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nextDegree -- Lifts the kernel to the next degree

Synopsis

Description

Let S be a polynomial ring and consider a quotient Q=S^p/N where N is a submodule generated in degrees at most d. If the graded component Q_d is free, then N_d is free as well, and N_d\otimes S_1 \to S^p_{d+1} \to Q_{d+1}\to 0 gives a free resolution of Q_{d+1}. The function nextDegree takes the matrix corresponding to the map N_d\to S^p_d and gives the matrix corresponding to the map N_d\otimes S_1\to S^p_{d+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

See also

Ways to use nextDegree:

For the programmer

The object nextDegree is a method function.