It is known how you compute the first syzygy module of a monomial ideal but it seems an hard work to do the same for binomial ones. I don't know any procedure to aim that, so I would like kindly if there exists something to find the first syzygy of a binomial ideal in terms of its minimal generators. For instance, I have this example in the context of determinantal rings. Let $R$ be the polynomial ring in the twelve variables$$x_{11},x_{22},x_{12},x_{21},x_{23},x_{13},x_{24},x_{14},x_{32},x_{31},x_{42},x_{41}$$ over a field $K$. Consider the binomial ideal $I$ generated by the following 2-minors: $$ f_1=x_{11}x_{22}-x_{12}x_{21},\\ f_2=x_{11}x_{23}-x_{13}x_{21},\\ f_3=x_{11}x_{24}-x_{14}x_{21},\\ f_4=x_{11}x_{32}-x_{12}x_{31},\\ f_5=x_{11}x_{42}-x_{12}x_{41},\\ f_6=x_{21}x_{32}-x_{22}x_{31},\\ f_7=x_{21}x_{42}-x_{22}x_{41},\\ f_8=x_{31}x_{42}-x_{32}x_{41},\\ f_9=x_{12}x_{23}-x_{13}x_{22},\\ f_{10}=x_{12}x_{24}-x_{14}x_{22},\\ f_{11}=x_{13}x_{24}-x_{14}x_{23} $$
The first syzygy module of $I$ is the kernel of the $R$-module homomorphism $R(-2)^{11} \to I$ with $e_i\to f_i$, where $\{e_i:i\in[11]\}$ is the canonical basis of $R(-2)^{11}$.
How can you compute manually with pen and paper the minimal set of generators of the first syzygy module of $I$?
Using an algebra program as Macaulay2, I find that the generators are given by the combinations of $e_i$ with the following polynomials read by columns
| x_(3,1) x_(2,1) 0 x_(1,1) 0 0 0
{2} | 0 0 0 0 0 0 0
{2} | 0 0 0 0 0 x_(2,4) x_(1,4)
{2} | -x_(1,4) 0 x_(1,2) 0 x_(1,1) 0 0
{2} | x_(2,4) 0 -x_(2,2) 0 -x_(2,1) 0 0
{2} | -x_(3,2) -x_(2,2) 0 -x_(1,2) 0 -x_(2,3) -x_(1,3)
{2} | 0 x_(2,4) x_(3,2) x_(1,4) x_(3,1) 0 0
{2} | 0 0 0 0 0 x_(2,1) x_(1,1)
{2} | 0 0 0 0 0 0 0
{2} | 0 0 0 0 0 0 0
{2} | 0 0 0 0 0 0 0
---------------------------------------------------------------------------
-x_(2,3) 0 -x_(1,3) 0 0 0 0 0
0 0 0 0 0 -x_(3,2) -x_(3,1) 0
0 -x_(2,2) 0 x_(3,2) -x_(1,2) 0 0 0
0 0 0 x_(1,3) 0 0 0 -x_(4,2)
0 0 0 -x_(2,3) 0 x_(4,2) x_(4,1) 0
0 0 0 0 0 0 0 0
0 x_(2,3) 0 0 x_(1,3) 0 0 0
x_(2,2) 0 x_(1,2) 0 0 0 0 0
x_(2,4) x_(2,1) x_(1,4) -x_(3,1) x_(1,1) 0 0 0
0 0 0 0 0 x_(1,2) x_(1,1) -x_(2,2)
0 0 0 0 0 0 0 x_(3,2)
---------------------------------------------------------------------------
-x_(4,1) 0 0 0 0 0
-x_(2,4) 0 -x_(2,3) -x_(2,2) -x_(2,1) 0
0 0 x_(4,2) 0 0 0
0 -x_(4,1) 0 0 0 0
0 0 0 0 0 0
x_(4,2) 0 0 0 0 0
0 0 0 x_(4,2) x_(4,1) 0
0 0 0 0 0 -x_(3,1)x_(4,2)+x_(3,2)x_(4,1)
0 0 -x_(4,1) 0 0 0
0 -x_(2,1) 0 0 0 x_(1,3)x_(2,4)-x_(1,4)x_(2,3)
x_(1,4) x_(3,1) x_(1,3) x_(1,2) x_(1,1) 0
o6 : Matrix $R^{11}$ <--- $R^{21}$
