Let \(\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\subseteq \mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\) denote the small full subcategory on the graded polynomial algebras \(k[x_1, \ldots, x_n]\) for \(n \geq 0\), with all \(x_i\) in degree \(2\). (Graded polynomial algebras are flat, see example 4.5.4, this hence indeed defines a full subcategory.) Since tensor products of polynomial algebras are polynomial algebras, this is in fact a symmetric monoidal subcategory and hence defines an object \[\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\in \mathrm{CAlg}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]).\]
6.5 The fiber functor on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\)[00D9]
We now construct the monoidal shift-preserving \(k\)-linear exact functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).
Recall from corollary 4.5.8.([0094]) the (large) derived Morita category \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{st}_k^{B\mathbb{Z}}]).\] Its objects can be understood as ordinary \(\mathbb{Z}\)-graded flat \(k\)-algebras, and its \(k\)-linear stable, idempotent-complete hom-category between two objects \(A\) and \(B\) is given by the full subcategory of the derived \(\infty\)-category \(\mathcal D( {}_{A}\mathrm{grbmod}_{B})\) of the abelian category \({}_{A}\mathrm{grbmod}_{B}\) of graded \(A\)–\(B\) bimodules on those objects which are graded-perfect as right \(B\)-modules, i.e. are quasi-isomorphic to bounded chain complexes of graded-compact-projective \(B\)-modules.
As before, it suffices to consider the small full subcategory of polynomial algebras:
In corollary 4.5.8.([0095]) we constructed a symmetric monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \[\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{\mathbb{Z}})\rightarrow\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\] which sends flat graded algebras to themselves and includes graded bimodules as the discrete objects into the derived \(\infty\)-category of graded bimodules, and hence restricts to a symmetric monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \[\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}}).\]
By definition 6.2.6, there is also a faithful monoidal shift-preserving \(k\)-linear functor \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\).
The monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \[\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\] factors through a monoidal \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched functor \[ H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}}).\]
Proof.
This follows immediately from the adjunction 
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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 \(H_{\mathrm{loc}}\).
The unpacking of the functor \(H_{\mathrm{loc}}\) from proposition 6.5.2 in observation 6.5.3 immediately implies the following:
The induced functor on homotopy categories \[h_1H_{\mathrm{loc}}\colon h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\] agrees with the functor \(h_1{H_{\mathrm{loc}}}\) from ([001H]).
Recall from observation 4.5.10 that \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\) has a symmetric monoidal fully faithful \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched functor \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}.\] We will abuse notation and also denote by \(H_{\mathrm{loc}}\) the monoidal \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched composite \[ H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\hookrightarrow \mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}.\] The composite sends a graded \(k\)-algebra \(A\) to the stable \(\infty\)-category of bounded chain complexes of graded-compact-projective right \(A\)-modules, with \(\mathbb{Z}\)-action given by the internal (i.e. non-homological) grading shift.
The composite \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) and its further composite \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}\) are faithful.
Proof.
We show that the former functor is faithful, the latter functor is a composite with a fully faithful functor and hence also faithful. By construction, the former functor factors as \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), the first of which is faithful by corollary 6.2.7, and the second is faithful since the induced map on hom-categories between two polynomial algebras \(A\) and \(B\) is given by the full inclusion. \[{}_{A}\mathrm{grbmod}^{\mathrm{gr-cp}}_{B} \hookrightarrow \mathcal D({}_{A} \mathrm{grbmod}_{B})^{\mathrm{gr-perf}}\] by corollary 4.5.8.([0095]). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2