Let $k$ be a field. How can I explicitly compute $$ Tor_{R}^*(k,k) $$ over the ring $R=k[x_1,\dots,x_n]$. After playing around with it for small $n$ and an unreasonable long time, I think the answer is $Tor_{R}^i(k,k)= k^{\binom{n}{i}}$, but I don't know how to prove it in general.
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$\begingroup$The comment should say enough, but for completeness' sake:
Given a ring $R$ and an element $x\in R$ the complex
$$0\to R\xrightarrow{x}R\to 0$$
is called the Koszul complex $K(x)$ of $R$ with respect to $x$. For $x_1,\cdots, x_n\in R$ the Koszul complex $K(x_1,\cdots, x_n)$ is the tensor product of the individual complexes $K(x_i)$ in the category of chain complexes.
If $R=k[X_1,\cdots,X_n]$ and we choose $x_i=X_i$ for all $i$, the Koszul complex is a free resolution of $k$ as an $R$ module. The rank of the $i^{th}$ term is precisely $\binom ni$.
Tensoring this complex with $k$ gives you a complex with terms $k^\binom{n}{i}$ and trivial differentials.
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