Foundations of Boij–Söderberg theory for Grassmannians

Abstract

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...

Decomposition of graded local cohomology tables

“…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.…”