The syzygies of the ideal [formula omitted] in the hypersurface ring defined by [formula omitted

Abstract

Let k be an arbitrary field and n, d, and r be non-negative integers with r at most n−1. Let N be the integer dn+r, P be the polynomial ring k[x1,x2,x3,x4], fn be the polynomial x1n+x2n+x3n+x4n in P, Cd,n,r be the ideal (x1N,x2N,x3N,x4N) of P, P¯n be the hypersurface ring P/(fn), Qd,n,r be the quotient ring P¯n/Cd,n,rP¯n and Ωd,n,ri be the i-th syzygy module of Qd,n,r as a P¯n-module. We prove that Ωd,n,r3 is isomorphic to the direct sum (Ω0,n,r3)a⊕(Ω0,n,r4)b⊕(P¯n)c, for some non-negative integers a, b, and c. (The parameters a, b, and c depend on d and the characteristic of k; however, they are independent of n and r.) Furthermore, if the characteristic of k is zero, then a=2d+1 and b=c=0.