Finding the syzygy (relation module) of a monomial ideal.

Question

I have read pages 322-323 of "Commutative algebra, with a view toward algebraic geometry" by David Eisenbud, but it is still not much clear what are the steps of finding the syzygy.

I am practicing for myself finding the syzygy (relation module) of the following monomial ideal:

$$I_2 = (x_1^{34} x_2^{7}, x_1^{23}x_2^{19})$$ in $S = k[x_1, x_2].$

I understood that one of the steps is to find the $\sigma_{ij}$ but even this step is unclear to me.

Any help will be greatly appreciated!

Answer 1

Your ideal $I_2$ is generated by two monomials $u=x_1^{23}x_2^7 u_1, v=x_1^{23}x_2^7 v_1$ where $u_1=x_1^{11}, v_1=x_2^{12}$. Then, the resolution of $I_2$ is $$0\to S\to S^2\to I_2\to 0.$$ The right side map is just $u,v$. The left side map is $(v_1,-u_1)$.