Advances in Mathematics ( IF 1.494 ) Pub Date : 2020-02-03 , DOI: 10.1016/j.aim.2020.107005
Harrison Chen

We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical z-fixed point stack ${\pi }_{0}\left({X}^{z}\right)/{G}^{z}$ become equivalent after completion along a semisimple parameter $\left[z\right]\in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\mathrm{Perf}\left(X/G\right)$. We then prove an analogue of the Atiyah-Segal completion theorem in the setting of periodic cyclic homology, where the completion of the periodic cyclic homology of $\mathrm{Perf}\left(X/G\right)$ at the identity $\left[e\right]\in G//G$ is identified with a 2-periodic version of the derived de Rham cohomology of $X/G$. Together, these results identify the completed periodic cyclic homology of a stack $X/G$ over a parameter $\left[z\right]\in G//G$ with the 2-periodic derived de Rham cohomology of its z-fixed points.

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