I have a problem
Let $R=k[[x,y]]$. Describe explicitly a minimal free resolution of $R$-module $R/(x^2,xy,y^3)$.
I don't have a clue what I should do. Normally I studied the abstract theory and just admit the existence of minimal free resolution. So I am clueless when it comes to a specific situation.
Thank you for your help
Let $R\rightarrow R/(x^2,xy,y^3)$ be the canonical projection. It's kernel is the ideal $(x^2,xy,y^3)$. Consider the map $$R^3\rightarrow (x^2,xy,y^3),\;\; e_1\mapsto x^2, e_2\mapsto xy, e_3\mapsto y^3,$$ where $\{e_1,e_2,e_3\}$ is the canonical basis of $R^3$. The kernel of this map is generated by $ye_1-xe_2$ and $y^2e_2-xe_3$. The map $$R^2 \rightarrow (ye_1-xe_2,y^2e_2-xe_3)R,\;\; f_1\mapsto ye_1-xe_2, f_2\mapsto y^2e_2-xe_3$$ is an isomorphism, where $\{f_1,f_2\}$ is the canonical basis of $R^2$.
So, the minimal free resolution of $R/(x^2,xy,y^3)$ is $$ 0 \rightarrow R^2 \rightarrow R^3 \rightarrow R \rightarrow R/(x^2,xy,y^3) \rightarrow 0.$$ (The matrix of the map $R^2\rightarrow R^3$ has entries in $(x,y,z)R$. Same for the map $R^3\rightarrow R$.)