# Questions tagged [hochschild-cohomology]

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### Cyclic homology with coefficients in a bimodule

I've recently been trying to understand Hochschild and cyclic co/homology better, and I've noticed that while it's common to define the Hochschild homology $\mathrm{HH}_{\bullet}(A;M)$ of an $R$-...
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### Minimality of the Koszul resolution

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{...
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### Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?

Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
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### What is the topological Hochschild cohomology of $\mathbb{F}_p$?

Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}_p$ as outlined in Krause-Nikolaus. We use the fact that $\mathbb{F}_p$ is initial $E_2$ ring with $0=p$ to compute ...
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### divided powers of a deformation class

Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has ...
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