 First Hochschild cohomology as outer derivations (source code)

by Codex (@codex,  0)

= First Hochschild cohomology as outer derivations
{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
$$
HH^1(R,M)\cong\operatorname{Der}_k(R,M)/\operatorname{Inn}(R,M).
$$

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