At the moment I’m writing my master thesis and need help to understand stuff from homological algebra. Concretely I search for a reference to the following problem:
Let $R = k[x_1, \dotsc, x_n]$ be a polynomial ring and $M$ a $R$-bimodule. Show that the Hochschild cohomology groups $H^p(R,M)$ are zero for $p > n$.
It would be nice if someone can give me some references which maybe are not too involved since I’m new in the field of homological algebra.
Recall that to compute $H^p(R,M)$, it suffices you resolve $R$ as an $R$-bimodule via say free bimodules. Now identify $R\otimes_k R^{\rm op}$ with $k[x_1,\ldots,x_n,x_1',\ldots,x_n']$ where $x_i' = x_i-y_i$, the $y_i$ being the original coordinates in $R^{\rm op}$. Then $R$ is the trivial $R[x_1',\ldots,x_n']$ module: $R$ acts by scalars, and the $x_i'$ act by zero.
This makes available the Koszul resolution of $R$ as a trivial $A = R[x_1',\ldots,x_n']$ module, which is precisely of length $n$: let $\mathcal K_i$ be the free $A$-module with basis the anti-symmetric tuples $e_{t_1}\wedge\cdots\wedge e_{t_i}$, so that it has rank $\binom ni$, and consider the complex
$$0 \to \mathcal K_n \longrightarrow \cdots \longrightarrow \mathcal K_1 \longrightarrow \mathcal K_0\to 0$$
where the map $\mathcal K_{i+1}(A)\to \mathcal K_i(A)$ is defined on basis elements by
$$d(e_{t_0}\wedge\cdots\wedge e_{t_i}) = \sum_{j=0}^i (-1)^j x_{t_j}' e_{t_0}\wedge\cdots\wedge \widehat{e_{t_j}} \wedge\cdots\wedge e_{t_i}.$$ and extended $A$-linearly. The map $\mathcal K_0 =A\to R$ that evaluates a polynomial at $(0,\ldots,0)$ gives an augmentation $\mathcal K \to R$ and makes $\mathcal K$ into a resolution of $R$. This last claim is definitely easy to find in the literature, along with a proof.
Having found a resolution of the bimodule $R$ of length $n$, it follows that for any other $R$-bimodule $M$, we have $H^p(R,M)$ for any $p>n$. This resolution shows that this bound is sharp, since it is easy to check that $\hom_A(\mathcal K,R)$ has trivial differential, and so $H^p(R,R)$ is nonzero for $p=0,\ldots,n$.