The resolution of (xN,yN,zN,wN)

Abstract

Let k be a field of characteristic zero, n<N be positive integers, P be the polynomial ring k[x,y,z,w], F be the homogeneous polynomial xn+yn+zn+wn, K be the ideal (xN,yN,zN,wN), and P‾ be the hypersurface ring P‾=P/(F). We describe the minimal multi-homogeneous resolution of P‾/KP‾ by free P‾-modules, the socle degrees of P‾/KP‾, and the minimal multi-homogeneous resolution of the Gorenstein ring P/(K:F) by free P-modules. Our arguments use Stanley's theorem that every Artinian monomial complete intersection over a polynomial ring with coefficients from a field of characteristic zero has the strong Lefschetz property as well as a multi-grading on P for which both ideals K and (F) are homogeneous. The resolution of P‾/KP‾ by free P‾-modules is obtained from a Differential Graded Algebra resolution of P/(K:F) by free P-modules, together with one homotopy map. The multi-grading is used to prove that the resulting resolution is minimal.