The Hochschild complex of an algebra or scheme is exhibited as the E1-operadic center of the structure sheaf, and the induced E2-algebra structure recovers the classical Gerstenhaber bracket and cup product.
On the derived category of an algebra over an operad
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$\mathbb{E}_2$-algebra structures on the derived center of an algebraic scheme
The Hochschild complex of an algebra or scheme is exhibited as the E1-operadic center of the structure sheaf, and the induced E2-algebra structure recovers the classical Gerstenhaber bracket and cup product.