BOIJ-SO¨DERBERG THEORY: INTRODUCTION AND SURVEY GUNNAR FLØYSTAD 2 1 Abstract. Boij-S¨oderberg theory describes the Betti diagrams of graded mod- 0 2 ules over the polynomial ring, up to multiplication by a rational number. Analog Eisenbud-Schreyer theory describes the cohomology tables of vector bundles on n projective spaces up to rational multiple. We give an introduction and survey of a J these newly developed areas. 7 1 ] C Contents A . Introduction 2 h t 1. The Boij-So¨derberg conjectures 5 a 1.1. Resolutions and Betti diagrams 5 m 1.2. The positive cone of Betti diagrams 7 [ 1.3. Herzog-Ku¨hl equations 8 2 1.4. Pure resolutions 8 v 1 1.5. Linear combinations of pure diagrams 9 8 1.6. The Boij-So¨derberg conjectures 11 3 0 1.7. Algorithmic interpretation 11 6. 1.8. Geometric interpretation 12 0 2. The exterior facets of the Boij-So¨derberg fan and their supporting 1 hyperplanes 12 1 : 2.1. The exterior facets 13 v i 2.2. The supporting hyperplanes 14 X 2.3. Pairings of vector bundles and resolutions 19 r a 3. The existence of pure free resolutions and of vector bundles with supernatural cohomology 22 3.1. The equivariant pure free resolution 22 3.2. Equivariant supernatural bundles 26 3.3. Characteristic free supernatural bundles 27 3.4. The characteristic free pure resolutions 27 3.5. Pure resolutions constructed from generic matrices 30 4. Cohomology of vector bundles on projective spaces 31 4.1. Cohomology tables 32 4.2. The fan of cohomology tables of vector bundles 33 4.3. Facet equations 34 1991 Mathematics Subject Classification. Primary: 13D02, 14F05; Secondary: 13C14, 14N99. Key words and phrases. Betti diagrams, cohomology of vector bundles, Cohen-Macaulay mod- ules, pure resolutions, supernatural bundles. 1 5. Extensions to non-Cohen-Macaulay modules and to coherent sheaves 37 5.1. Betti diagrams of graded modules in general 37 5.2. Cohomology of coherent sheaves 38 6. Further topics 40 6.1. The semigroup of Betti diagrams of modules 40 6.2. Variations on the grading 43 6.3. Poset structures 44 6.4. Computer packages 44 6.5. Three basic problems 45 References 46 Introduction In November 2006 M.Boij and J.So¨derberg put out on the arXiv a preprint ”Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjec- ture”. The paper concerned resolutions of graded modules over the polynomial ring S = k[x ,...,x ] over a field k. It put forth two striking conjectures on the form 1 n of their resolutions. These conjectures and their subsequent proofs have put the greatest floodlight on our understanding of resolutions over polynomial rings since the inception of the field in 1890. In this year David Hilbert published his syzygy theorem stating that a graded ideal over the polynomial ring in n variables has a resolution of length less than or equal to n. Resolutions of modules both over the polynomial ring and other rings have since then been one of the pivotal topics of algebraic geometry and commutative algebra, and more generally in the field of associative algebras. For the next half a year after Boij and S¨oderberg put out their conjectures, they were incubating in the mathematical community, and probably not so much exposed to attacks. The turning point was the conference at MSRI, Berkeley in April 2007 in honor of David Eisenbud 60’th birthday, where the conjectures became a topic of discussion. For those familiar with resolutions of graded modules over the polynomial ring, a complete classification of their numerical invariants, the graded Betti numbers (β ), ij seemed a momentous task, completely out of reach (and still does). Perhaps the central idea of Boij and S¨oderberg is this: We don’t try to determine if (β ) are the ij graded Betti numbers of a module, but let us see if we can determine if m·(β ) are ij the graded Betti numbers of a module if m is some big integer. This is the idea of stability which has been so successful in stable homotopy theory in algebraic topology and rational divisor theory in algebraic geometry. Another way to phrase the idea of Boij and S¨oderberg is that we do not determine the graded Betti numbers (β ) but rather the positive rays t · (β ) where t is a ij ij positive rational number. It is easy to see that these rays form a cone in a suitable vector space over the rational numbers. The conjectures of Boij and S¨oderberg considered the cone B of such diagrams comingfrommodulesofcodimensioncwiththeshortestpossiblelengthofresolution, 2 c. This isthe class ofCohen-Macaulay modules. The first conjecture states precisely what the extremal rays of the cone B are. The diagrams on these rays are called pure diagrams. They are the possible Betti diagrams of pure resolutions, (1) S(−d0)β0,d0 ← S(−d1)β1,d1 ← ··· ← S(−dc)βc,dc of graded modules, where the length c is equal to the codimension of the module. To prove this conjecture involved two tasks. The first is to show that there are vectors on these rays which actually are Betti diagrams of modules. The second is to show that these rays account for all the extremal rays in the cone B, in the sense that any Betti diagram is a positive rational combination of vectors on these rays. This last part was perhaps what people found most suspect. Eisenbud has said that his immediate reaction was that this could not be true. Boij and S¨oderberg made a second conjecture giving a refined description of the cone B. There is a partial order on the pure diagrams, and in any chain in this partial order the pure diagrams are linearly independent. Pure diagrams in a chain therefore generate a simplicial cone. Varying over the different chains we then get a simplicial fan of Betti diagrams. The refinement of the conjectures states that the realization of this simplicial fan is the positive cone B. In this way each Betti diagram lies on a unique minimal face of the simplicial fan, and so we get a strong uniqueness statement on how to write the Betti diagram of a module. After the MSRI conference in April 2007, Eisenbud and the author independently started to look into the existence question, to construct pure resolutions whose Betti diagram is a pure diagram. They came up with the construction of the GL(n)- equivariant resolution described in Subsection 3.1. Jerzy Weyman was instrumental in proving the exactness of this resolution and the construction was published in a joint paper in September 2007 on the arXiv, [11]. In the same paper also appeared another construction of pure resolutions described in Subsection 3.5. After this success D. Eisenbud and F.-O. Schreyer went on to work on the other part of the conjectures. And in December 2007 they published on the arXiv the proof of the second part of the conjectures of Boij and S¨oderberg, [12]. But at least two moreinteresting thingsappearedinthispaper. They gaveaconstructionofpure resolutions that worked in all characteristics. The constructions above, [11] work only in characteristic zero. But most startling, they discovered a surprising duality with cohomology tables of algebraic vector bundles on projective spaces. And fairly parallel to the proof of the second Boij-So¨derberg they were able to give a complete description of all cohomology tables of algebraic vector bundles on projective spaces, up to multiplication by a positive rational number. In the wake of this a range of papers have followed, most of which are discussed in this survey. But one thing still needs to be addressed. What enticed Boij and S¨oderberg to come up with their conjectures? The origin here lies in an observation by Huneke and Miller from1985, that if a Cohen-Macaulay quotient ring of A = S/I has pure resolution (1) (so d = 0 and β = 1), then its multiplicity e(A) is equal 0 0,d0 3 to the surprisingly simple expression c 1 · d . i c! i=1 Y This led naturally to consider resolutions F of Cohen-Macaulay quotient rings A = • S/I in general. In this case one has in each homological term F in the resolution a i maximal twist S(−a ) (so a is minimal) and a minimal twist S(−b ) occurring. The i i i multiplicity conjecture of Herzog, Huneke and Srinivasan, see [25] and [26], stated that the multiplicity of the quotient ring A is in the following range c c 1 1 a ≤ e(A) ≤ b . i i c! c! i=1 i=1 Y Y Over the next two decades a substantial number of papers were published on this treating various classes of rings, and also various generalizations of this conjecture. But efforts in general did not succeed because of the lack of a strong enough under- standing of the (numerical) structure of resolutions. Boij and S¨oderberg’s central idea is to see the above inequalities as a projection of convexity properties of the Betti diagrams of graded Cohen-Macaulay modules: The pure diagrams generate the extremal rays in the cone of Betti diagrams. Notation. The graded Betti numbers β (M) of a finitely generated module M are ij indexed by i = 0,...,n and j ∈ Z. Only a finite number of these are nonzero. By a diagram we shall mean a collection of rational numbers (β ), indexed as above, ij with only a finite number of them being nonzero. The organization of this paper is as follows. In Section 1 we give the important notions, like the graded Betti numbers of a module, pure resolutions and Cohen- Macaulay modules. Such modules have certain linear constraints on their graded Betti numbers, the Herzog-Ku¨hl equations, giving a subspace LHK of the space of diagrams. We define the positive cone B in LHK of Betti diagrams of Cohen- Macaulay modules. An important technical convenience is that we fix a “window” on the diagrams, considering Betti diagrams where the β are nonzero only in a ij finite range of indices (i,j). This makes the Betti diagrams live in a finite dimen- sional vector space. Then we present the Boij-So¨derberg conjectures. We give both the algorithmic version, concerning the decomposition of Betti diagrams, and the geometric version in terms of fans. In Section 2 we define the simplicial fan Σ of diagrams. The goal is to show that its realization is the positive cone B, and to do this we study the exterior facets of Σ. The main work of this section is to find the equations of these facets. They are the key to the duality with algebraic vector bundles, and the form of their equations are derived from suitable pairings between Betti diagrams and cohomology tables of vector bundles. The positivity of the pairings proves that the cone B is contained in the realization of Σ, which is one part of the conjectures. The other part, that Σ is contained in B, is shown in Section 3 by providing the existence of pure resolutions. We give in 3.1 the construction of the equivariant pure resolution of [11], in 3.4 the characteristic free resolution of [12], and in 3.5 4 the second construction of [11]. For cohomology of vector bundles, the bundles with supernatural cohomology play the analog role of pure resolutions. In 3.2 we give the equivariant construction of supernatural bundles, and in 3.3 the characteristic free construction of [12]. In Section 4 we first consider the cohomology of vector bundles on projective spaces, and give the complete classification of such tables up to multiplication by a positive rational number. The argument runs analogous to what we do for Betti diagrams. We define the positive cone of cohomology tables C, and the simplicial fan of tables Γ. We compute the equations of the exterior facets of Γ which again are derived from the pairings between Betti diagrams and cohomology tables. The positivity of these pairings show that C ⊆ Γ, and the existence of supernatural bundles that Γ ⊆ C, showing the desired equality C = Γ. Section 5 considers extensions of the previous results. First in 5.1 we get the classificationofgradedBettinumbersofallmodulesuptopositiverationalmultiples. For cohomology of coherent sheaves there is not yet a classification, but in 5.2 the procedure to decompose cohomology tables of vector bundles is extended to cohomology tables of coherent sheaves. However this procedure involves an infinite number of steps, so this decomposition involves an infinite sum. Section 6 gives more results that have followed in the wake of the conjectures and their proofs. The ultimate goal, to classify Betti diagrams of modules (not just up to rational multiple) is considered in 6.1, and consists mainly of examples of diagrams which are or are not the Betti diagrams of modules. So far we have considered k[x ,...,x ] to be standard graded, i.e. each degx = 1. In 6.2 we consider other 1 n i gradings and multigradings on the x . Subsection 6.3 considers the partial order i on pure diagrams, so essential in defining the simplicial fan Σ. In 6.4 we inform on computer packages related to Boij-So¨derberg theory, and in 6.5 we give some important open problems. Acknowledgement. We thank the referee for several corrections and useful sugges- tions for improving the presentation. 1. The Boij-So¨derberg conjectures We work over the standard graded polynomial ring S = k[x ,...,x ]. For a 1 n graded module M over S, we denote by M its graded piece of degree d, and by d M(−r) the module where degrees are shifted so that M(−r) = M . d d−r Note. We shall always assume our modules to be finitely generated and graded. 1.1. Resolutions and Betti diagrams. A natural approach to understand such modules is to understand their numerical invariants. The most immediate of these is of course the Hilbert function: h (d) = dim M . M k d Another set of invariants is obtained by considering its minimal graded free resolu- tion: (2) F ← F ← F ← ··· ← F 0 1 2 l 5 Here each F is a graded free S-module ⊕ S(−j)βij. i j∈Z Example 1.1. Let S = k[x,y] and M be the quotient ring S/(x2,xy,y3). Then its minimal resolution is y 0   −x y2   x2 xy y3  0 −x S ←h−−−−−−−−−i S(−2)2 ⊕S(−3) ←−−−−−−−− S(−3)⊕S(−4). The multiple β of the term S(−j) in the i’th homological part F of the res- i,j i olution, is called the i’th graded Betti number of degree j. These Betti numbers constitute another natural set of numerical invariants, and the ones that are the topic of the present notes. By the resolution (2) we see that the graded Betti num- ber determine the Hilbert function of M. In fact the dimension dim M is the k d alternating sum (−1)idim (F ) . The Betti numbers are however more refined k i d numerical invariants of graded modules than the Hilbert function. P Example 1.2. Let M′ be the quotient ring S/(x2,y2). Its minimal free resolution is S ← S(−2)2 ← S(−4). Then M of Example 1.1 and M′ have the same Hilbert functions, but their graded Betti numbers are different. The Betti numbers are usually displayed in an array. The immediate natural choice is to put β in the i’th column and j’th row, so the diagram of Example 1.1 i,j would be: 0 1 2 0 1 0 0 1 0 0 0   2 0 2 0 30 1 1   40 0 1   However, to reduce the number ofrows, one uses the convention that the i’th column is shifted i steps up. Thus β is put in the i’th column and the j−i’th row. i,j Alternatively, β is put in the i’th column and j’th row. So the diagram above is i,i+j displayed as : 0 1 2 0 1 0 0 (3) 1 0 2 1   2 0 1 1   A Betti diagram has columns indexed by 0,...,n and rows indexed by elements of Z, but any Betti diagram (of a finitely generated graded module) is nonzero only in a finite number of rows. Our goal is to understand the possible Betti diagrams that can occur for Cohen-Macaulay modules. This objective seems however as of 6 yet out of reach. The central idea of Boij-So¨derberg theory is rather to describe Betti diagrams up to a multiple by a rational number. I.e. we do not determine if a diagram β is a Betti diagram of a module, but we will determine if qβ is a Betti diagramforsome positive rationalnumber q. ByHilbert’s Syzygy Theorem weknow that the length l of the resolution (2) is ≤ n. Thus we consider Betti diagrams to live in the Q-vector space D = ⊕ Qn+1, with the β as coordinate functions. An j∈Z ij element in this vector space, a collection of rational numbers (β ) where ij i=0,...,n,j∈Z only a finite number is nonzero, is called a diagram. 1.2. The positive cone of Betti diagrams. We want to make our Betti diagram live in a finite dimensional vector space, so we fix a “window” in D as follows. Let c ≤ n and Zc+1 be the set of strictly increasing integer sequences (a ,...,a ) in deg 0 c Zc+1. Such an element is called a degree sequence. Then Zc+1 is a partially ordered deg set with a ≤ b if a ≤ b for all i = 0,...,c. i i Definition 1.3. For a,b in Zc+1 let D(a,b) be the set of diagrams (β ) deg ij i=0,...,n,j∈Z such that β may be nonzero only in the range 0 ≤ i ≤ c and a ≤ j ≤ b . ij i i We see that D(a,b) is simply the Q-vector space with basis elements indexed by the pairs (i,j) in the range above determined by a and b. The diagram of Example 1.1, displayed above in (3), lives in the window D(a,b) with a = (0,1,2) and b = (0,3,4) (or a any triple ≤ (0,1,2) and b any triple ≥ (0,3,4)). If the module M has codimension c, equivalently its Krull dimension is n−c, the depth of M is ≤ n−c. By the Auslander-Buchsbaum theorem, [8], the length of the resolution is l ≥ c. To make things simple we assume that l has its smallest possible value l = c or equivalently that M has depth equal to the dimension n − c. This gives the class of Cohen-Macaulay (CM) modules. Definition 1.4. Let a and b be in Zc+1. deg • L(a,b) is the Q-vector subspace of the window D(a,b) spanned by the Betti diagrams of CM-modules of codimension c, whose Betti diagrams are in this window. • B(a,b) is the set of non-negative rays spanned by such Betti diagrams. Lemma 1.5. B(a,b) is a cone. Proof. We must show that if β and β are in B(a,b) then q β +q β is in B(a,b) 1 2 1 1 2 2 for all positive rational numbers q and q . 1 2 This is easily seen to be equivalent to the following: Let M and M be CM- 1 2 modules of codimension c with Betti diagrams in D(a,b). Show that c β(M ) + 1 1 c β(M ) is in B(a,b) for all natural numbers c and c . But this linear combination 2 2 1 2 is clearly the Betti diagram of the CM-module Mc1 ⊕ Mc2 of codimension c. And 1 2 clearly this linear combination is still in the window D(a,b). (cid:3) Our main objective is to describe this cone. 7 1.3. Herzog-Ku¨hl equations. Now given a resolution (2) of a module M, there are natural relations its Betti numbers β must fulfil. First of all if the codimension ij c ≥ 1, then clearly the alternating sum of the ranks of the F must be zero. I.e. i (−1)iβ = 0. ij i,j X When the codimension c ≥ 2 we get more numerical restrictions. Since M has dimension n−c, its Hilbert series is of the form h (t) = p(t) , where p(t) is some M (1−t)n−c polynomial. This may be computed as the alternating sum of the Hilbert series of each of the terms in the resolution (2): β tj β tj β tj h (t) = j 0j − j 1j +···+(−1)l j lj . M (1−t)n (1−t)n (1−t)n P P P Multiplying with (1−t)n we get (1−t)cp(t) = (−1)iβ tj. ij i,j X Differentiating this successively and setting t = 1, gives the equations (4) (−1)ijpβ = 0, p = 0,...,c−1. ij i,j X These equations are the Herzog-Ku¨hl equations for the Betti diagram (β ) of a ij module of codimension c. We denote by LHK(a,b) the Q-linear subspace of diagrams in D(a,b) fulfilling the Herzog-Ku¨hl equations (4). Note that L(a,b) is a subspace of LHK(a,b). We shall show that these spaces are equal. 1.4. Pure resolutions. Now we shall consider a particular case of the resolution (2). Let d = (d ,...,d ) be a strictly increasing sequence of integers, a degree 0 l sequence. The resolution (2) is pure if it has the form S(−d0)β0,d0 ← S(−d1)β1,d1 ← ··· ← S(−dl)βl,dl. By a pure diagram (of type d) we shall mean a diagram such that for each column i there is only one nonzero entry β , and the d form an increasing sequence. We i,di i see that a pure resolution gives a pure Betti diagram. When M is CM of codimension c, the Herzog-Ku¨hl equations give the following set of equations 1 −1 ··· (−1)c β 0,d0 d −d ··· (−1)cd β 0 1 c 1,d1  . .  . . . . . . . . dc−1 −dc−1 ··· (−1)cdc−1β   0 1 c  c,dc    This is a c×(c+1)matrix of maximal rank. Hence there is only a one-dimensional Q-vector space of solutions. The solutions may be found by computing the maximal 8 minors which are Vandermonde determinants and we find 1 β = (−1)i ·t· i,di (d −d ) k i k6=i Y wheret ∈ Q. Whent > 0allthesearepositive. Letπ(d)bethediagramwhich isthe smallest integer solution to the equations above. As we shall see pure resolutions and pure diagrams play a central role in the description of Betti diagrams up to rational multiple. 1.5. Linear combinations of pure diagrams. The rays generated by the π(d) turnouttobeexactlytheextremal raysintheconeB(a,b). ThusanyBettidiagram is a positive linear combination of pure diagrams. Let us see how this works in an example. Example 1.6. If the diagram of Example 1.1 0 1 0 0 β = 1 0 2 1   2 0 1 1   is a positive linear combination of pure diagrams π(d), the only possibilities for these diagrams are 1 0 0 1 0 0 1 0 0 π(0,2,3) = 0 3 2 , π(0,2,4) = 0 2 0 , π(0,3,4) = 0 0 0 .       0 0 0 0 0 1 0 4 3       Note that by the natural partial order on degree sequences we have (0,2,3) < (0,2,4) < (0,3,4). To find this linear combination we proceed as follows. Take the largest positive multiple c of π(0,2,3) such that β −c π(0,2,3) is still non-negative. We see that 1 1 c = 1/2 and get 1 1/2 0 0 1 β = β − π(0,2,3) = 0 1/2 0 . 1 2   0 1 1 Then take the largest possible multiple c ofπ(0,2,4) suchthat β −c π(0,2,4) is 2 1 2 non-negative. We see that c = 1/4 and get 2 1/4 0 0 1 1 β = β − π(0,2,3)− π(0,2,4) = 0 0 0 . 2 2 4   0 1 3/4   Taking the largest multiple c of π(0,3,4) such that β −c π(0,3,4) is non-negative, 3 2 3 we see that c = 1/4 and the last expression becomes the zero diagram. Thus we 3 get β as a positive rational combination of pure diagrams 1 1 1 β = π(0,2,3)+ π(0,2,4)+ π(0,3,4). 2 4 4 9 The basic part of Boij-So¨derberg theory says that this procedure will always work: It gives a non-negative linear combination of pure diagrams. We proceed to develop this in more detail. With Zc+1 equipped with the natural partial order, we get for deg a,b ∈ Zc+1 the interval [a,b] consisting of all degree sequences d with a ≤ d ≤ deg deg b. The diagrams π(d) where d ∈ [a,b] are the pure diagrams in the window deg determined by a and b. Example 1.7. If a = (0,2,3) and b = (0,3,4), the vector space D(a,b) consists of the diagrams which may be nonzero in the positions marked by ∗ below. ∗ 0 0 0 ∗ ∗ ,   0 ∗ ∗ and so is five-dimensional. The Herzog-Ku¨hlequations for the diagrams (c = 2) are the following two equations β −(β +β )+(β +β ) = 0 0,0 1,2 1,3 2,3 2,4 0·β −(2β +3β )+(3β +4β ) = 0 0,0 1,2 1,3 2,3 2,4 These are linearly independent and so LHK(a,b) will be three-dimensional. On the other hand the diagrams π(0,2,3),π(0,2,4) and π(0,3,4) are clearly linearly independent in this vector space and so they form a basis for it. This is a general phenomenon. The linear space LHK(a,b) (and as will turn out L(a,b)) may be described as follows. Proposition 1.8. Given any maximal chain a = d1 < d2 < ··· < dr = b in [a,b] . The associated pure diagrams deg π(d1),π(d2),...,π(dr) form a basis for LHK(a,b). The length of such a chain, and hence the dimension of the latter vector space is r = 1+ (b −a ). i i Proof. Let β be a solution of the HPK-equations contained in the window D(a,b). The vectors d1 and d2 differ in one coordinate, suppose it is the i’th coordinate, so d1 = (··· ,d1,···)andd2 = (··· ,d1+1,···). Let c besuch thatβ = β−c π(d1) is i i 1 1 1 zero in position (i,d1). Then β is contained in the window D(d2,b) and d2,...,dr i 1 is a maximal chain in [d2,b] . We may proceed by induction and in the end get deg β contained in [b,b] . Then β is pure and so is a multiple of π(dr). In r−1 deg r−1 conclusion r β = c π(di). i i=1 X To see that the π(di) are linearly independent, note that π(d1) is not a linear combination of the π(di) for i ≥ 2 since π(d1) is nonzero in position (i,d1) while i 10