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The fundamental example of Gerstenhaber algebra is the space$T_{poly}({\mathbb R}^d)$ of polyvector fields on $\mathbb{R}^d$, equipped withthe wedge product and the Schouten bracket. In this paper, we explicitelydescribe what is the enveloping $G_\infty$ algebra of a Gerstenhaber algebra$\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrisoncohomology operator for $\mathcal{G}$. We finally show the nontriviality of aChevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in$T_{poly}({\mathbb R}^d)$.

Alg\`ebres et cog\`ebres de Gerstenhaber et cohomologie de Chevalley-Harrison