A non flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0$ for all $n\ge 1$

Question

I want to find a non-flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0 \,\, \forall n\ge 1$, where $R=k[x,y]/(xy)$ and $k$ is field.

Answer 1

In my answer to your previous question I have constructed a free resolution of $k$ as an $R$-module $$\cdots\longrightarrow R^2\stackrel{v}\longrightarrow R^2\stackrel{u}\longrightarrow R^2\stackrel{v}\longrightarrow R^2\stackrel{u}\longrightarrow R^2\stackrel{f}\longrightarrow R\longrightarrow 0$$ where $f(r_1,r_2)=r_1x+r_2y$, $u(a_1,a_2)=(a_1y,a_2x)$, and $v(b_1,b_2)=(b_1x,b_2y)$.

Now consider $M=R/(x,y-1)$. Tensoring the above resolution by $M$ we get $$\cdots\longrightarrow M^2\stackrel{\bar v}\longrightarrow M^2\stackrel{\bar u}\longrightarrow M^2\stackrel{\bar v}\longrightarrow M^2\stackrel{\bar u}\longrightarrow M^2\stackrel{\bar f}\longrightarrow M\longrightarrow 0$$ where $\bar f(r_1,r_2)=r_2$, $\bar u(a_1,a_2)=(a_1,0)$, and $\bar v(b_1,b_2)=(0,b_2)$.