Foundations of Boij-Soderberg Theory for Grassmannians

algebraic geometry;combinatorics;commutative algebra;syzygies;Mathematics;Science;Mathematics

Levinson, JakeSnowden, Andrew ;

University of Michigan

关键词: algebraic geometry;    combinatorics;    commutative algebra;    syzygies;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/137146/jakelev_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Boij-Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This thesis is concerned with extending the theory to the setting of modules with an action of the general linear group, and sheaves on Grassmannians Gr(k,n). Algebraically, we study modules over a polynomial ring in kn variables, thought of as the entries of a k-by-n matrix. The goal is to characterize equivariant Betti tables of such modules and, dually, cohomology tables of sheaves on Gr(k,n).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 and fundamental base case, we consider resolutions and certain more general complexes for the case of square matrices.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.