# nextDegree -- Lifts the kernel to the next degree

## Synopsis

• Usage:
nextDegree M,d,S
• Inputs:
• Outputs:
• ,

## 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

• gaussCol -- Makes column reductions of a matrix
• affinePart -- Replaces columns in a matrix with an identity matrix
• co1Fitting -- Calculates the (n-1)'th Fitting ideal of a finite module

## Ways to use nextDegree :

• "nextDegree(Matrix,ZZ,Ring)"
• "nextDegree(Module,ZZ,Ring)"

## For the programmer

The object nextDegree is .