I am asked to find $\operatorname{Tor}_{*}^{k[x]}(M,M)$, with $M=k[x,x^{-1}]/xk[x]$.
I start with finding a projective resolution for $M$. An arbitrary element of $M$ is $\sum_{n \leq 0}a_nX^{-n}$, so I was looking at a surjection $\oplus_{i=1}^{\infty} k[x] \to M$, where we map $$ (1,0,0,\ldots) \mapsto 1 $$ $$(0,1,0,0,\ldots) \mapsto x^{-1}$$ $$(0,0,1,0,0,\ldots) \mapsto x^{-2}$$ and more generally $$e_i \mapsto x^{-i+1}$$ and extend this linearly.
The kernel of this map is generated by elements of the form $x(1,0,0,\ldots) ,x^2(0,1,\ldots),\ldots , x^{i}e_i$.
We continue the projective resolution by finding a map surjecting on this kernel, ie $\oplus_{i=1}^{\infty} k[x] \to \oplus_{i=1}^{\infty}k[x]$, where we define $(1,0,0,\ldots) \mapsto x(1,0,0,\ldots)$ $$(0,1,0,\ldots) \mapsto x^2(0,1,0,\ldots)$$ and more generally $e_i \mapsto x^{i}e_i$ and then extend this linearly. We thus have $P_{*} : 0 \to \oplus_{i=1}^{\infty} k[x] \to \oplus_{i=1}^{\infty} k[x] \to 0$.
We can then tensor this with $M$, so $P_{*}$ becomes $ 0 \to \oplus_{i=1}^{\infty} M \to \oplus_{i=1}^{\infty} M \to 0$ with the induced map being the zero map. So I guess this gives $\operatorname{Tor}_{*}^{k[x]}(M,M)$.
Something doesn't seem right to me, but I am not sure I can find where this argument is flawed. Any ideas?
EDIT: I guess I have found the first error in this argument. It's the kernel of the first map $\oplus_{i=1}^{\infty} k[x] \to M$ , i.e. $e_1 -xe_2$ is in the kernel, but that is not generated by $\{xe_1 , x^2e_2,....\}$.