Let $R=k[x,y]$, $\mathfrak{a}=(x,y)$ be an ideal of $R$, how to compute $\text{Tor}_i^R(k,\mathfrak{a}^n)$ for all $i\ge 0$ and $n\ge 1$? Here $\mathfrak{a}^n$ denotes the ideal obtained by taking product of $\mathfrak{a}$ with itself, $n$ times.
Thanks for your help.
The Koszul complex gives a free resolution of $k$: $$0\to R\to R^2\to R\to k\to0$$ where the map $R^2\to R$ is given by $(f,g)\mapsto xf+yg$ and $R\to R^2$ is given by $h\mapsto (y,-x)$. So $\textrm{Tor}_*^R(k,M)$ is the homology of the complex $$0\to M\to M^2\to M\to0$$ where the maps are $m\mapsto(ym,xm)$ and $(m_1,m_2)\mapsto xm_1+ym_2$. Now you just have to set $M=\mathfrak a^n$.