Algebra/Topology seminar

Speaker: Sonja Farr

Title: Hochschild Cohomology and Higher Centers

Abstract: In Higher Algebra, J. Lurie developed a theory of derived centers for algebras over ∞-operads. Similar to how the classical center of an associative 𝕜-algebra is a commutative 𝕜-algebra, the derived center of an 𝒪-algebra is automatically an 𝔼_1-algebra object in the category of 𝒪-algebras. In the case of 𝔼_1 -algebras, the Dunn Additivity Theorem promotes this to a structure of an 𝔼_1-algebra. By defining the Hochschild complex of an 𝔼_2-algebra object as its derived center, we hence obtain a built-in solution of Deligne’s conjecture on Hochschild cochains. We show that for an associative 𝕜-algebra, this definition recovers the classical Hochschild complex, including the correct Gerstenhaber algebra structure in cohomology. Globalizing to schemes, we show that the derived 𝔼_1 -center of the structure sheaf is indeed glued from the local centers, and that for a smooth scheme we recover the sheaf of polydifferential operators. The motivation for this work has its origin in Kontsevich’s description of the action of the Grothendieck-Teichmüller group on the Hochschild cohomology of a smooth algebraic variety.