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, extending the theory to the setting of $G{L}_k$-equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in $kn$ variables, thought of as the entries of a $k\times n$ 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 previous results, 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 $2\times 3$ matrices, exhibiting certain classes of extremal rays on the cone of Betti tables.

33 pages; comments welcome. v2 has minor revisions and one corrected statement on simple Betti tables in the final section

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