The colorful Helly theorem and colorful resolutions of ideals
aa r X i v : . [ m a t h . A C ] A ug THE COLORFUL HELLY THEOREM AND COLORFULRESOLUTIONS OF IDEALS
GUNNAR FLØYSTAD
Abstract.
We demonstrate that the topological Helly theorem and thealgebraic Auslander-Buchsbaum may be viewed as different versions ofthe same phenomenon. Using this correspondence we show how thecolorful Helly theorem of I.Barany and its generalizations by G.Kalaiand R.Meshulam translates to the algebraic side. Our main results arealgebraic generalizations of these translations, which in particular givea syzygetic version of Helly’s theorem.2000 MSC : Primary 13D02. Secondary 13F55, 05E99. Introduction
The classical Helly theorem is one of the basic results of convex geometry.It was subsequently generalized by Helly to a topological setting. In the lastdecade topological techniques has been infused into the study of resolutionsof a monomial ideal I in a polynomial ring S by the technique of cellularresolutions [11]. Such resolutions are constructed from a cell complex withmonomial labellings on the cells satisfying certain topological conditions.We show that these topological conditions are the same as the hypothesisof Helly’s theorem, and that the conclusion of Helly’s theorem correspondsto the weaker version of the Auslander-Buchsbaum theorem one gets if onedoes not involve the concept of depth, but only that of dimension of thequotient ring S/I .The Helly theorem was generalized by I.B´ar´any [2] to a colorful version.By considering nerve complexes this has again recently been abstracted andgeneralized by G.Kalai and R.Meshulam [10]. Their result involves a pair M ⊆ X wher M is a matroidal complex and X is a simplicial complex ona vertex set V . In the case when M is a transversal matroid on a partition V = V ∪ · · · ∪ V d +1 it corresponds to the colorful version of Barany. Thislatter result may be translated to the algebraic side in two ways. Either bythe notion of cellular resolutions, or by the Stanley-Reisner ring of the nervecomplex. In fact these two translations are related by Alexander duality.The translation by the Stanley-Reisner ring of the nerve complex admitsalgebraic generalizations to multigraded ideals. We consider a polynomialring S whose set of variables X is partioned into a disjoint union X = X ∪ · · · ∪ X r and the variables in X i have multidegree e i , the i ’th coordinatevector. (One may think of the variables in X i as having color i .) We investigate N r -graded ideals I of S and their resolutions. The following isour first main result. (The notion of ( d + 1)-regular ideal is explained in thebeginning of Section 4.) Theorem 4.1.
Let S be a polynomial ring where the variables have r col-ors. Let I be a ( d + 1) -regular ideal in S , homogeneous for the N r -grading.Suppose I has elements of pure color i for each i = 1 , . . . , r . Then for eachcolor vector ( a , . . . , a r ) where P a i = d + 1 , there exists an element of I with this color vector. This provides a generalization of the result of Kalai and Meshulam [10]in the case of transversal matroids, since in our situation it applies to multi-graded ideals and not just to squarefree monomial ideals. Also the numberof colors r and the regularity d + 1 may be arbitrary, and not the same asin [10]. (But in the monomial case this generalization would have been arelatively easy consequence of [10].) The theorem above is however a specialcase of our second main result which is the following syzygetic version. Theorem 4.3.
Let S be a polynomial ring where the variables have r colors.Let I be an ideal in S which is homogeneous for the N r -grading, is generatedin degree d + 1 and is d + 1 -regular, so it has linear resolution. Suppose I haselements of pure color i for each i = 1 , . . . , r , and let Ω l be the l ’th syzygymodule in an N r -graded resolution of S/I (so Ω = I ). For each l = 1 , . . . , r and each color vector a = ( a , . . . , a r ) where P a i = d + l and s the numberof nonzero coordinates of a , the vector space dimension of (Ω l ) a is greateror equal to (cid:0) s − l − (cid:1) . It is noteworthy that we do not prove Thorem 4.1 directly and do notknow how to do it. Rather we show Theorem 4.3 by descending inductionon the Ω l , starting from Ω r , and then Theorem 4.1 is easily deduced fromthe special case of Ω .Letting T = k [ y , . . . , y r ] be the polynomial ring in r variables, our tech-nique involves comparing the ideal I ⊆ S and its natural image in T bymapping the variables in X i to sufficiently general constant multiples of y i .Along this vein we also prove the following. Theorem 4.7.
Let I ⊆ S be a multigraded ideal with elements of pure color i for each i , and let J ⊆ T be an ideal containing the image of I by the map S → T (described above). Then the regularity of I is greater than or equalto the regularity of J . While we study colored homogeneous ideals, until now they have mostlybeen studied in the monomial setting. One of the first is perhaps Stanley[15] studying balanced simplicial complexes. Then followed Bj¨orner, Frankl,Stanley [4] which classified flag f -vectors of a -balanced Cohen-Macaulaysimplicial complexes where a ∈ N r . A more recent paper is Babson, Novik[1] where they develop shifting theory for colored homogeneous ideals, butrather focus on monomial ideals to give another approach to [4]. Nagel and OLORFUL HELLY THEOREM AND COLORFUL RESOLUTIONS OF IDEALS 3
Reiner [12] give a very nice construction of the cellular resolution of coloredmonomial ideals associated to strongly stable ideals generated in a singledegree.To give some more perspective on our approach, in studying graded idealsin a polynomial ring much attention has been given to homogeneous (i.e. N -graded) ideals and to monomial (i.e. N n -graded) ideals. One way to ap-proach intermediate cases is toric ideals where the variables are attacheddegrees in some N r . A toric ideal is uniquely determined by this associationof degrees in N r , but their class does not encompass the classes of mono-mial or homogeneous ideals. Our approach is probably the simplest way ofbuilding a bridge encompassing these two extremal cases: let the variablesonly have the unit vectors in N r as degrees.The organization of the paper is as follows. In Section 2 we show that thetopological Helly theorem and the algebraic Auslander-Buchsbaum theoremare basically different versions of the same phenomenon. In Section 3 weconsider the colorful Helly theorem and how it translates to the algebraicside, either via cellular resolutions or via the nerve complex. In Section 4we state and prove our main results, Theorems 4.1, 4.3, and 4.7. Acknowledgements.
We thank B.Sturmfels for comments on the resultsand in particular for providing the idea to Theorem 4.7.2.
The topological Helly theorem and theAuslander-Buchsbaum theorem
The Helly theorem.
The classical Helly theorem is one of the found-ing theorems in convex geometry, along with Radon’s theorem and Carath´eodory’stheorem, and is the one which has been generalized in most directions.
Theorem 2.1 (Helly 1908) . Let { K i } i ∈ B be a family of convex subsets of R d . If ∩ i ∈ B K i is empty, then there is A ⊆ B of cardinality d + 1 such that ∩ i ∈ A K i is empty. Helly generalized this result in 1930 to a topological version. We shall beinterested in it in the context of polyhedral complexes. Let X be a boundedpolyhedral complex. So X is a finite family of convex polytopes with thefollowing properties. • If P is in X , then all the faces of P are in X . • If P and Q are in X , then P ∩ Q is a face in both P and Q .Let V be the vertices of X , i.e. the 0-dimensional faces. Let { V i } i ∈ B be acollection of subsets of V , and let C i be the subcomplex of X induced fromthe vertex set V i , consisting of the faces of X whose vertices are all in V i .Now if X can be embedded in R d , and for every B ′ ⊆ B the intersection ∩ i ∈ B ′ C i is either empty or acyclic, then the topological realizations | C i | fulfillthe conditions of the topological Helly theorem. GUNNAR FLØYSTAD
Theorem 2.2 (Helly 1930) . Let { K i } i ∈ B be a family of closed subsets in R d with empty interesection, such that for each B ′ ⊆ B the intersection ∩ i ∈ B ′ K i is either empty or acyclic over the field R . Then there is A ⊆ B ofcardinality ≤ d + 1 such that ∩ i ∈ A K i is empty. Corollary 2.3.
Every minimal subset A of B such that ∩ i ∈ A K i is empty,has cardinality ≤ d + 1 .Remark 2.4. Note that in Theorem 2.2 we only require the existence of an A with ∩ i ∈ A K i empty while in Corollary 2.3 we say that every minimal A with ∩ i ∈ A K i is empty has cardinality ≤ d + 1. Given the hypothesis thesetwo properties are really equivalent.2.2. Cellular resolutions.
We now make a monomial labelling of the ver-tices in X by letting the vertex v be labeled by(1) x a v = Π v C i x i . In other words the variable x i is distributed to all vertices outside C i .Before proceeding we recall some basic theory of monomial labellings ofcell complexes and their associated cellular complexes of free modules overa polynomial ring, following [11, Ch.4].Let k be a field and let ˜ C · ( X ; k ) be the reduced chain complex of X over k . The term ˜ C i ( X ; k ) is the vector space ⊕ dim F = i k F with basis consistingof the i -dimensional polytopes of X , and differential ∂ i ( F ) = X facets G ⊆ F sign( G, F ) · G, where sign( G, F ) is either 1 or − ∂ a differential.Now given a polynomial ring S = k [ x i ] i ∈ B we may label each vertex v of X by a monomial x a v . Each face F is then labeled by x a F which isthe least common multiple lcm v ∈ F { x a v } . Now we construct the cellularcomplex F ( X ; k ) consisting of free S -modules. The term F i ( X ; k ) is the free S -module ⊕ dim F = i SF with basis consisting of the i -dimensional polytopesof X . The basis element F is given degree a F . The differential is given by ∂ i ( F ) = X facets F ⊆ G sign( G, F ) x a F x a G · G, which makes it homogeneous of degree 0.We are interested in the case that F ( X ; k ) gives a free resolution ofcoker ∂ = S/I where I is the monomial ideal generated by the monomi-als x a v . For each b ∈ N r , let X ≤ b be the subcomplex of X induced on thevertices v such that x a v divides x b . This is the subcomplex consisting of allfaces F such that x a F divides x b . According to [11, Ch.4], F ( X ; k ) is a freecellular resolution iff X ≤ b is acyclic over k or empty for every b ∈ N r . Let e i be the i ’th unit coordinate vector in N r and let = P ri =1 e i . OLORFUL HELLY THEOREM AND COLORFUL RESOLUTIONS OF IDEALS 5
The following provides an unexpected connection between Helly’s theoremand resolutions of square free monomial ideals.
Theorem 2.5.
Let X be an acyclic polyhedral complex. There is a one-to-one correspondence between the following.a. Finite families { C i } i ∈ B of induced subcomplexes of X such that foreach B ′ ⊆ B , the intersection ∩ i ∈ B ′ C i is either empty or acyclic.b. Monomial labellings of X with square free monomials in k [ x i ] i ∈ B suchthat F ( X ; k ) gives a cellular resolution of the ideal generated by these mono-mials.The correspondence is given as follows. Given a monomial labeling define C i , i ∈ B , by letting C i = X ≤ − e i , and given a family { C i } define themonomials x a v , by x a v = Π v C i x i .Proof. Assume part a. To show part b. we must show that X ≤ b is eitheracyclic or empty for b ∈ N B . It is enough to show this for b ∈ { , } B sincewe have a square free monomial labelling. If b = , then X ≤ b = X whichis acyclic. If b < we have an intersection X ≤ b = ∩ { i ; b i =0 } X ≤ − e i and C i = X ≤ − e i by (1).Now given part b., then ∩ i ∈ B ′ C i is equal to ∩ i ∈ B ′ X ≤ − e i which is X ≤ − P i ∈ B ′ e i and therefore is empty or acyclic. (cid:3) The Auslander-Buchsbaum theorem.
We shall now translate thestatements of the Helly theorem to statements concerning the monomialideal. First we have the following.
Lemma 2.6.
Let A ⊆ B . Then ∩ i ∈ A C i is empty if and only if the ideal I is contained in h x i , i ∈ A i .Proof. Since C i = X ≤ − e i , that the intersection is empty means that foreach monomial x a v there exists i ∈ A such that x i divides x a v . But thismeans that x a v is an element of h x i ; i ∈ A i and so I is included in this. (cid:3) Condition a. in Theorem 2.5 is the same as the condition in the topo-logical Helly theorem. Taking Corollary 2.3 and the lemma above into con-sideration, the topological Helly theorem, Theorem 2.2, translates to thefollowing.
Theorem 2.7.
Suppose the monomial labelling (1) gives a cellular resolutionof the ideal I generated by these monomials. Then every minimal prime idealof I has codimension ≤ d + 1 . On the other hand we have the following classical result.
Theorem 2.8 (Auslander-Buchsbaum 1955) . Let I be an ideal in the poly-nomial ring S . Thenprojective dimension ( S/I ) + depth ( S/I ) = | B | . GUNNAR FLØYSTAD
Without the concept of depth but using only the concepts of dimensionand projective dimension this takes the following form.
Corollary 2.9. codim I ≤ projective dimension ( S/I ) . We see that in the situation we consider where X gives a cellular resolutionof I , the projective dimension of I is less or equal to d +1, with equality if theresolution is minimal. We thus see that the Auslander-Buchsbaum theoremand Helly’s theorem are simply different versions of the same phenomenon.Note however that not every monomial ideal has a cellular resolution givinga minimal free resolution of the ideal, see [16].2.4. The nerve complex.
Recall that a simplicial complex ∆ on a subset B is a family of subsets of B such that if F ∈ ∆ and G ⊆ F then G ∈ ∆.Simplicial complexes on B are in one-to-one correspondence with square freemonomial ideals in k [ x i ] i ∈ B , where ∆ corresponds to the ideal I ∆ generatedby x τ where τ ranges over the nonfaces of ∆. If R ⊆ B the restriction ∆ R consist of all faces F in ∆ which are contained in R . The simplicial complexis called d -Leray if the reduced homology groups ˜ H i (∆ R , k ) vanish whenever R ⊆ B and i ≥ d .The nerve complex of the family { C i } i ∈ B is the simplicial complex con-sisting of the subsets A ⊆ B such that ∩ i ∈ A C i is nonempty. The followingare standard facts. Proposition 2.10.
Let { C i } i ∈ B be a family of induced subcomplexes of apolyhedral complex X , with nerve complex N . Suppose that any intersectionof elements in this family is empty or acyclic.a. The union ∪ i ∈ B C i is homotopy equivalent to N .b. If X has dimension d and ˜ H d ( X, k ) = 0 , the nerve complex N is d -Leray.Proof. Part a. follows for instance from [3, Theorem 10.7]. Part b. followsbecause for R ⊆ B , the restriction N R is the nerve complex of { C i } i ∈ R , andso N R is homotopy equivalent to ∪ i ∈ R C i which is a subcomplex of X . Hence˜ H i ( ∪ i ∈ R C i ) must vanish for i ≥ d . (cid:3) The
Alexander dual simplicial complex of ∆ consist of those subsets F of B such that their complements, F c , in B are not elements of ∆. Proposition 2.11.
Given any finite family { C i } i ∈ B of induced subcomplexesof X , let I ∆ be generated by the monomials in (1). The nerve complex N ofthe family { C i } i ∈ B is the Alexander dual of the simplicial complex ∆ .Proof. We must show that a square free monomial x A is in I ∆ iff A c is in N .Given a generator x a v , we have v ∈ C i for every i supp a v (the supportis the set of positions of nonzero coordinates). Hence v ∈ ∩ i ∈ (supp a v ) c C i sothis intersection is not empty and hence (supp a v ) c is in the nerve complex.Conversely, if v is in ∩ i ∈ A c C i , then a v ≤ A so x A is in I ∆ . (cid:3) OLORFUL HELLY THEOREM AND COLORFUL RESOLUTIONS OF IDEALS 7
Corollary 2.12. If I ∆ is a Cohen-Macaulay monomial ideal of codimension d + 1 , then I N has ( d + 1) -linear resolution.Proof. This follows from [9], since N is the Alexander dual of ∆. (cid:3) The colorful Helly theorem
Helly’s theorem was generalized by I.Barany [2] to the so called colorfulHelly theorem. This was again generalized by G.Kalai and R. Meshulam[10]. The following version of the colorful Helly theorem is a specializationof their result, which we recall in Theorem 4.4. It follows from Corollary 4.5by letting Y there be the nerve complex of the C i and letting the V p be B p . Theorem 3.1.
Let X be a polyhedral complex of dimension d with ˜ H d ( X ; k ) =0 . Let { C i } i ∈ B p for p = 1 , . . . , d + 1 be d + 1 finite families of induced sub-complexes of X , such that for every A ⊆ ∪ p B p the intersection ∩ i ∈ A C i isempty or acyclic. If every ∩ i ∈ B p C i is empty, there exists i p ∈ B p for each p = 1 , . . . , d + 1 such that ∩ d +1 p =1 C i p is empty. Note that we think of the index sets as disjoint although C i may be equalto C j for i and j in distinct index sets, or even in the same index set. Remark 3.2.
The topological Helly theorem follows from the colorful theo-rem above by letting all the families { C i } i ∈ B p be equal.Let B be the disjoint union of the B p and let S = k [ x i ] i ∈ B . (One maythink of the variables x i for i ∈ B p as having a given color p .) Let I ∆ ⊆ S be the associated Stanley-Reisner ideal of the family { C i } i ∈ B , given by thecorrespondence in Theorem 2.5. The following is then equivalent to thetheorem above. Theorem 3.3.
Suppose I ∆ is contained in each ideal h x i ; i ∈ B p i generatedby variables of the same color p , for p = 1 , . . . , d + 1 . Then there are i p ∈ B p for each p such that I ∆ is contained in the ideal h x i , . . . , x i d +1 i generated byvariables of each color.Proof. Immediate from the theorem above when we take into considerationLemma 2.6 and Theorem 2.5. (cid:3) If A ⊆ B let a p be the cardinality | A ∩ B p | . We say that the ideal h x i ; i ∈ A i has color vector ( a , . . . , a d +1 ). Corollary 3.4.
Suppose I ∆ is a Cohen-Macaulay ideal of codimension d +1 .If I ∆ has associated prime ideals of pure color ( d + 1) · e i for each i =1 , . . . , d + 1 , then it has an associated prime ideal with color vector P d +1 i =1 e i .Proof. In this case all associated prime ideals are generated by d +1 variables. (cid:3) The following is an equivalent formulation of Theorem 3.1 and is the formwhich will inspire the results in the next Section 4. Say that a square free
GUNNAR FLØYSTAD monomial has color vector ( a , . . . , a d +1 ) if it contains a i variables of color i . Theorem 3.5.
Let N be the nerve complex of the family { C i } i ∈ B of Theorem3.1. If I N has monomials of pure color p for p = 1 , . . . , d + 1 , then I N hasa monomial of color P d +1 i =1 e i .Proof. The intersection ∩ i ∈ A C i is empty iff Π i ∈ A x i is a monomial in I N . (cid:3) Colorful resolutions of ideals
Main results.
Let X = X ∪ X ∪· · ·∪ X r be a partition of the variablesin a polynomial ring S . Letting the variables in X i have multidegree e i ∈ N r ,the i ’th coordinate vector, we get an N r -grading of the polynomial ring. Ahomogeneous polynomial for this grading with multidegree a = ( a , . . . , a r )will be said to have color vector ( a , . . . , a r ). A polynomial with color vector r i e i where r i is a positive integer is said to have pure color i .If I is an ideal in S , homogeneous for this grading and containing polyno-mials of pure color i for i = 1 , . . . , r , then it may be a complete intersectionof these, in which case I does not have more generators. However if we putconditions on the regularity of I this changes. We recall this notion. Let F • be a minimal free resolutions of I with terms F p = ⊕ i ∈ Z S ( − i ) β p,i . We say I is m -regular if i ≤ m + p for every nonzero β p,i . This may be shown tobe equivalent to the truncated ideal ⊕ p ≥ m I p having linear resolution. For asimplicial complex ∆ it follows by the description of the Betti numbers of I ∆ by reduced homology groups, see [11], Corollar 5.12, that I ∆ is ( d +1)-regularif and only if ∆ is d -Leray. The latter hypothesis is fulfilled in Theorem 3.5for the nerve complex N , and the conclusion is that I N contains an elementwith color vector P d +1 i =1 e i . The following generalizes this. Theorem 4.1.
Let S be a polynomial ring where the variables have r col-ors. Let I be a ( d + 1) -regular ideal in S , homogeneous for the N r -grading.Suppose I has elements of pure color i for each i = 1 , . . . , r . Then for eachcolor vector ( a , . . . , a r ) where P a i = d + 1 , there exists an element of I with this color vector.Example 4.2. Let there be r = 3 colors and suppose I contains polynomialsof multidegree (3 , , , , , , I is generated by them, it isa complete intersection and the regularity is 3+ 4+ 5 − , ,
7, and so on. What can be said of the generatorsof I ?If the regularity is 9, by the above theorem it must contain a generatorof multidegree ≤ (2 , ,
4) (for the partial order where a ≤ b if in eachcoordinate a i ≤ b i ).If the regularity is 8, it must by the above contain a generator of multi-degree ≤ (1 , , ≤ (2 , , ≤ (2 , , I if its regularity is 7 ,
6, and 5 also.
OLORFUL HELLY THEOREM AND COLORFUL RESOLUTIONS OF IDEALS 9
The most obvious example of an ideal as in the theorem above is of coursethe power m d +1 = ( x , . . . , x r ) d +1 in k [ x , . . . , x r ]. Let a = ( a , . . . , a r )be a multidegree with P a i = d + j and support s = supp( a ) defined asthe number of coordinates a i which are nonzero. Then the multigradedBetti number β j, a ( S/ m d +1 ) is equal to (cid:0) s − j − (cid:1) . Motivated by this we have asyzygetic version of the colorful Helly theorem. Theorem 4.3.
Let S be a polynomial ring where the variables have r colors.Let I be an ideal in S which is homogeneous for the N r -grading, is generatedin degree d + 1 and is ( d + 1) -regular, so it has linear resolution. Suppose I has elements of pure color i for each i = 1 , . . . , r , and let Ω l be the l ’thsyzygy module in an N r -graded resolution of S/I (so Ω = I ). For each l = 1 , . . . , r and each color vector a = ( a , . . . , a r ) where P a i = d + l and s = supp ( a ) , the vector space dimension of (Ω l ) a is greater than or equal to (cid:0) s − l − (cid:1) . Theorem 4.1 may be deduced from this. Simply apply Theorem 4.3 to thetruncated ideal I ≥ d +1 = ⊕ p ≥ d +1 I p . Then Theorem 4.1 is the special case l = 1. Our goal is therefore now to prove Theorem 4.3.4.2. The result of Kalai and Meshulam.
Let M be a matroid on a finiteset V (see Oxley [13] or, relating it more directly to simplicial complexes,Stanley [17, III.3]). This gives rise to a simplicial complex consisting of theindependent sets of the matroid. If ρ is the rank function of the matroid,this simplicial complex consist of all S ⊆ V such that ρ ( S ) = | S | . Kalai andMeshulam [10, Thm. 1.6] show the following. Theorem 4.4 (Kalai, Meshulam 2004) . Let Y be a d -Leray complex on V and M a matroid complex on V such that M ⊆ Y . Then there is a simplex τ ∈ Y such that ρ ( V − τ ) ≤ d . In case we have a partition V = V ∪ · · · ∪ V d +1 we get the following. Corollary 4.5.
Let M be the transversal matroid on the sets V , . . . , V d +1 ,i.e. the bases consist of all S ⊆ V such that the cardinality of each S ∩ V i isone, and let Y be a d -Leray complex containing M . Then there is a simplex τ such that ( V − τ ) ∩ V i is empty for some i , or in other words τ ⊇ V i . In the monomial case it is not difficult, by polarizing, to prove Theorem4.1 as a consequence of Corollary 4.5. However our result is a simultaneousgeneralization of this fact to arbitrary multigraded ideals, to two parameters, r the number of colors, and d + 1 the regularity, and to higher syzygies ofthe ideal.It is particularly worth noting that we do not prove Theorem 4.1 in anydirect way, and we do not know how to do it. Rather, it comes out as aspecial case of Theorem 4.3, which is proved by showing that it holds forthe Ω l by descending induction on l , starting from Ω r . Comparing the resolution of I to resolutions of monomialideals. Let T = k [ y , . . . , y r ]. To start with we will look at monomial ideals J in T such that T /J is artinian, i.e. J contains an ideal K = ( y a , . . . , y a r r )generated by powers of variables. There is then a surjection T /K p −→ T /J which lifts to a map of minimal resolutions B · ˜ p −→ A · . Lemma 4.6.
Let
T e be the last term in the minimal resolution B · of T /K (so e has multidegree ( a , . . . , a r ) ), and let A r = ⊕ n T e i be the last term inthe minimal free resolution A · of T /J . Suppose in the lifting B · ˜ p −→ A · of T /K → T /J that e P m i e i . Then each m i = 0 .Proof. Let ω T ∼ = S ( − ) be the canonical module of T . Dualizing the map ˜ p we get a map Hom T ( A · , ω T ) Hom T (˜ p,ω T ) −→ Hom T ( B · , ω T ) . Since
T /K and
T /J are artinian, this map will be a lifting of the mapExt rT ( T /J, ω T ) → Ext rT ( T /K, ω T )to their minimal free resolutions. But this map is simply(2) ( T /J ) ∗ → ( T /K ) ∗ , where () ∗ denotes the vector space dual Hom k ( − , k ), and so this map isinjective.Now Hom T ( A r , ω T ) is ⊕ n T u i where the u i are a dual basis of the e i , andcorrespondingly let u be the dual basis element of e . Then we will have u i Hom T (˜ p,ω T ) m i u. Each u i maps to a minimal generator of ( T /J ) ∗ . If m i was 0, this generatorwould map to 0 in ( T /K ) ∗ , but since (2) is injective this does not happen. (cid:3) Let λ : X → k be a function associating to each variable in X a consantin k . There is then a map(3) p λ : S = k [ X ] → k [ y , . . . , y r ] = T sending each element x in X i to λ ( x ) y i . Given a multigraded ideal I in S with elements of pure color i for each i , we can compare its regularity to theregularity of ideals in T . Theorem 4.7.
Let I ⊆ S be a multigraded ideal with elements of pure color i for each i . The image p λ ( I ) by the map (3) is then a monomial ideal. Let λ be sufficiently general so that for each i some element of pure color i in I has nonzero image. If J ⊆ T is an ideal containing the image p λ ( I ) , itsregularity is less than or equal to the regularity of I . OLORFUL HELLY THEOREM AND COLORFUL RESOLUTIONS OF IDEALS 11
Proof.
Let P i be elements of pure color i in I for i = 1 , . . . , r , such that P i maps to a nonzero multiple of y a i i . Let H be the ideal ( P , . . . , P r ). We geta commutative diagram S/H −−−−→
T /K y y
S/I −−−−→
T /J.
In the minimal free resolution A · of T /J let A r = ⊕ m T e i . Since T /J isartinian, its regularity is max { deg( e i ) − r } . Let F · be the minimal freeresolution of S/I and let F r = ⊕ m ′ Sf i . The maps in the diagram abovelift to maps of free resolutions. In homological degree r of the resolution of S/H we have a free S -module of rank one. If we consider its image in T /J and use Lemma 4.6 together with the commutativity of the diagram, we seethat by the map F r ˜ p r −→ A r there must for every e i exist an f j such that thecomposition Sf j → F · → A · → T e i is nonzero. But thenmax { deg( f j ) − r } ≥ max { deg( e i ) − r } so we get our statement. (cid:3) Corollary 4.8. If I has linear resolution, then p λ ( I ) has linear resolution(with the generality assumption on λ ) .Question 4.9. Is the corollary above true without the hypothesis on I thatit contains elements of pure color i for each i ?Now we shall proceed to prove Theorem 4.3 and thereby also the specialcase Theorem 4.1. Proof of Theorem 4.3.
Let J = m d +1 be the power of the maximal ideal in T .There is then a completely explicit form of the minimal free resolution A · of m d +1 , the Eliahou-Kervaire resolution, [8] or [14], which we now describe inthis particular case. For a monomial m let max( m ) = max { q | y q divides m } .Let A p be the free T -module with basis elements ( m ; j , . . . , j p − ) where m is a monomial of degree d + 1 and 1 ≤ j < · · · < j p − < max( m ). Thesebasis elements are considered to have degree d + p . The differential of A · isgiven by sending a basis element ( my k ; j , . . . , j p − ) with max( my k ) = k to X q ( − q y j q ( my k ; j , . . . , ˆ j q , . . . , j p − )(4) − X q ( − q y k ( my j q ; j , . . . , ˆ j q , . . . , j p − ) . (If a term in the second sum has max( my j q ) ≥ j p − , the term is consideredto be zero.) In homological degree r we know by the proof of Theorem 4.7 that for each basis element e = ( my r ; 1 , , . . . , r −
1) of A r there is a basiselement f of F r such that the composition Sf → F r → A r → T e maps f to ne where n is a nonzero monomial. Since the f and e havethe same total degree (the resolution is linear), we may assume that themonomial n is 1. Since the map F r ˜ p r −→ A r is multihomogeneous and everytwo of the basis elements of A r have different multidegrees, we will in facthave f ˜ p r e . This shows that the theorem holds when l = r .We will now show Theorem 4.3 by descending induction on l . Let ( my k ; J ′ )be a given basis element of A l where |J ′ | = l − my k ) = k > max( J ′ ).1. Suppose k > l . Then there is an r < k not in J ′ , and let J = J ′ ∪ { r } .Consider now the image of ( my k ; J ) given by (4). No other basis elementinvolved in this image has the same multidegree as ( my k ; J ′ ). By inductionwe assume there is an element f in F l +1 , that forms part of a basis of F l +1 ,such that f ˜ p l +1 ( my k ; J ). The differential in F · maps f to P | X | x i f i whereeach f i may be considered a basis element of F l if nonzero. Since the map˜ p l is homogeneous, a scalar multiple of some f i must map to ( my k ; J ′ ).2. Suppose k = l . Then J ′ = { , , . . . , k − } and let J = J ′ ∪ { k } . Theimage of ( my k +1 ; J ) by the differential in A · will be(5) k X q =1 ( − q y q ( my k +1 ; J \{ q } ) − k X q =1 ( − q y k +1 ( my q ; J \{ q } ) . Since max( my q ) ≤ k and max J \{ q } = k when q = k , all the terms in thesecond sum become zero save( − k y k +1 ( my k ; J ′ )and this term is the only one in (5) involving a basis element with themultidegree of ( my k ; J ′ ). By the same argument as in 1. above, there is abasis element f in F l that maps to ( my k ; J ′ ). This concludes the proof ofthe theorem. (cid:3) The following is a consequence of Theorem 4.1.
Corollary 4.10.
Let J in k [ y , . . . , y r ] be a monomial ideal with linear res-olution generated in degree d + 1 . If J contains the pure powers y d +1 i foreach i , then J is ( y , . . . , y r ) d +1 . This also follows from the fact that the vector space dimensions of thegraded pieces of the socle of
S/J is determined by the last term of theresolution of
S/J . The resolution being linear means that the socle of
S/J is concentrated in degree one lower than the generators of J .Let ∆( d + 1) be the geometric simplex in R r defined as the convex hullof all r -tuples ( a , . . . , a r ) of non-negative integers with P a i = d + 1. The OLORFUL HELLY THEOREM AND COLORFUL RESOLUTIONS OF IDEALS 13 monomials of degree d + 1 of a monomial ideal in k [ y , . . . , y r ] may be iden-tified with a subset of ∆( d + 1). Under the hypothesis of d + 1-linearity thecorollary above says that if the ideal contains the extreme points of ∆( d + 1)then it contains all of its lattice points. This suggests the following moregeneral problem. Problem 4.11.
Let J in k [ y , . . . , y r ] be a monomial ideal with linear res-olution generated in degree d + 1 . What can be said of the “topology” of thegenerating monomials of J considered as elements of ∆( d + 1) ? References [1] E.Babson, I.Novik
Face numbers of nongeneric initial ideals , Electronic Journal ofCombinatorics, Research paper 25, , no.2, (2004-2006), 1-23.[2] I.B´ar´any, A generalization of Caratheodory’s theorem , Discrete Math. (1982), 141-152.[3] A.Bj¨orner, Topological methods , in: R.Graham, M.Gr¨otschel, L.Lovasz(Eds.), Hand-book of Combinatorics, North-Holland, Amsterdam, 1995, 1819-1872.[4] A.Bj¨orner, P.Frankl, R.Stanley,
The number of faces of balanced Cohen-Macaulaycomplexes and a generalized Macaulay theorem , Combinatorica (1987), 23-34.[5] W.Bruns and J.Herzog, Cohen-Macaulay rings , Cambridge studies in advanced math-ematics 39, Cambridge University Press 1993.[6] D.Eisenbud,
Commutative algebra with a view towards algebraic geometry , GTM 150,Springer-Verlag, 1995.[7] J. Eckhoff,
Helly, Radon and Carath´eodory type theorems , in: P.M. Gruber, J.M.Wills (Eds.), Handbook of Convex Geometry, North-Holland, Amsterdam, 1993.[8] S.Eliahou, M.Kervaire,
Minimal resolutions of some monomial ideals , Journal of Al-gebra (1990), 1-25.[9] J.Eagon, V.Reiner,
Resolutions of Stanley-Reisner rings and Alexander duality , Jour-nal of Pure and Applied Algebra (1998), 265-275.[10] G.Kalai, R.Meshulam,
A topological colorful Helly theorem , Advances in Mathematics, (2005), 305-311.[11] E.Miller, B.Sturmfels,
Combinatorial commutative algebra , GTM 2005, Springer-Verlag, 2005.[12] U.Nagell, V.Reiner
Betti numbers of monomial ideals and shifted skew shapes , Re-search paper 3, , no.2, 59 pp.[13] J.G.Oxley Matroid theory , Oxford Graduate Texts in Mathematics 1992.[14] I.Peeva, M.Stillman
The minimal free resolution of a Borel ideal , Expositiones Math-ematicae, , no.3 (2008), 237-247.[15] R.Stanley Balanced Cohen-Macaulay complexes , Trans. Amer. Math. Soc. , no.1,(1979), 139-157.[16] M.Velasco
Minimal free resolutions that are not supported on a CW-complex , Journalof Algebra, , no.1 (2008), 102-114.[17] R.Stanley,
Combinatorics and Commutative Algebra , Second Edition, Birkh¨auser1996.
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