The cone of Betti diagrams over a hypersurface ring of low embedding dimension

Berkesch, Christine; Burke, Jesse; Erman, Daniel; Gibbons, Courtney R.

doi:10.1016/j.jpaa.2012.03.007

Cited by 8 publications

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“…The theory has rapidly expanded to other settings. More recent work has considered modules over multigraded and toric rings [EE17,BES17], and some homogeneous coordinate rings [BBEG12,GS16,KS15], as well as more detailed homological questions [NS13,BEKS13,EES13]. A good survey of the field is [Flø12].…”

Section: Boij-söderberg Theorymentioning

confidence: 99%

Foundations of Boij–Söderberg theory for Grassmannians

Ford¹,

Levinson²

2018

Compositio Math.

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Boij-Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with S. Sam [FLS16], extending the theory to the setting of GL k -equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in kn variables, thought of as the entries of a kˆn matrix.We give equivariant analogues of two important features of the ordinary theory: the Herzog-Kühl equations and the pairing between Betti and cohomology tables. As a necessary step, we also extend the result of [FLS16], concerning the base case of square matrices, to cover complexes other than free resolutions.Our statements specialize to those of ordinary Boij-Söderberg theory when k " 1. Our proof of the equivariant pairing gives a new proof in the graded setting: it relies on finding perfect matchings on certain graphs associated to Betti tables.Finally, we give preliminary results on 2ˆ3 matrices, exhibiting certain classes of extremal rays on the cone of Betti tables.thought of as the entries of a kˆn matrix. The group GL k acts on R k,n , and we are interested in (finitely-generated) equivariant modules M , i.e., those with a compatible GL k action. Aside from the inherent interest of understanding sheaf cohomology and syzygies on Grassmannians, there is hope that this setting might avoid some obstacles faced in other extensions of Boij-Söderberg theory, e.g. to products of projective spaces. For example, in the 'base case' of square matrices pn " kq, the 'irrelevant ideal' is the principal ideal generated by the determinant, and the Boij-Söderberg cone has an especially elegant structure (see below, Section 1.3.2).We define equivariant Betti tables βpM q using the representation theory of GL k . Let S λ pC k q denote the irreducible GL k representation of weight λ, where S λ is the Schur functor. There is a corresponding free module, namely S λ pC k q b C R k,n , and every equivariant free module is a direct sum of these. Then βpM q is the collection of numbers β i,λ pM q :" # copies of S λ pC k q in the generators of the i-th syzygy module of M. -SÖDERBERG THEORY FOR GRASSMANNIANS 3 Thus, by definition, the minimal equivariant free resolution of M has the form FOUNDATIONS OF BOIJNext, for E a coherent sheaf on Grpk, C n q, we will define the GL-cohomology table γpEq, generalizing the usual cohomology table:where S is the tautological vector bundle on Grpk, C n q of rank k. We write BT k,n :" À i,λ Q for the space of abstract Betti tables. Similarly, we write CT k,n :" À i ś λ Q for the space of abstract GL-cohomology tables.Remark 1.1. The case k " 1 reduces to the ordinary Boij-Söderberg theory, since an action of GL 1 is formally equivalent to a grading; the module Rr´js is just S pjq pCq b R. Note also that S " Op´1q on projective space.The initial questions of Boij-Söderberg theory concerned finite-length graded modules M , i.e. those annihilated by a power of the homogeneous maximal ideal, and more generally Cohen-Macaulay modules. Similarly, we r...

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Section: Boij-söderberg Theorymentioning

confidence: 99%

Foundations of Boij–Söderberg theory for Grassmannians

Ford¹,

Levinson²

2018

Compositio Math.

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“…Then the proofs of [BBEG,Lemmas 2.7,2.8] show that B Q (B) has a triangulation coming from the simplicial cones spanned by the rays corresponding to the elements of maximal chains in this partial order.…”

Section: The Cone Of Betti Diagramsmentioning

confidence: 99%

The cone of Betti tables over three non-collinear points in the plane

Gheorghita¹,

Sam²

2016

J. Commut. Algebra

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“…For the hypersurface ring R, this partial order provides a simplicial fan structure, as illustrated in [BBEG11] and discussed in Example 8.1. The partial order is determined by an analog of Theorem 1.1. over R is described in detail in [BBEG11]. The extremal rays still correspond to Cohen-Macaulay modules with pure resolutions, though some of the degrees are infinite in length.…”

Section: Remarks On Other Graded Ringsmentioning

confidence: 99%

“…A second implication involves the extension of Boij-Söderberg theory to more complicated projective varieties or graded rings. For instance, the cone of free resolutions over a quadric hypersurface ring of k[x, y] is described in [BBEG11]. The extremal rays in this case correspond to pure resolutions of finite or infinite length.…”

Section: Introductionmentioning

confidence: 99%

Poset Structures in Boij–Söderberg Theory

Berkesch¹,

Erman²,

Kummini³

et al. 2011

International Mathematics Research Notices

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Boij-Söderberg theory is the study of two cones: the cone of Betti diagrams of standard graded minimal free resolutions over a polynomial ring and the cone of cohomology tables of coherent sheaves over projective space. We provide a new interpretation of these partial orders in terms of the existence of nonzero homomorphisms, for both the general and equivariant constructions. These results provide new insights into the families of modules and sheaves at the heart of Boij-Söderberg theory: Cohen-Macaulay modules with pure resolutions and supernatural sheaves. In addition, they suggest the naturality of these partial orders and provide tools for extending Boij-Söderberg theory to other graded rings and projective varieties.Outline. In this paper, we first focus on the cone of Betti diagrams for S. In Section 2, we prove the reverse implications of Theorems 1.1 and 1.3. We then construct nonzero morphisms between modules with pure resolutions. Sections 3 and 4, respectively, address the forward directions of Theorems 1.1 and 1.3. We next address the cone of cohomology tables for P n−1 . In Section 5, we prove the reverse implications of Theorems 1.2 and 1.4. We then turn to the construction of nonzero morphisms between supernatural sheaves: Sections 6 and 7, respectively, address the forward directions of Theorems 1.2 and 1.4. Finally, we provide in Section 8 a brief discussion of how Theorem 1.1 has been applied in the study of Boij-Söderberg theory over other graded rings. We suggest the survey [ES10b] to the reader seeking additional background on Boij-Söderberg theory. The poset of degree sequencesLet M be a finitely generated graded S-module. The (i, j)th graded Betti number of M, denoted β i,j (M), is dim k Tor S i (k, M) j . The Betti diagram of M is a table, with rows indexed by Z and columns by 0, . . . , n, such that the entry in column i and row j is length of d, denoted ℓ(d), is the largest integer t such that d t is finite. Definition 2.1. A graded S-module M is said to have a pure resolution of type d if a minimal free resolution of M has the form 0 ← M ← S(−d 0 ) β 0,d 0 ← S(−d 1 ) β 1,d 1 ← · · · ← S(−d ℓ(d) ) β ℓ(d),d ℓ(d) ← 0.

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The cone of Betti diagrams over a hypersurface ring of low embedding dimension

Cited by 8 publications

References 21 publications

Foundations of Boij–Söderberg theory for Grassmannians

Foundations of Boij–Söderberg theory for Grassmannians

The cone of Betti tables over three non-collinear points in the plane

Poset Structures in Boij–Söderberg Theory

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