Inner derivation

For

m \in M

d_{m} (r) = r m - m r

defines an inner derivation

R \to M

.

Table of contents
- First Hochschild cohomology as outer derivations Inner derivation

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

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

H H^{1} (R, M) ≅ Der_{k} (R, M) / Inn (R, M) .

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