − 1a − 1

)(λn − 1

k − a + 1

)

=k∑

j=0

(n − 1

j

)(λn − 1k − j

)=

(n + λn − 2

k

).

The latter equality is nothing but the Vandermonde convolution [25, Formula 3.1].

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1382 A. CORSO AND U. NAGEL

(II) The proof of Theorem 2.1 shows that there is a partition λ′ such that|λ′| = |λ| − 1,

Iλ = Iλ′ + (xnyλn) and Iλ′ : xnyλn

= (x1, . . . , xn−1, y1, . . . , yλn−1).

This provides the exact sequence

0 → R/(x1, . . . , xn−1, y1, . . . , yλn−1)[−2]·xnyλn−→ R/Iλ′ −→ R/Iλ → 0.

(III) Let ε be the incidence function of Xλ that gives the signs in FXλ. Then

its restriction to Xλ′ is an incidence function too, which we use to define the cellcomplex FXλ′ .

Observe that, for each variable l ∈ R and each non-empty face P ∈ Xλ, there isa unique facet Q of P such that mP = l · mQ. We denote this facet Q by P/l. LetP denote a k-dimensional face of Xλ involving the monomial xnyλn

and observethat ∂k(eP ) can be written as

∂k(eP ) =∑l|mP

l · ε(P, P/l)eP/l

=∑l|mP

l�xnyλn

l · ε(P, P/l)eP/l + xnε(P, P/xn)eP/xn+ yλn

ε(P, P/yλn)eP/yλn

= ϕk−1(eP ) + (−1)kδk−1(eP ),

where

ϕk−1(eP ) =∑l|mP

l�xnyλn

l · ε(P, P/l)eP/l,

δk−1(eP ) = (−1)kxnε(P, P/xn)eP/xn+ (−1)kyλn

ε(P, P/yλn)eP/yλn

.

Note that ϕk−1(eP ) is in Gk−2. Thus, we get a sequence of graded R-modules:

G• : 0 −→ Gn+λn−3ϕn+λn−3−→ . . . −→ G1

ϕ1−→ G0 −→ 0,

where the image of ϕ1 is the ideal (x1, . . . , xn−1, y1, . . . , yλn−1). In fact, it is not toodifficult to see that G• is actually the Koszul complex on x1, . . . , xn−1, y1, . . . , yλn−1,where the degrees are shifted by −2.

(IV) Set H ′k = RFk(Xλ′ ). Then RFk(Xλ) = H ′

k ⊕ Gk−1. Moreover, for eachgenerator eP ∈ Gk−1, δk−1(eP ) is in H ′

k−1. Hence, we get the following square:

Gk−2δk−2

H ′k−2

Gk−1

ϕk−1

δk−1H ′

k−1.

∂′k−1

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1383

We claim that it is commutative, i.e. δk−2 ◦ ϕk−1 = ∂′k−1 ◦ δk−1. Indeed, we have

that

δk−2(ϕk−1(eP )) = δk−2

( ∑l| mP

xnyλn

l ε(P, P/l) eP/l

)

= (−1)k−1

( ∑l| mP

xnyλn

xnl ε(P, P/l)ε(P/l, P/lxn)eP/lxn

+yλnl ε(P, P/l)ε(P/l, P/lyλn

)eP/lyλn

).

On the other hand, we get

∂′k−1(δk−1(eP ))

= ∂′k−1

((−1)kxn ε(P, P/xn)eP/xn

+ (−1)kyλnε(P, P/yλn

)eP/yλn

)= (−1)kxn ε(P, P/xn)

∑l|mP

xn

l ε(P/xn, P/lxn)eP/lxn

+ (−1)kyλnε(P, P/yλn

)∑

l| mPyλn

l ε(P/yλn, P/lyλn

)eP/lyλn

= (−1)kxn ε(P, P/xn)( ∑

l| mPxnyλn

l ε(P/xn, P/lxn)eP/lxn

+ yλnε(P/xn, P/xnyλn

)eP/xnyλn

)

+(−1)kyλnε(P, P/yλn

)( ∑

l| mPxnyλn

l ε(P/yλn, P/lyλn

)eP/lyλn

+ xn ε(P/yλn, P/xnyλn

)eP/xnyλn

)

= (−1)kxn ε(P, P/xn)( ∑

l| mPxnyλn

l ε(P/xn, P/lxn)eP/lxn

)

+ (−1)kyλnε(P, P/yλn

)( ∑

l| mPxnyλn

l ε(P/yλn, P/lyλn

)eP/lyλn

)

= (−1)k−1

( ∑l| mP

xnyλn

xnl ε(P, P/l)ε(P/l, P/lxn)eP/lxn

+ yλnl ε(P, P/l)ε(P/l, P/lyλn

)eP/lyλn

).

Observe that the last two equalities follow from one of the properties of incidencefunctions:

ε(F, F/l) · ε(F/l, F/lh) + ε(F, F/h) · ε(F/h, F/lh) = 0.

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1384 A. CORSO AND U. NAGEL

(V) By Step (IV), δ• : G• → FXλ′ is a morphism of chain complexes. It allowsus to apply the mapping cone procedure to the exact sequence in Step (II), whichprovides the desired free resolution of R/Iλ. �

Remark 3.3. In [20], Eisenbud, Green, Hulek, and Popescu consider more generallya projective subscheme that is the union of linear subspaces and that has a 2-linearfree resolution. They construct a free resolution of such a scheme X. However, itis not in general minimal though it gives the exact number of minimal generatorsof the homogeneous ideal IX . Our Theorem 3.2 treats the special case where IX isa monomial ideal, but our conclusion is stronger.

4. Characterization of Ferrers graphs

In this brief section we establish an intrinsic characterization of Ferrers graphs(not referring to a suitable labeling) by proving the converse of Theorem 2.1. Inother words, we characterize Ferrers ideals as essentially the only edge ideals witha 2-linear free resolution among the ones arising from bipartite graphs. To this endwe will use a result of Froberg, which has been recently refined in [19].

Let G be a finite graph on the vertex set V = {v1, . . . , vn}. We recall that thecomplementary graph G of G is the graph (on the same vertex set V as G) suchthat, for vertices vi, vj ∈ V , the pair (vi, vj) is an edge of G if and only if (vi, vj) isnot an edge of G. Furthermore, the graph G is called chordal if every cycle of G oflength at least 4 has a chord. With this notation, a result of Froberg says:

Theorem 4.1 (Froberg [21]). The edge ideal of a graph G has a 2-linear freeresolution if and only if the complementary graph G is chordal.

Note that adding an isolated vertex to a given graph G does not change thegenerating set nor the graded Betti numbers of the edge ideal of G. Thus, it isharmless to assume that the graph does not have isolated vertices. We are nowready to show:

Theorem 4.2. Let G be a bipartite graph without isolated vertices. Then its edgeideal has a 2-linear free resolution if and only if G is (up to a relabeling of thevertices) a Ferrers graph.

Proof. We have shown in Theorem 2.1 that the condition is sufficient. We nowestablish its necessity. Thus, let G be a bipartite graph on two distinct sets ofvertices, say {x1, . . . , xn} and {y1, . . . , ym}, and assume that its edge ideal has a2-linear resolution. Let λi be the degree of xi. By relabeling the vertices, we mayalso assume that m ≥ λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 1 and that the edges connected to x1

are labeled y1, . . . , yλ1 .For 1 ≤ i ≤ n, we now claim first that the λi vertices connected to xi are among

the vertices in {y1, . . . , yλi−1} and second that in fact they are exactly the firstconsecutive λi vertices contained in {y1, . . . , yλi−1}. Indeed, there is nothing toprove if i = 1. Let i > 1 and first assume that xi is connected to some yk withk > λi−1. Thus there exists some j, with 1 ≤ j ≤ λi−1, such that xiyj is notan edge in G. Moreover, the induction hypothesis provides that xi−1yk is not anedge of G either. It follows that the complementary graph G contains the cycleΓ = {xiyj , xi−1yk, xi−1xi, yjyk} of length 4. However, none of the chords of Γ,namely xiyk and xi−1yj , belongs to G. This contradicts Froberg’s theorem. Now,

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1385

by relabeling the vertices of G, we may assume that the vertices connected to xi

are exactly the first consecutive λi vertices contained in {y1, . . . , yλi−1}.As a by-product of our argument, we observe that λ1 is exactly m. It also shows

that if (xp, yq) is an edge of G, then so is (xh, yk), provided 1 ≤ h ≤ p and 1 ≤ k ≤ q.In addition, (x1, ym) and (xn, y1) are edges of G. Hence G is a Ferrers graph asclaimed. �

The following example shows that there are edge ideals with a 2-linear resolutionwhich do not arise from a bipartite graph. However, this ideal can be obtained asa specialization of a suitable Ferrers ideal.

Example 4.3. Let R = K[x1, x2, x3] be a polynomial ring over a field K andlet I be the edge ideal corresponding to a cycle of length three, that is, I =(x1x2, x1x3, x2x3). Clearly, the ideal I does not arise from a bipartite graph.On the other hand, it is a height two Cohen-Macaulay ideal with the following2-linear resolution:

0 → R2[−3]ϕ−→ R3[−2] −→ R −→ R/I → 0.

We notice though that it can be obtained as a specialization of the Ferrers idealIλ = (x1y1, x1y2, x2y1), by setting y1 := x3 and y2 := x2 (see [13]).

Notice that the combination of Theorems 4.2 and 2.1 provides a complete de-scription of the possible Betti numbers of edge ideals of bipartite graphs with a2-linear free resolution.

5. Toric rings associated to Ferrers ideals

Let T := Tλ denote the Ferrers tableau associated to a Ferrers graph G withpartition λ = (λ1, . . . , λs, 1, . . . , 1). We now define an associated tableau T′ ob-tained from T by deleting all boxes in the first row beyond the λ2 one, and allboxes in the first column beyond the s one. Hence the partition λ′ associated toT′ is (λ2, λ2, λ3, . . . , λs). Observe that, in this manner, the thickness of the outerborder of T′ is at least 2. From a combinatorial point of view, we removed fromG all the vertices (and, a fortiori, the corresponding edges) having degree 1. Anexample is illustrated in Figure 5.

Ferrers tableau T Ferrers tableau T′

x1

x2

x3

x4

x5

y1 y2 y3 y4 y5

x1

x2

x3

x4

y1 y2 y3 y4

••

Figure 5. Ferrers tableaux T and T′

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1386 A. CORSO AND U. NAGEL

According to [46], the Rees algebra R[It], the associated graded ring grI(R) andthe special fiber ring F(I) of the edge ideal I of every bipartite graph are normalCohen-Macaulay domains. Since edge ideals are generated in one degree, the specialfiber ring is also isomorphic to the toric ring of the graph.

Proposition 5.1. Let X = {x1, . . . , xn} and Y = {y1, . . . , ym} be distinct setsof variables. Set R = K[X,Y], where K is a field, and let Iλ be the edge idealcorresponding to a Ferrers graph Gλ with associated tableaux T and T′, and par-tition λ = (λ1, λ2, . . . , λs, 1, . . . , 1). Then the special fiber ring F(Iλ) of Iλ has thefollowing properties:

(a) F(Iλ) is a Cohen-Macaulay normal domain of dimension n + m − 1;(b) F(Iλ) is the ladder determinantal ring k[T]/I2(T′);(c) F(Iλ) is Gorenstein if and only if λ2 = s and all the inside corners (if

any) of the Ferrers tableau T′ lie on the main anti-diagonal of T′, i.e.{(i, j) ∈ T′ | i + j = λ2 + 1}.

Proof. The result stated in (a) is due to Simis, Vasconcelos and Villarreal and holdsfor the special fiber ring of the edge ideal of every connected bipartite graph [46].It is also recovered by part (b), as ladder determinantal rings are known to havesuch properties (see [43, 32, 10]).

In order to prove (b), observe that F(Iλ) ∼= K[xiyj ’s] = K[Gλ], as Iλ is generatedby homogeneous polynomials of the same degree. Moreover, since Gλ is a bipartitegraph its dimension is m + n − 1 (see [46] or [51, 8.2.13]). Let T and T′ denotethe Ferrers tableaux associated to Gλ. Let Tij , for (i, j) ∈ T, be distinct variables:each variable is associated to the corresponding box of the tableau T (and T′,respectively). By abuse of notation we also let T (and T′, respectively) denote thecollection of these new variables. We now consider the following epimorphism

π : K[T] � K[Gλ] ∼= F(Iλ),

where π(Tij) = xiyj . We claim that the kernel of π is the determinantal idealI2(T′) = I2(T′) · k[T] generated by the 2× 2 minors of the one-sided ladder T′. Itis clear that the ideal I2(T′) · k[T] is contained in the ideal ker(π). On the otherhand, we now show that these ideals have the same height. Hence they coincide, asthey are both prime ideals (see [43] for the primeness of I2(T′)). Indeed, we have

ht ker(π) = dim k[T] − dim k[Gλ]= (λ1 + λ2 + · · · + λs + n − s) − (n + m − 1)= λ2 + · · · + λs − s + 1

(as λ1 = m), whereas

ht I2(T′) · k[T] = ht I2(T′) · k[T′]= dim k[T′] − dimR2(T′)= λ2 + λ2 + · · · + λs − (s + λ2 − 1)= λ2 + · · · + λs − s + 1.

As far as (c) is concerned, the Gorensteiness of F(Iλ) now follows from work ofConca [10, 2.5]. �Corollary 5.2. The special fiber ring F(Iλ) is Gorenstein if and only if there is apartition µ such that the Ferrers ideal Iµ is unmixed and there are variables suchthat the polynomial rings over F(Iλ) and F(Iµ), respectively, are isomorphic.

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1387

Proof. Assume that F(Iλ) is Gorenstein. Then define µ := (λ2 + 1, λ2, . . . , λs, 1) ∈Zλ2+1. It follows that the ladder determinantal ideals determined by λ and µ,respectively, have the same generators. Moreover, Proposition 5.1 and Corollary 2.7provide that Iµ is unmixed.

Conversely, if Iµ is unmixed, then we see that F(Iµ) is Gorenstein using firstCorollary 2.7 and then Proposition 5.1. �

As announced earlier, we now turn our attention to the computation of theHilbert series of the toric ring K[Gλ]. This is a highly investigated area of research;see, for example, [1, 8, 10, 11, 12, 26, 32, 34, 35, 36, 37, 38, 43, 45, 52]. Whilemost of these works involve — to a different extent — path counting arguments,we offer here a new and self-contained approach based on Gorenstein liaison theory.Proposition 5.1 implies that, for each partition λ ∈ N

n, there is a unique polynomialpλ ∈ Z[t] such that the Hilbert series of K[Gλ] ∼= F(Iλ) can be written as

P (K[Gλ], t) =pλ(t)

(1 − t)n+m−1.

Note that the multiplicity of K[Gλ] is e(K[Gλ]) = pλ(1). With this notationand using Gorenstein liaison theory methods, we establish the following key result,which provides a simple recursive formula for the Hilbert series.

Lemma 5.3. Let λ = (λ1, . . . , λn) ∈ Nn be a partition with λn ≥ 2. Set λ′′ =

(λ1, . . . , λn−1, λn − 1) ∈ Nn and λ′ = (λ1 − λn + 1, . . . , λn−1 − λn + 1) ∈ Nn−1. Ifn ≥ 3, then there is the following relation among Hilbert series:

pλ(t) = pλ′′(t) + t · pλ′(t).

Proof. We need some more notation. Given the partition λ ∈ Zn, we define S :=K[T] as the polynomial ring in the λ1 + . . .+λn variables Tij and the ideal Jλ ⊂ S

by S/Jλ := K[Gλ]. Let T′′ be the Ferrers tableau associated to λ′′ and let T bethe Ferrers tableau for the partition λ := (λ1, . . . , λn−1) ∈ Zn−1. Furthermore,

denote by N and ˜T the subtableaux of T consisting of the first (λn − 1) and theremaining columns, respectively. Let I1(N) be the ideal generated by the entries

of N and let I2(˜T) be the ideal generated by the 2× 2 minors whose entries are in˜T. Finally, let V, W, V ′ ⊂ Proj (S) be the subvarieties that are defined by Jλ, Jλ′′ ,

and Jλ + I1(N), respectively.In [24, proof of Theorem 2.1], Gorla shows that V is an elementary biliaison of

V ′ on W . Thus, V is linearly equivalent to a basic double link of V ′ on W . Inparticular, both have the same Hilbert function. If follows (see, for instance, [33,Lemma 4.8]) that the Hilbert functions satisfy for all integers j:

hV (j) = hV ′(j − 1) + hW (j) − hW (j − 1).

In terms of Hilbert series this reads as

P (S/Jλ, t) = t · P (S/IV ′ , t) + (1 − t) · P (S/Jλ′′S, t).

Thus we get using Proposition 5.1:

(5.1)pλ(t)

(1 − t)m+n−1= t · P (S/IV ′ , t) + (1 − t) · pλ′′(t)

(1 − t)m+n

because P (S/Jλ′′S, t) =1

1 − t· P (K[Gλ′′ ], t) =

pλ′′(t)(1 − t)m+n

.

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1388 A. CORSO AND U. NAGEL

The definition of the homogeneous ideal of V ′ implies

IV ′ = I2(˜T) + I1(N).

It follows that S/IV ′ is isomorphic to a polynomial ring in λn variables over

K[ ˜T]/I2(˜T) ∼= K[Gλ′ ]. Since dim K[Gλ′ ] = m + n − λn − 1, we get

P (S/IV ′ , t) =1

(1 − t)λn· P (K[Gλ′ ], t) =

1(1 − t)λn

· pλ′(t)(1 − t)m+n−λn−1

=pλ′(t)

(1 − t)m+n−1.

Substituting in Equation (5.1), the claim follows. �

As a first consequence, we derive an explicit formula for the Hilbert series. Ob-serve that all terms are non-negative.

Theorem 5.4. Let λ = (λ1, . . . , λn) ∈ Nn be a partition with n ≥ 2. Then the

numerator of the normalized Hilbert series of K[Gλ] is

pλ(t) = 1 + h1(λ) · t + · · · + hn−1(λ) · tn−1,

where

h1(λ) =n∑

j=2

(λj − 1)

and

hk(λ) =∑

2≤i1<i2<...<ik≤n

λi1−k∑jk−1=λi1−λik

−k+2

jk−1∑jk−2=λi1−λik−1−k+3

. . .

j2∑j1=λi1−λi2

j1,

for k ≥ 2.

Proof. We use the notation introduced in Lemma 5.3 and its proof. This resultimplies for all k ∈ N:

(5.2) hk(λ) = hk(λ′′) + hk−1(λ′).

It is easy to see that this recursion provides the formula for h1(λ).We now induct on n ≥ 2. If n = 2, the minimal free resolution of F(Iλ) is given

by an Eagon-Northcott complex. This implies in particular that

pλ(t) = 1 + (λ2 − 1) · t,

as claimed. Let n ≥ 3. Now we induct on k ≥ 2. Since the case k = 2 is similar,but easier than the general case, we assume k ≥ 3. We now induct on λn ≥ 1. If

λn = 1, then the sumλi1−k∑

jk−1=λi1−λn−k+2

vanishes. Thus in the formula for hk(λ) all

sums with ik = n vanish. This implies that we have to show hk(λ) = hk(λ), whereλ = (λ1, . . . , λn−1). But this is true because the ideals Jλ and Jλ have the samegenerators.

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1389

Finally, we may assume that λn ≥ 2. Then the induction hypotheses and For-mula (5.2) provide by distinguishing the cases ik < n and ik = n:

hk(λ)

=∑

2≤i1<...<ik≤n−1

λi1−k∑jk−1=λi1−λik

−k+2

jk−1∑jk−2=λi1−λik−1−k+3

. . .

j2∑j1=λi1−λi2

j1

+∑

2≤i1<...<ik−1≤n−1

λi1−k∑jk−1=λi1−λn−k+3

jk−1∑jk−2=λi1−λik−1−k+3

. . .

j2∑j1=λi1−λi2

j1

+∑

2≤i1<...<ik−1≤n−1

λi1−λn−k+2∑jk−2=λi1−λik−1−k+3

jk−2∑jk−3=λi1−λik−2−k+4

. . .

j2∑j1=λi1−λi2

j1

=∑

2≤i1<...<ik≤n−1

λi1−k∑jk−1=λi1−λik

−k+2

jk−1∑jk−2=λi1−λik−1−k+3

. . .

j2∑j1=λi1−λi2

j1

+∑

2≤i1<...<ik−1≤n−1

λi1−k∑jk−1=λi1−λn−k+2

jk−1∑jk−2=λi1−λik−1−k+3

. . .

j2∑j1=λi1−λi2

j1

=∑

2≤i1<i2...<ik≤n

λi1−k∑jk−1=λi1−λik

−k+2

jk−1∑jk−2=λi1−λik−1−k+3

. . .

j2∑j1=λi1−λi2

j1.

This completes the proof. �

Remark 5.5. It is well known that the coefficient hi(λ) of ti in pλ(t) has a combina-torial interpretation. In fact, interpreting the Ferrers tableau as a bounded regionin the lattice Z2, hi(λ) is the number of lattice paths inside Tλ that start in thesouthwest corner, end in the northeast corner, and have exactly i east-north turns(see [1, 22, 32, 36]).

For the multiplicity of K[Gλ] we obtain a somewhat simpler formula:

Corollary 5.6.

e(K[Gλ]) =λ2∑

jn−2=λ2−λn+1

jn−2∑jn−3=λ2−λn−1+1

. . .

j2∑j1=λ2−λ3+1

j1.

Proof. Since e(F(Iλ)) = pλ(1), this follows from Theorem 5.4. However, compu-tationally, it is easier to use more directly Lemma 5.3, which implies e(K[Gλ]) =e(K[Gλ′ ]) + e(K[Gλ′′ ]). �

The method of proof, using Gorenstein liaison theory, applies to all ladder de-terminantal ideals [24]. However, here we restrict ourselves to the ideals relatedto Ferrers graphs, i.e. to one-sided ladder determinantal ideals generated by 2 × 2minors.

We recall that for a finitely generated graded module M (over an affine K-algebra) a suitable measure for the complexity of its resolution (hence of M itself) isgiven by the Castelnuovo-Mumford regularity reg(M), that is, max{j− i | βij = 0},where βij are the graded Betti numbers of M . On the other hand, the a-invariant

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1390 A. CORSO AND U. NAGEL

a(M) of M is the degree of the Hilbert series of M as a rational function. In generalthese numbers are related by a(M) ≤ reg(M) − depth(M), with equality if M isCohen-Macaulay. In the latter case, we thus have that reg(M) equals the degreeof the numerator pM (t) of the Hilbert series of M . One can also interpret a(M)in terms of non-vanishing of the top local cohomology of M . The approach wefollowed thus far allows us to easily compute the Castelnuovo-Mumford regularityand the a-invariant of the toric ring K[Gλ]. Before stating our results, we recallthat for a partition λ, we set s := s(λ) := λ∗

2. Note that λs ≥ 2.

Proposition 5.7. Let λ be a partition such that λ2 ≥ 2. Then the Castelnuovo-Mumford regularity of the toric ring of the Ferrers graph Gλ is

reg(K[Gλ]) = min{λ∗2 − 1, {λj + j − 3 | 2 ≤ j ≤ λ∗

2 =: s}}

={

s − 1 if λs ≥ 3min{j − 1 | λj = 2} if λs = 2.

Proof. The second equality follows simply by evaluating the minimum using λs +s − 3 > s − 1 if λs ≥ 3. In order to show the first equality, we note that theCohen-Macaulayness of K[Gλ] implies that reg(K[Gλ]) = deg pλ(t). Now for thisproof denote by rλ −1 the right-hand side of the claim. Then we have to show thatdeg pλ = rλ − 1. This follows directly from Theorem 5.4. Alternatively, we can useLemma 5.3, and it suffices to show (using its notation):

rλ = max{rλ′′ , 1 + rλ′}.

But this can be easily checked. �

Corollary 5.8. Let λ be a partition such that λ2 ≥ 2. Then the a-invariant of thetoric ring of the Ferrers graph Gλ is

a(K[Gλ]) = −(n + m − 1) + min{λ∗2 − 1, {λj + j − 3 | 2 ≤ j ≤ λ∗

2}}.

Proof. This result follows by applying Theorem 5.1, the equality a(K[Gλ]) =−dim(K[Gλ]) + reg(K[Gλ]) and Proposition 5.7. �

Remark 5.9. We find it noteworthy, at this stage, to highlight an existing connectionbetween a-invariants and Integer Programming techniques, as pointed out by Va-lencia and Villarreal in [49]. In fact, for the type of ideals considered in this paper,the computation of the a-invariant amounts to the computation of the maximumnumber of edge disjoint directed cuts or equivalently to the minimum cardinalityof the edge set that contains at least one edge of each directed cut. The latter isnot easily computable using combinatorial techniques; thus it is remarkable thatCorollary 5.8 provides an explicit formula.

We conclude this section by discussing particular classes of Ferrers graphs wherethe formulas simplify considerably. In Example 5.10 we recover in a simple way theexpression for the multiplicity (due to Herzog and Trung [32]) and the coefficientsof the Hilbert series of the 2 × 2 minors of a generic n × m matrix (due to Concaand Herzog [12]). In the same simple fashion, we recover in Example 5.11 a resultthat appears in [53].

Example 5.10. Let 2≤n, m be integers and consider the partition λ=(λ1, . . . , λn),where λi := m; i.e. Gλ is the complete bipartite graph Kn,m. Then the coefficients of

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1391

the polynomial in the Hilbert series of the toric ring K[Gλ] as well as its multiplicityare

hk(λ) =(

m − 1k

)(n − 1

k

)and e(K[Gλ]) =

(n + m − 2

m − 1

).

Proof. In the calculations that will follow we will make repeated use of the combi-natorial identity

jt+1∑jt=1

(jt + t − 1

jt − 1

)=

(jt+1 + (t + 1) − 1

jt+1 − 1

),

which can be found in [25, Formula 1.49]. According to Theorem 5.4 we only needto show the formula for hk when k ≥ 2, as in the other two cases the expression istrivially verified. In this particular case the expression in Theorem 5.4 reduces to

hk(λ) =∑

2≤i1<i2<...<ik≤n

m−k∑jk−1=−k+2

jk−1∑jk−2=−k+3

. . .

j3∑j2=−1

j2∑j1=0

j1

=∑

2≤i1<i2<...<ik≤n

m−k∑jk−1=−k+2

jk−1∑jk−2=−k+3

. . .

j3∑j2=−1

(j2 + 1

2

)

=∑

2≤i1<i2<...<ik≤n

m−k∑jk−1=−k+2

jk−1∑jk−2=−k+3

. . .

j4∑j3=−2

(j3 + 2

3

)

=∑

2≤i1<i2<...<ik≤n

m−k∑jk−1=−k+2

(jk−1 + (k − 1) − 1

jk−1 − 1

)

=∑

2≤i1<i2<...<ik≤n

(m − 1

k

)=

(m − 1

k

)(n − 1

k

).

Finally, the expression for the multiplicity of K[Gλ] follows from Corollary 5.6 byperforming similar computations. �

Example 5.11. Let 2 ≤ n ≤ m be integers and consider the partition λ =(λ1, . . . , λn) where λk := m + 1 − k. Then the Hilbert series of the toric ringK[Gλ] is

P (K[Gλ], t) =1 + h1(λ) · t + · · · + hn−1(λ) · tn−1

(1 − t)m+n−1,

where

hk(λ) =(

n − 1k

)(m − 2

k

)−

(n − 1k + 1

)(m − 2k − 1

)for all k = 0, . . . , n − 1. In particular, the multiplicity is:

e(K[Gλ]) =m − n + 1

m·(

m + n − 2m − 1

).

Proof. We induct on n ≥ 2. Theorem 5.4 immediately provides the claim abouth0(λ) and h1(λ). Thus we may assume n ≥ 3. Consider the partition λ =(m, . . . , m+2−n, j) where 1 ≤ j ≤ m+1−n; i.e. λ differs from the given partitionλ at most in the last entry. Then, using the notation of Lemma 5.3, if j ≥ 2, we getλ′′ = (m, . . . , m+2−n, j−1) ∈ Nn and λ′ = (m− j +1, . . . , m− j +3−n) ∈ Nn−1.

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1392 A. CORSO AND U. NAGEL

Note that the induction hypothesis applies to λ′. Letting j vary between 1 andm + 1 − n, Lemma 5.3 and the induction hypothesis provide

hk(λ) = hk(m, m − 1, . . . , m + 2 − n)

+m+1−n∑

j=2

hk−1(m − j + 1, . . . , m − j + 3 − n)

=(

n − 2k

)(m − 2

k

)−

(n − 2k + 1

)(m − 2k − 1

)

+m+1−n∑

j=2

[(n − 2k − 1

)(m − j − 1

k − 1

)−

(n − 2

k

)(m − j − 1

k − 2

)]

=(

n − 2k

)(m − 2

k

)−

(n − 2k + 1

)(m − 2k − 1

)

+(

n − 2k − 1

)·[(

m − 2k

)−

(n − 2

k

)]−

(n − 2

k

)·[(

m − 2k − 1

)−

(n − 2k − 1

)]

=(

n − 1k

)(m − 2

k

)−

(n − 1k + 1

)(m − 2k − 1

),

as claimed. Finally, as noted earlier, e(K[Gλ]) = pλ(1). Thus, using [25, Formula3.20], we get

e(K[Gλ]) =n−1∑k=0

[(n − 1

k

)(m − 2

k

)−

(n − 1k + 1

)(m − 2k − 1

)]

=(

m + n − 3n − 1

)−

(m + n − 3

n − 3

)

=m − n + 1

m·(

m + n − 2m − 1

),

where the last equality is easy to verify. �

As reflected in the coefficients hk, one should observe that the roles of m andn are not symmetric in the previous corollary, since we consider the partition λ =(m, m− 1, . . . , m− n + 1). However, in the case m = n, the above formulæ greatlysimplify (becoming symmetric!) and the rings have particularly good properties:

Example 5.12. Consider the partition λ = (n, n − 1, . . . , 2, 1) ∈ Zn. The ringR/Iλ cogenerated by the edge ideal Iλ of the associated Ferrers graph Gλ is Cohen-Macaulay (by Corollary 2.8) with multiplicity n + 1 and Hilbert series

P (R/Iλ, t) =1 + nt

(1 − t)n.

The toric ring K[Gλ] is Gorenstein with Hilbert series:

P (K[Gλ], t) =

n−2∑k=0

( n−2k ) ( n−1

k )k + 1

tk

(1 − t)2n−1.

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MONOMIAL AND TORIC IDEALS ASSOCIATED TO FERRERS GRAPHS 1393

In particular, the multiplicity of K[Gλ] is the Catalan number:

e(K[Gλ]) =

(2(n−1)

n−1

)n

.

We refer the interested reader to [47, 48] for a wealth of information about Catalannumbers.

Acknowledgement

The authors would like to thank the anonymous referee for the careful readingof the original manuscript and helpful comments.

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Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

E-mail address: [email protected]

URL: www.ms.uky.edu/~corso

Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

E-mail address: [email protected]

URL: www.ms.uky.edu/~uwenagel

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