The syzygies of the ideal (xN1, xN2, xN3, xN4) in the hypersurface ring defined by xn1+xn2+xn3+xn4

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 xn1+xn2 +xn3 +xn4 in P, Cd,n,r be the ideal (xN1, xN2 , xN3 , xN4 ) of P, P over line n be the hypersurface ring P/(fn), Qd,n,r be the quotient ring module of Qd,n,r as a P over line n/Cd,n,rP over line n and Omega id,n,r be the i-th syzygy P over line n-module. We prove that Omega 3d,n,r is isomorphic to the direct sum (Omega 30,n,r)a (R) (Omega 40,n,r)b (R) (P over line 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.(c) 2022 Elsevier Inc. All rights reserved.