The Eisenbud - Koh - Stillman Conjecture on Linear Syzygies

Although the relationship between minimal free resolutions and Koszul cohomology has been known for a long time, it has been difficult to find a way to fully utilize the " exterior " nature of the Koszul classes. The technique used here seems to be one way to begin to do this. We prove a conjecture of Eisenbud-Koh-Stillman on linear syzygies and in consequence a conjecture of Lazarsfeld and myself on points in projective space. The main novelties in the proof are the use of " exterior minors, " explained below, and showing that certain kinds of linear syzygies in the exterior algebra are impossible. I will work over a field of arbitrary characteristic. It is a pleasure to acknowledge David Eisenbud for many highly useful conversations regarding this work. In particular, he simplified several arguments in the original version, some of which had only worked in characteristic 0. DEFINITION. Consider two vector spaces A,B of dimensions a,b respectively, and let V be a vector space of dimension n. Consider a b×a matrix of linear forms, which we think of as a linear map M : A → B ⊗ V. By a generalized column of M we mean, for some non-zero α ∈ A, the map M (α): B * → V , and by the rank of a generalized column α we mean the rank of the map M (α); similarly an element β * ∈ B * gives a generalized row which is a map M (β *): A → V whose rank is the rank of M (β *).