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 of rank n, 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}. Let K be the matrix corresponding to the map N_d\otimes S_1\to S^p_{d+1}. The function co1Fitting calculates the (n-1)'th Fitting ideal of Q_{d+1} assuming that the basis of Q_d was given by a Gotzmann set.
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 : K2=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
|
i5 : co1Fitting(K2)
o5 = ideal(a a - a )
2 3 4
o5 : Ideal of R
|
The object co1Fitting is a method function.