Unpacked, the functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow \mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) sends an object \(n\) to the polynomial algebra \(R_n:=k[x_1, \ldots, x_n]\), and a bounded chain complex of Soergel bimodules to the induced object in \(\mathcal D({}_{R_n} \mathrm{grbmod}_{R_n})^{\mathrm{gr-perf}}\), i.e. the chain complex considered up to quasi-isomorphism. Thinking of this as a homotopy coherent version of ‘taking homology’ motivates the notation \(H_{\mathrm{loc}}\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2