The Resolution of the Ideal of 2×2 Minors of a 2×nMatrix of Linear Forms

AbstractLetkbe an algebraically closed field and letS=k[x1,…,xm]. LetMbe a 2×nmatrix of linear forms ofSand letI2(M) denote the ideal generated by the determinants of the 2×2 minors ofM. We study in this paper the minimal finite free resolution ofS/I2(M) as anS-module.Mcorresponds to a certain 2-dimensional vector spaceLofm×nmatrices, that is, to a matrix pencil. The Kronecker–Weierstrass theory of such matrix pencils provides a normal form forL, and we characterize the resolution ofS/I2(M) in terms of this normal form. In particular, if the general element ofLis injective, we explicitly construct the minimal resolution ofS/I2(M) by repeated application of the horseshoe lemma. For anyM, we express the regularity ofS/I2(M) as a function of the invariants ofL