The polynomial diagonal Koszul bimodule complex

Let $k$ be a field and let $R=k[x_1,\dots ,x_n]$ for $n\ge 0$. Since $R$ is commutative, identify its enveloping algebra $R^e=R{\otimes }_k{R}^{\mathrm{o}\mathrm{p}}$ (Enveloping algebra and the bimodule–module dictionary) with $R{\otimes }_{k}R$. Write $x_i^L=x_i\otimes 1$ and $x_i^R=1\otimes x_i$ for the two copies of each polynomial generator, and put

$u_i:=x_i^L-x_i^R\mathrm{.}$

The diagonal Koszul bimodule complex is the Koszul complex $K(u_1,\dots ,u_n;R^e)$ defined in Koszul Complex Of A Sequence With Coefficients, augmented by the multiplication map $\mu :R^e\to R$. Its degree-$p$ term is free over $R^e$ on symbols

${\theta }_{i_1}\wedge \cdots \wedge {\theta }_{i_p}(1\le i_1<\cdots <i_p\le n),$

and its differential is

$d({\theta }_{i_1}\wedge \cdots \wedge {\theta }_{i_p})={\sum }_{r=1}^p(-1{)}^{r-1}{u}_{i_r}{\theta }_{i_1}\wedge \cdots \wedge \widehat{{\theta }_{i_r}}\wedge \cdots \wedge {\theta }_{i_p}\mathrm{.}$

The augmentation is a chain map because $\mu (u_i)=0$ for every $i$; the Koszul differential squares to zero by the defining alternating deletion formula. There are $\left(\genfrac{}{}{0px}{}{n}{p}\right)$ displayed basis symbols in degree $p$, and no terms above degree $n$.

When ${\mathrm{deg}}_{\mathrm{i}\mathrm{n}\mathrm{t}}{x}_i=2$, assign each ${\theta }_i$ homological degree $1$ and internal degree $2$. Then the differential lowers homological degree by $1$ and preserves internal degree; degree $p$ is a direct sum of $\left(\genfrac{}{}{0px}{}{n}{p}\right)$ copies of $R^e\{2p\}$ under the shift convention $M\{r{\}}_d=M_{d-r}$ from Associative graded algebras, bimodules, and internal shifts. Internal grading adds no super sign.

For $n=0$, the sequence and exterior generators are empty, $R=k$ and $R^e=k$; the complex is $k$ in degree zero and $\mu :k\to k$ is the identity.