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Let $R = \mathbb{C}[x,y]$ and $V = \mathbb{C}x\oplus\mathbb{C}y$. Then, the Koszul resolution of $R$ (as an $R$-bimodule) is given by \begin{align*} 0\to R\otimes_{\mathbb{C}}\wedge^2V\otimes_{\mathbb …

Just like Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras, it'll be nice (or disappointing?) if Harrison cohomology can be expressed in terms of Ext' …

The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \wedge …

Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely …

Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of Gerst …

Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{Ho …