Let S be a polynomial ring and consider a quotient Q=Sp/N where N is a submodule generated in degrees at most d. If the graded component Qd is free, then Nd is free as well, and Nd⊗S1 →Spd+1 →Qd+1→0 gives a free resolution of Qd+1. The function nextDegree takes the matrix corresponding to the map Nd→Spd and gives the matrix corresponding to the map Nd⊗S1→Spd+1.
i1 : S=ZZ[x_0,x_1]; |
i2 : R=S[a_1..a_4]; |
i3 : K=gens ker matrix{{1,a_2,a_3,a_4}} o3 = {0, 0} | a_2 a_3 a_4 | {1, 0} | -1 0 0 | {1, 0} | 0 -1 0 | {1, 0} | 0 0 -1 | 4 3 o3 : Matrix R <--- R |
i4 : nextDegree(K,1,S) o4 = {-1, 0} | a_2 0 a_3 0 a_4 0 | {-1, 0} | -1 a_2 0 a_3 0 a_4 | {0, 0} | 0 -1 0 0 0 0 | {0, 0} | 0 0 -1 0 0 0 | {0, 0} | 0 0 0 -1 -1 0 | {0, 0} | 0 0 0 0 0 -1 | 6 6 o4 : Matrix R <--- R |