I want to find the syzygies of the following monomial ideal $I = (x_1^4, x_1^3x_2, x_1^2x_2^2, x_1x_2^3, x_2^4)$ in $S = k[x_1, x_2]$.
To do this I will use Lemma 15.1 on pg. 322 in Eisenbud "Commutative algebra, with a view toward algebraic geometry" shown below:
I know that I should calculate the $\sigma_{ij}$'s
But my questions are:
1- How can I be sure to choose a pair of indices $i,j$ such that $m_i$ and $m_j$ involve the same basis element of $F$? what are $F,M$ and $S$ in my case here?
My guess is:
$F = k[x_1, x_2], S = k$ and $M = I$ is that correct?
2- I am guessing that the $m_i$'s in my case are just $x_1^4, x_1^3x_2, x_1^2x_2^2, x_1x_2^3, x_2^4$ am I correct? but what about the $\epsilon_i,$ what are those? how can I calculate them, if that can be done?
3- How many $\sigma_{ij}$'s should I get?
4- Could someone show me the calculation of one of the sigmas, please?
Could anyone help me answer those questions, please?
Take $S=k[x_{1},x_{2}],$ $F = S,$ $M=I.$ Then $M$ is a submodule of $F$ and is generated by $m_{1}=x_{1}^{4},$ $m_{2}=x_{1}^{3}x_{2},$ $\ldots,$ $m_{5} = x_{2}^{4},$ as you suggest; in particular, $t=5$ here. The $\epsilon_{j}$ are just the elements of a basis of a newly introduced free module on $t=5$ elements.
Here is the computation of $\sigma_{12}.$ We know that $m_{1}=x_{1}^{4}$ and $m_{2}=x_{1}^{3}x_{2},$ so $\mathrm{GCD}(m_{1},m_{2})=x_{1}^{3}.$ Therefore $m_{12}=m_{1}/x_{1}^{3}=x_{1}$ and $m_{21}=m_{2}/x_{1}^{3}=x_{2}.$ Now, $$\sigma_{12}=m_{21}\epsilon_{1}-m_{12}\epsilon_{2} = x_{2}\epsilon_{1}-x_{1}\epsilon_{2}.$$ That's it! This is an element of a free $k[x_{1},x_{2}]$-module generated by the $\epsilon_{j},$ so there is no simpler expression.
Strictly speaking there are $t^{2}=25$ different $\sigma_{ij},$ but on closer inspection you will notice that $\sigma_{ji}=-\sigma_{ij},$ so you only need $t(t-1)/2$ of them (and without thinking about it too hard, I guess there could be more redundancy even after that).