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Hochschild cohomology

Derivation into a bimodule

Inner derivation

{c}
{title2=$HH^1(R,M)=\operatorname{Der}_k(R,M)/\operatorname{Inn}(R,M)$}

The degree-one Hochschild cocycles are exactly the maps that are a <derivation into a bimodule>, while degree-one coboundaries are exactly the <inner derivations>. Therefore
$$
HH^1(R,M)\cong\operatorname{Der}_k(R,M)/\operatorname{Inn}(R,M).
$$

First Hochschild cohomology as outer derivations

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 148 / 5 / Solution

First Hochschild cohomology as outer derivations ( $H H^{1} (R, M) = Der_{k} (R, M) / Inn (R, M)$ )

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First Hochschild cohomology as outer derivations (HH1(R,M)=Derk​(R,M)/Inn(R,M))

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First Hochschild cohomology as outer derivations ( $H H^{1} (R, M) = Der_{k} (R, M) / Inn (R, M)$ )

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