nLab
Noether identities

Definition

For $S:C\to {𝔸}^1$ an action functional, the $𝒪(C)$-module of its Noether identities is the kernel $N_S\hookrightarrow \mathrm{Der}(𝒪(C))$ of the canonical morphism

$${\iota }_{dS}:\mathrm{Der}(𝒪(S))\to 𝒪(S).$$\iota_{d S} : Der(\mathcal{O}(S)) \to \mathcal{O}(S) \,.

See BRST-BV complex for background and details.

Examples

Assume that $C$ is sufficiently well behaved and of finite dimension such that the module of derivations has (locally) a basis $\{{\partial }_i{\}}_i$. Then the Noether identities are (locally) tuples of functions $\{v^i\}$ such that

$$\sum _i{v}^i{\partial }_{i}S=0.$$\sum_i v^i \partial_i S = 0 \,.