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

Gheorghita, Iulia; Sam, Steven V

doi:10.1216/jca-2016-8-4-537

Cited by 3 publications

(3 citation statements)

References 15 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%

“…The key early observation was that it is easier to determine which tables arise up to scalar multiple, and so the initial results [BS08, ES09, EFW11] consisted of characterizing the Boij–Söderberg cone of positive scalar multiples of Betti tables: 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: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Foundations of Boij–Söderberg theory for Grassmannians

Ford¹,

Levinson²

2018

Compositio Math.

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“…Boij-Söderberg theory (initiated in [BS08]) seeks to characterize the possible Betti tables of graded modules over polynomial rings, with the key insight that it is easier to study these tables only up to positive scalar multiple. The theory has been broadly successful: while the earliest results concerned Betti tables of Cohen-Macaulay modules (stratified by their codimension) [EFW11,ES09], the theory was extended to all modules [BS12], to certain modules over multigraded and toric rings [BBEG12,EE12], and more [KS15,GS16]. For some surveys, see [Flø12,FMP16].…”

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confidence: 99%

Towards Boij–Söderberg theory for Grassmannians : the case of square matrices

2018

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We characterize the cone of GL-equivariant Betti tables of Cohen-Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for 'Boij-Söderberg theory for Grassmannians', with the goal of characterizing the cones of GL k -equivariant Betti tables of modules over the coordinate ring of kˆn matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Grpk, C n q. The proof uses Hall's Theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman's geometric technique to certain graded pure complexes of Eisenbud-Fløystad-Weyman.

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Extremal rays of Betti cones

Ananthnarayan

Kumar

2019

J. Algebra Appl.

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We study the Betti cone of modules of a standard graded algebra over a field, and find a class of modules whose Betti diagrams span extremal rays of the Betti cone. We also identify a class of one-dimensional Cohen–Macaulay rings with certain Hilbert series, for which the Betti cone is spanned by these extremal rays. These results lead to an algorithm for the decomposition of Betti table of modules, into the extremal rays, over such rings, and also help to obtain bounds for the multiplicity of the given module.

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

Abstract: We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.Comment: 7 page

Cited by 3 publications

References 15 publications

Foundations of Boij–Söderberg theory for Grassmannians

Foundations of Boij–Söderberg theory for Grassmannians

Towards Boij–Söderberg theory for Grassmannians : the case of square matrices

Extremal rays of Betti cones

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