Resolutions of De Concini-Procesi ideals of hooks

Resolutions of De Concini-Procesi ideals of hooks Riccardo Biagioli∗Sara Faridi†Mercedes Rosas‡ Abstract We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity. 1 Introduction A nilpotent matrix of size n, over a field kof characteristic 0, can be labelled with a partition of n, say λ= (λ1, λ2, . . . , λ`), where the λiare the sizes of its Jordan blocks; let Oλdenote the conjugacy class of such a matrix. The problem of finding a generating set for the defining ideal of the Zariski closure of Oλ was suggested by De Concini and Procesi [DP], and many authors since then conjectured or proved various generating sets (Eisenbud and Saltman [ES], Tanisaki [T], and Weyman [W]). The schematic intersection of Oλ0(where λ0denotes the conjugate of λ), with the set of all diagonal matrices, whose defining ideal we denote by Iλ, has also been studied by the authors mentioned above. In this case, however, the generating set is simpler to understand. De Concini and Procesi [DP] produced a generating set for Iλ, and proved that the quotient of the polynomial ring R=k[x1, . . . , xn]by Iλ, is isomorphic to the cohomology ring of a certain subvariety of the flag variety. Moreover, as a representation of the symmetric group Sn,R/Iλis isomorphic to the induction of the trivial representation of the Young subgroup Sλ1× · · · × Sλ`. Garsia and Procesi [GP] studied the graded character of this representation, and showed that it could be expressed in terms of Kostka-Foulkes polynomials, leading the way to more investigations in this subject by Aval and Bergeron [AB], Bergeron and Garsia [BG], among others. These investigations, were facilitated by Tanisaki’s work [T], where a simpler generating set for Iλ, in terms of elementary partially symmetric functions, is defined. In this paper we study the ideals Iλ, which we call De Concini-Procesi ideals. We reduce Tanisaki’s generating set in the case when λis a hook partition, and obtain a minimal generating set. This reduced generating set allows us to compute the bigraded Poincar´ e series associated to such an ideal (i.e. the generating function encoding the ranks of the free modules appearing in a minimal free resolution of the ideal), by using only relatively basic algebraic techniques. We also give a description of the Hilbert series of R/Iλ. We end the paper with a couple of combinatorial results. We compute the generating function of the single Poincar´ e series, and a combinatorial recurrence for the bigraded one. This paper is organized as follows. In Section 2 contains a review of resolutions, Cohen-Macaulay rings, and the other commutative algebra tools that we use in the paper. Section 3, we give the basic definitions of partitions and the language used in the paper. We then introduce De Concini-Procesi ideals, and compute a new generating set for them in the case of hooks; we show later in Section 5 that this generating set is minimal. In Section 4 we study the resolutions of such ideals, and conclude with the formula of the corresponding bigraded Poincar´ e series. Finally, in Section 5 we compute the regularity and we give an explicit formula for the Hilbert series of the module R/Iλ. ∗Institut Camille Jordan, Universit´ e Claude Bernard Lyon 1, Villeurbanne, FRANCE, [email protected]v-lyon1.fr †Department of Mathematics, Dalhousie University, Halifax, CANADA, [email protected] (research supported by NSERC) ‡Department of Mathematics and Statistics, York University, York, CANADA, [email protected] 1 Weyman and Shimozono have brought to our attention that a resolution for Iλcan be obtained also with a different technique, namely by using Lascoux resolution and the Koszul complex. Some details on this construction can be found in [W]. Acknowledgments: All the test examples that supported this research were run using the computer algebra program Macaulay2 [GS]. We would like to thanks Franc¸ois Bergeron, Emmanuel Briand, Tony Geramita and the referee for useful comments, and Mark Shimozono and Jerzy Weyman for telling us about resolutions of nilpotent closures, and many helpful remarks and suggestions. 2 Commutative algebra tools Let R=k[x1, . . . , xn]be a polynomial ring over a field kof characteristic 0, with the standard grading deg xi= 1, for all i. Let m= (x1, . . . , xn)be the (irrelevant) homogeneous maximal ideal of R. We are interested in the quotient S=R/I where Iis an ideal of Rgenerated by homogeneous polynomials. Definition 2.1 (Minimal free resolution). Afree resolution of R/I is an exact complex F 0−→ · · · δi+1 −→ Fi δi −→ Fi−1 δi−1 −→ · · · δ2 −→ F1 δ1 −→ Rδ0 −→ R/I −→ 0. of free R-modules Fi(F0=R). The resolution is minimal if δi(Fi)⊆mFi−1for i > 0. If each Fiis a free module of rank βi, the βiare called the Betti numbers of R/I; these are independent of which minimal resolution one considers. In the case where Iis a homogeneous ideal, and therefore R/I is graded, we define the graded Betti numbers of R/I. This is done by making the maps δihomogeneous, so that they take a degree jelement of Fito a degree jelement of Fi−1. To serve this purpose the degree of each generator of Fiis adjusted. So we can write the free module Fi=Rβias Rβi=LjR(−j)βi,j where for a given integer a,R(a)is the same as Rbut with a new grading: R(a)d=Ra+d.So the resolution shown in above becomes 0−→ M j R(−j)βm,j δm −→ M j R(−j)βm−1,j δm−1 −→ · · · δ2 −→ M j R(−j)β1,j δ1 −→ Rδ0 −→ R/I −→ 0.(1) This is called the graded minimal free resolution of R/I, and the βi,j are the graded Betti numbers of R/I. Clearly, X j βi,j =βi. Definition 2.2 (Bigraded Poincar´ e series). The bigraded Poincar´ e series of an ideal Iis the generating function for the graded Betti numbers of I: PR/I (q, t) = X i,j βi,jqitj. Definition 2.3 (Linear resolution). The graded resolution described in (1) is a linear resolution, if for some u,βi,j = 0 unless j=u+i−1. Discussion 2.4 (Resolutions using mapping cones). The mapping cone technique provides a way to build a free resolution of an ideal by adding generators one at a time. A resolution obtained using mapping cones is not in general minimal. However, we will be focusing only on the special case of multiplication by a nonzerodivisor, in which case we obtain a minimal free resolution. Suppose that Iis an ideal in the polynomial ring R, and e∈mis a nonzerodivisor in R/I (i.e. eis a regular element mod I). The goal is to build a minimal free resolution of R/(I+(e)) starting from a minimal free resolution of R/I. Consider the short exact sequence 0−→ R/(I: (e)) .e −→ R/I −→ R/(I+ (e)) −→ 0 2 where I: (e)is the quotient ideal consisting of all elements x∈Rsuch that xe ∈I. Since eis a nonzerodivisor in R/I, we have I: (e) = I, and so our short exact sequence turns into 0−→ R/I .e −→ R/I −→ R/(I+ (e)) −→ 0. Suppose we have a minimal free resolution of R/I 0−→ · · · δi+1 −→ Ai δi −→ Ai−1 δi−1 −→ · · · δ2 −→ A1 δ1 −→ Rδ0 −→ R/I −→ 0.(2) Then we can obtain the following minimal free resolution of R/(I+ (e)) 0−→ · · · di+1 −→ Fi di −→ Fi−1 di−1 −→ · · · d2 −→ F1 d1 −→ Rd0 −→ R/(I+ (e)) −→ 0(3) where for each i > 0, as a free R-module Fi=Ai⊕Ai−1and di(x, y) = (ey +δi(x),−δi−1(y)). We now focus on the grading of each Fi. Suppose that the element e∈Ris homogeneous of degree m, and for each i, each of the free modules Aiin (2) are of the form Ai=M j R(−j)βi,j where the βi,j are the graded Betti numbers. We would like to compute the graded Betti numbers of R/(I+ (e)). Below we give an explicit description of the grading for each Fi; the gist of the argument, which can be found in Schenck’s book [Sc], is that we need to twist the graded resolution of R/I in (2) by mto obtain a resolution of R/(I: (e)) that make the maps that produce the mapping cone resolution homogeneous. So each Ai−1-component of Fiis a twist of Ai−1appearing in (2). Lemma 2.5. Consider the minimal free resolutions (2) of R/I, and (3) of R/(I+ (e)) obtained by mapping cones. For each i > 0we have Fi=M j R(−j)βi,j ⊕M j R(−j−m)βi−1,j . Proof. In the case where i= 1, we have the homogeneous map d1:A1⊕R−→ Rwhere d1(x, y) = ey +δ1(x). In particular, if x∈A1is a homogeneous element of degree t, then d1(x, 0) = δ1(x)is also a degree thomogeneous element of R. If y∈Ris a homogeneous element of degree t, then d1(0, y) = ey has degree t+m. In order to make d1a homogeneous (degree 0) map, we shift the grading of the component R of Fiby m, so that F1=M j R(−j)β1,j ⊕R(−m). The same argument applies, by induction, to each step iof the resolution. Corollary 2.6. Let Ibe an ideal of the polynomial ring Rand e∈mbe a homogeneous element of degree mwhich is a nonzerodivisor in R/I. Then PR/(I+(e))(q, t) = (1 + qtm)PR/I (q, t). Proof. By Lemma 2.5, if for a fixed i,Ai= bi M j=0 R(−j)βi,j then Fi= bi M j=0 R(−j)βi,j ⊕ bi−1 M j=0 R(−j−m)βi−1,j . 3 So we have PR/(I+(e))(q, t) = 1 + X i≥1  bi X j=0 βi,jtj+ bi−1 X j=0 βi−1,jtj+m qi =X i≥0 bi X j=0 βi,jtjqi+tmX i≥0 bi X j=0 βi,jtjqi+1 = (1 + qtm)X i≥0 bi X j=0 βi,jtjqi= (1 + qtm)PR/I (q, t). Recall that a (square-free) monomial ideal is an ideal generated by (square-free) monomials in the variables x1, . . . , xn. If Iand Jare two ideals of R, their quotient is the ideal defined as I:J={x∈R|xJ ⊆I}. Definition 2.7 (linear quotients). If I⊂k[x1, . . . , xn]is a monomial ideal and G(I)is its unique minimal set of monomial generators, then Iis said to have linear quotients if there is an ordering M1, . . . , Mmon the elements of G(I)such that for every i= 2, . . . , m, the quotient ideal (M1, . . . , Mi−1) : Mi is generated by a subset of the variables x1, . . . , xn. Lemma 2.8. Let Ibe an ideal in the polynomial ring R=k[x1, . . . , xn]generated by all square-free monomials of a fixed degree m. Then 1. Ihas linear quotients; 2. R/I has a linear resolution; 3. R/I is Cohen-Macaulay. Proof. Statements (1) and (3) follow from [HH]. Statement (2) is true because of the Eagon-Reiner [ER] criterion for Cohen-Macaulayness of square-free monomial ideals, and the fact that the Alexander dual of I is also generated by all square-free monomials of a fixed degree. 3 De Concini-Procesi Ideals We now introduce a family of ideals {Iλ}λof the polynomial ring R=k[x1, . . . , xn]indexed by partitions λof n. These ideals were first introduced by De Concini-Procesi in [DP]. They showed that for any partition λof n,R/Iλis the coordinate ring of the diagonal matrices which are in the closure of the conjugacy class of a nilpotent matrix of Jordan block structure given by the partition λ0, the conjugate of λ. We start with some definitions and notation about partitions, that will be used in the rest of this paper. We let N+={1,2, . . .}, and N=N+∪ {0}. The cardinality of a set Sis denoted by |S|. We define a partitionofn∈Ntobeafinitesequenceλ= (λ1, . . . , λk)∈Nk, such thatPk i=1 λi=nandλ1≥. . . ≥λk. If λis a partition of nwe write λ`n. The nonzero terms λiare called parts of λ. The number of parts of λ is called the length of λ, denoted by `(λ). The Young diagram of a partition (λ1, . . . , λk)`n, is the diagram with λisquares in the ith-row. We use the symbol λfor both a partition and its associated Young diagram. For example, the diagram of λ= (5,4,2,1) is illustrated in Figure 1. For a partition λ= (λ1, . . . , λk)denote the conjugate partition λ0:= (λ0 1, . . . , λ0 h), where for each i≥1, λ0 iis the number of parts of λthat are bigger than or equal to i. The diagram of λ0is obtained by flipping the diagram of λacross the diagonal. 4 Figure 1: The partition λ= (5,4,2,1) A partition is said to be a hook if it is of the form λ= (a, 1b), with a, b ∈N. It will often be useful to denote hook partitions using a different notation. The hook λ= (a+ 1,1b)in Frobenius’s notation [M, page 3] will be denoted by λ= (a|b). Note that its conjugate is λ0= (b|a). From now on, we shall assume that a partition of nhas nterms. So we will add enough zero terms to any partition until we have the right number of terms. Let λ= (λ1, . . . , λn)be a partition of n, and λ0= (λ0 1. . . , λ0 n)its conjugate partition. For any 1≤k≤n, we define δk(λ) := λ0 n+λ0 n−1+. . . +λ0 n−k+1. Recall that for any 1≤r≤n, the elementary symmetric polynomial [M] is defined by er(x1, . . . , xn) := X 1≤i1<...<ir≤n xi1xi2· · · xir. Given a subset S⊆ {x1, . . . , xn}, let er(S)be the rth elementary symmetric polynomial in the variables in S. Clearly, every er(S)is a homogeneous polynomial in Rof degree r. We are now ready to introduce the ideals originally defined by De Concini and Procesi [DP]. We use a different and simpler set of generators with respect to the original one, which was defined by Tanisaki [T]. Definition 3.1 (De Concini-Procesi ideal). We let Cλdenote the collection of partial elementary symmetric polynomials Cλ={er(S)|S⊆ {x1, . . . , xn},|S|=k≥1, k ≥r > k −δk(λ)}.(4) The De Concini-Procesi ideal Iλis the homogeneous ideal generated by the elements of Cλ, in symbols, Iλ:= (Cλ). Example 3.2. Let λ= (3,1,0,0) `4and λ0= (2,1,1,0). Then (δ1(λ), . . . , δ4(λ)) = (0,1,2,4). Hence (1 −δ1(λ), . . . , 4−δ4(λ)) = (1,1,1,0),and the collection Cλconsists of the following elements. For k= 1 there is no admissible er(S). For k= 2 we get the set of monomials: x1x2, x1x3, x1x4, x2x3, x2x4, x3x4. For k= 3, we get x1x2+x1x3+x2x3, x1x2+x1x4+x2x4, x1x3+x1x4+x3x4, x2x3+x2x4+x3x4 x1x2x3, x1x2x4, x1x3x4, x2x3x4. Finally for k= 4, we get the complete set of the elementary symmetric functions er(x1, x2, x3, x4), for 1≤r≤4. Remark 3.3. Note that δn(λ) = n, for any partition λof n. Hence when we set k=nin (4), we obtain that Iλcontains the ideal generated by the elementary symmetric polynomials in all the variables. It is well known that e1(x1, . . . , xn), . . . , en(x1, . . . , xn)are algebraically independent (this is due to Gauss; see [M]), and hence they form a regular sequence over R. Therefore R/Iλis an Artinian ring. When the indexing partition λis a hook, the ideal Iλcan be split in two parts. We have the following result. 5 Proposition 3.4 (A reduced generating set for hook partitions). Let λ= (a|b)`nbe a hook. Then the ideal associated to λin the polynomial ring k[x1, . . . , xn]is Iλ=Mb+1 +Eb, where Mb+1 = (xi1· · · xib+1 |1≤i1< . . . < ib+1 ≤n)(5) is the ideal generated by all square-free monomials in x1, . . . , xnof degree b+ 1, and Eb= (ei(x1, . . . , xn)|1≤i≤b)(6) is the ideal generated by all elementary symmetric polynomials of degree ≤bin the variables x1, . . . , xn. Proof. The partition λ= (a|b)is of size n=a+b+ 1. We can write λ0= (b|a) = (b+ 1,1, . . . , 1 | {z } a ,0, . . . , 0 | {z } b ). Then we have (δ1(λ), δ2(λ), . . . , δn(λ)) = (0, . . . , 0 | {z } b ,1,2, . . . , a, n), and so (1 −δ1(λ),2−δ2(λ), . . . , n −δn(λ)) = (1,2,3, . . . , b, b, . . . , b | {z } a ,0). The definition of Cλin (4) implies that no k, with 1≤k≤b, contributes a generator to the ideal Iλ. The first index making a nontrivial contribution to the set Cλis k=b+ 1, which adds to Cλall eb+1(S), with |S|=b+ 1, or in other words all the square-free monomials of degree b+ 1 in the variables x1, . . . , xn. We denote by Mb+1 the ideal generated by these square-free monomials. Now all the indices k, with b+ 2 ≤k≤n−1add to Cλelements of the form er(S), with k≥r≥b+ 1, and |S|=k. Each such er(S)is a homogeneous polynomial of degree r, which we can write as the sum of square-free monomials of degree r. Since r≥b+ 1, and all square-free monomials of degree b+ 1 or more are already in Iλ, such er(S)do not contribute any new generators to Iλ. Finally, for k=nwe obtain all the elementary symmetric polynomials in all the variables. For the same reasons as above, the only new contributions are e1(x1, . . . , xn), e2(x1, . . . , xn), . . . , eb(x1, . . . , xn). We denote the ideal generated by these elementary symmetric polynomials by Eb. We conclude that Iλ= Mb+1 +Eb. Example 3.5. Let λ= (2 |1) `4. It follows from the computations in Example 3.2, that the ideal Iλsplits into two parts Iλ= (x1x2, x1x3, x1x4, x2x3, x2x4, x3x4)+(x1+x2+x3+x4). The first part is generated by all monomials of degree 2 in the variables x1, x2, x3, x4, and the second is generated by e1(x1, x2, x3, x4), the elementary symmetric polynomial of degree 1. For hooks, the reduced generating set described in Proposition 3.4 is much smaller than that described in Definition 3.1 (it is in fact minimal), and hence simpler to understand. In the rest of the paper, we use this presentation of Iλto describe the Betti numbers and other numerical information of the algebra R/Iλ. 6 4 Bigraded Poincar´ e series of De Concini-Procesi ideals of hooks In this section we study the minimal free resolutions of the De Concini-Procesi ideal Iλof a hook λ= (a|b). We have seen that Iλis the sum of two ideals Iλ=Mb+1 +Eb where Mb+1 is generated by monomials, and Ebis generated by elementary symmetric functions. Below we show how we can recover the resolution of Iλusing the resolutions of each one of the summands. Since Mb+1 is generated by all square-free monomials of R=k[x1, . . . , xn]that have degree b+ 1, by Lemma 2.8, Mb+1 is a Cohen-Macaulay ideal with linear resolutions and linear quotients. On the other hand, it is easy to see that all the minimal primes of Mb+1 have uniform height n−b. This is because every generator of Mb+1 is a product of exactly b+ 1 variables in the set {x1, . . . , xn}, and so a minimal subset of {x1, . . . , xn}that shares at least one variable with each one of these generators must have n−belements. Such an ideal will have height equal to n−b, and so it follows that dim R/Mb+1 =b. We have thus shown that Corollary 4.1. For a hook λ= (a|b), the ideal Mb+1 of Rhas linear quotients, linear resolution, and R/Mb+1 is Cohen-Macaulay of (Krull) dimension b. Remark 4.2. Let G(Mb+1)denote the minimal monomial generating set for Mb+1. We can arrange the elements of G(Mb+1)in descending lexicographic order as M1, . . . , Mm. Take such a monomial Mi= xj1· · · xjb+1 , written so that j1< j2< . . . < jb+1. Since (M1, . . . , Mi−1)is a monomial ideal, and Miis also a monomial, the quotient ideal (M1, . . . , Mi−1) : Miis generated by monomials. Observe that 1. If s < jtfor some jt∈ {j1, . . . , jb+1}and s6∈ {j1, . . . , jb+1}, then xs∈(M1, . . . , Mi−1) : Mi. This is because the monomial xsMi xjtis a degree b+ 1 monomial that is lexicographically larger than Mi, that is, xsMi xjt∈ {M1, . . . , Mi−1}. 2. If uis a monomial in (M1, . . . , Mi−1) : Mi, then Ml|uMifor some l < i. Since Ml>lex Mi, there exists xs, such that xs|Ml,xs-Miand s < jtfor some jt∈ {j1, . . . , jb+1}. It follows that xs|u, and (M1, . . . , Mi−1) : Miis generated by the set of variables xs, with s<jb+1 and s /∈ {j1, . . . , jb+1}as described in part 1. This proves that Mb+1 has linear quotients. Next, we focus on the ideal Eb, which is generated by the first belementary symmetric functions. As observed in Remark 3.3, Iλcontains the regular sequence e1(x1, . . . , xn), . . . , en(x1, . . . , xn), and hence R/Iλis of (Krull) dimension 0. Proposition 4.3. For a hook λ= (a|b), the set of generators e1(x1, . . . , xn), . . . , eb(x1, . . . , xn) of Ebform a regular sequence over the quotient ring R/Mb+1. Proof. Let S=R/Mb+1. We know by Corollary 4.1 that Sis a Cohen-Macaulay ring, and dim S=b. To show that e1(x1, . . . , xn), . . . , eb(x1, . . . , xn)forms a regular sequence in S, by Theorem 2.1.2 of [BH], it is enough to show that dim S/Eb= 0. Now, S/Eb=R/Iλ, and the latter is an Artinian ring, and hence of dimension 0. We are now ready to state our central claim. Theorem 4.4 (Main theorem). Let λ= (a|b)be a hook. Then the bigraded Poincar´ e series for the ideal Iλis the following PR/Iλ(q, t) = b Y k=1 (1 + qtk)·1 + qtb+1 a X i=0 b+i b(1 + q t)i.(7) 7 Proof. As usual, let Iλ=Mb+1 +Eb. Step 1. The ideal Mb+1 has linear quotients (Corollary 4.1). It follows from Corollary 1.6 of [HT] that the bigraded Poincar´ e series of Mb+1 is the following: PR/Mb+1 (q, t) = 1 + X M∈G(Mb+1) (1 + qt)|set(M)|qtdeg(M)(8) where, if we arrange the elements of G(Mb+1)in descending lexicographic order as M1, . . . , Mm, then for i= 1, . . . , m set(Mi) = {j∈ {1, . . . , n} | xj∈(M1, . . . , Mi−1) : Mi}. As the degree of each of the monomials generating Mb+1 is b+ 1, Equation (8) turns into PR/Mb+1 (q, t) = 1 + qtb+1 X M∈G(Mb+1) (1 + qt)|set(M)|.(9) If Mi=xi1· · · xib+1 , by Remark 4.2 set(Mi) = {u≤ib+1 |xu-Mi}, so |set(Mi)|=ib+1 −(b+ 1).(10) We have shown that, if Mis any degree b+ 1 square-free monomial with highest index u(that is, xu|Mand xv-Mfor v > u), then |set(M)|=u−(b+1). So to compute the sum in (9), all we have to do is count the number of square-free degree b+ 1 monomials with highest index u, for any given u. This number is clearly u−1 b. So for a given i, the number of degree b+ 1 square-free monomials Mwith |set(M)|=iis exactly b+i b. Therefore, PR/Mb+1 (q, t)is equal to 1 + qtb+1 n−b−1 X i=0 b+i b(1 + qt)i= 1 + qtb+1 a X i=0 b+i b(1 + qt)i(11) since by Equation (10), ican reach at most n−b−1, which by definition is equal to a. Step 2. Since Ebis generated by a regular sequence over R/Mb+1 (Proposition 4.3), we can use a mapping cone construction to find its minimal graded resolution (see Discussion 2.4). We do this by adding the generators of Eb, one at a time, to Mb+1, and applying Corollary 2.6. As the generators e1(x1, . . . , xn), . . . , eb(x1, . . . , xn)of Ebhave degrees 1, . . . , b, respectively, each time we add a ei(x1, . . . , xn), the Poincar´ e series gets multiplied by a factor of (1 + qti), and hence from (11) we obtain that PR/Iλ(q, t)equals b Y k=1 (1 + qtk)·PR/Mb+1 (q, t) = b Y k=1 (1 + qtk)·1 + qtb+1 a X i=0 b+i b(1 + q t)i. 8 5 Some consequences of the Main Theorem We study some of the consequences of our main theorem. We prove that our new generating set is indeed minimal. Corollary 5.1 (The set of generators of Iλis minimal). Let λ= (a|b)be a hook. The generating set for Iλdescribed in Proposition 3.4 is minimal. Proof. The cardinality of the generating set of Iλdescribed in Proposition 3.4 is n b+1+b. On the other hand, the minimal number of generators of Iλis the first Betti number β1of R/Iλ, which is the coefficient of qin the Poincar´ e series PR/Iλ(q, 1). It is easy to see by Theorem 4.4 that this coefficient is b+1+ a X i=1 b+i b. So all we have to show is that n b+1+b=b+1+Pa i=1 b+i bwhich is equivalent to showing that n b+ 1= n−b−1 X i=0 b+i b. This last equation follows easily from induction on n. Regularity of Hooks Definition5.2(Castelnuovo-Mumfordregularity). LetIbeanidealofa R=k[x1, . . . , xn]. The CastelnuovoMumford regularity or simply regularity of R/I, denoted by reg(R/I)is defined as the maximum value of of j−iwhere the graded Betti number βi,j 6= 0 in a minimal free resolution of R/I. Corollary 5.3 (Regularity of hooks). Let λ= (a|b)be a hook. Then reg(R/I) = b(b+ 1)/2. Proof. The graded Betti numbers βi,j appear as the coefficients of the Poincar´ e series PR/Iλ(q, t) = b Y k=1 (1 + qtk) | {z } Factor 1 ·1 + qtb+1 a X i=0 b+i b(1 + q t)i | {z } Factor 2 . So the question is to find the term qitjin this polynomial, where the coefficient βi,j is nonzero and j−iis maximum. The terms with nonzero coefficients in each factor are of the following forms: Factor 1: qmtb1+...+bmwhere 1≤b1< . . . < bm≤b, 0≤m≤b, Factor 2: qe+1te+b+1 where 0≤e≤a. To show that reg(R/I) = b(b+ 1) 2, we need to show that this bound is achieved by the possible choices of j−i, and is the maximum possible bound. Consider the terms in Factor 1. We have b1+. . . +bm−m≤(b−(m−1)) + (b−(m−2)) + . . . +b−m =(1+2+. . . +b)−(1+2+. . . + (b−m))−m≤b(b+ 1) 2−b. Similarly, for terms in Factor 2, since b≥0, we have e+b+ 1 −(e+ 1) = b. Hence, for the product of a term in Factor 1 and a term in Factor 2 we have b1+. . . +bm+e+b+ 1 −(m+e+ 1) ≤b(b+ 1) 2. The bound is achieved if m=b, so that b1= 1, . . . , bm=b, and for any e, so that we have the term with nonzero coefficient qe+b+1t(1+...+b)+e+b+1 =qe+b+1tb(b+1) 2+e+b+1 which clearly has the property that j−i=b(b+ 1) 2, as desired. 9