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

Ford, Nicolas; Levinson, Jake; Sam, Steven V

doi:10.2140/ant.2018.12.285

Cited by 4 publications

(12 citation statements)

References 21 publications

(21 reference statements)

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“…The corresponding Grassmannian is a point, so there is no dual geometric picture or cone. We will recall the description of BS k,k due to [FLS18]; we then describe the derived cone BS D k,k . When k = 1, the ring is just C[t], and its torsion graded modules are essentially trivial to describe.…”

Section: Square Matrices and Perfect Matchingsmentioning

confidence: 99%

“…It is, by contrast, easy to establish the following inequalities on BS k,k . Recall that a subset S of a poset (P, ) is downwards closed if, whenever x ∈ S and y x, it follows that y ∈ S. (4.2) (The terminology of antichains is due to [FLS18], where the inequality (4.2) is stated in terms of the maximal elements of S, which form an antichain in Y ± . )…”

Section: Prior Work On Bs Kk [Fls18]mentioning

confidence: 99%

“…Theorem 4.5[FLS18, Theorem 3.8]. The cone BS k,k ⊆ BT k,k is defined by the rank equation (4.1), the conditions β i,λ 0 for i = 0, 1 and β i,λ = 0 for i = 0, 1, and the antichain inequalities (4.2).…”

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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: Square Matrices and Perfect Matchingsmentioning

confidence: 99%

Section: Prior Work On Bs Kk [Fls18]mentioning

confidence: 99%

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Foundations of Boij–Söderberg theory for Grassmannians

Ford¹,

Levinson²

2018

Compositio Math.

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“…Further results on categorification for the decomposition of cohomology tables were proved by Erman and Sam in [9]. Recently, there has been interest in extending the theory to other settings: for example, [10,11] develop a Boij-Söderberg theory for coherent sheaves on Grassmannians.…”

Section: Introductionmentioning

confidence: 99%

Decomposition of graded local cohomology tables

Stefani

Smirnov

2020

Math. Z.

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Let R be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded R-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets.

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“…Applications include the proof of the multiplicity conjecture [HS98,ES09], a special case of Horrocks' conjecture [Erm10], and constraints on regularity [McC12]. There are current efforts to extend Boij-Söderberg theory to Grassmannians [FLS16] as well as expository notes on open questions and the state of the field [ES16,Flø12]. In the case of complete intersections, it was shown in [AGHS17] that the diagrams of complete intersections behave similarly to pure diagrams, creating a non-trivial sub-cone of the Boij-Söderberg cone.…”

Section: Introductionmentioning

confidence: 99%

Recursive strategy for decomposing Betti tables of complete intersections

Gibbons

Huben

Stone

2019

Int. J. Algebra Comput.

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We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that we use to investigate stability and compatibility of the Boij-Söderberg decompositions of related diagrams; indeed, when the biggest generating degree is sufficiently large, the alternative algorithm produces the Boij-Söderberg decomposition. We also provide a detailed analysis of the Boij-Söderberg decomposition for Betti diagrams of codimension four complete intersections where the largest generating degree satisfies the size condition.

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

Cited by 4 publications

References 21 publications

Foundations of Boij–Söderberg theory for Grassmannians

Foundations of Boij–Söderberg theory for Grassmannians

Decomposition of graded local cohomology tables

Recursive strategy for decomposing Betti tables of complete intersections

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