Corollary 4.5.8 (0092) · ScalingStacks

Let \(k\) be an ordinary commutative ring and \(Z\) a discrete commutative monoid.

definition 4.5.7 defines a large symmetric monoidal \(\mathrm{add}_{Hk}^{BZ}\)-enriched \(\infty\)-category \[\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{add}_{Hk}^{BZ}])\] equipped with a symmetric monoidal surjective-on-objects functor \[\mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}}) \rightarrow\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z}).\] The additive \(k\)-linear hom-category between algebras \(A, B \in \mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}})\) is given by the ordinary category \({}_A\mathrm{grbmod}_B^{\mathrm{gr-cp}}\) with \(Z\)-action by grading shift.

Composition is given by the ordinary relative tensor product, and the monoidal structure by the ordinary tensor product over \(k\).
definition 4.5.7 defines a large symmetric monoidal \(\mathrm{st}_{Hk}^{BZ}\)-enriched \(\infty\)-category \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathrm{st}_{Hk}^{BZ}])\] equipped with a symmetric monoidal surjective-on-objects functor \[\mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}}) \rightarrow\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z}).\] The stable \(k\)-linear hom-category between algebras \(A, B \in \mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}})\) is \(\mathcal D(_A\mathrm{grbmod}_B)^{\mathrm{gr-perf}}\) with \(Z\)-action by grading shift.

Composition is given by the derived relative tensor product, and the monoidal structure by the derived tensor product over \(k\).
The functor from corollary 4.4.5.([008R]) restricts to a symmetric monoidal \(\mathrm{add}_{Hk}^{BZ}\)-enriched functor \[\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\rightarrow\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z}).\] On objects this functor acts via the identity on \(\mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}})^{\simeq}\); on hom-categories between \(A, B \in \mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}})\) it is given by the additive \(k\)-linear \(Z\)-equivariant fully faithful inclusion \[_A\mathrm{grbmod}_B^{\mathrm{gr-cp}} \hookrightarrow \mathcal D(_A\mathrm{grbmod}_B)^{\mathrm{gr-perf}}.\]

Proof.

Since the derived tensor product of discrete flat \(Z\)-graded algebras is again a discrete and flat algebra, the functor \(\mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}}) \rightarrow\mathrm{Alg}(\mathrm{Mod}_{Hk}^{\geq 0, \mathcal Z})\) is symmetric monoidal, and hence the full subcategories \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) and \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\) are closed under the tensor product in \(\mathrm{Morita}^{\mathrm{cp}}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) and \(\mathrm{Morita}^{\mathrm{c}}(\mathrm{Mod}_{Hk}^{ Z})\), respectively. Denoting the derived and underived Day convolution tensor product by \(\otimes^{L, \mathrm{gr}}\) and \(\otimes^{\mathrm{gr}}\), respectively, and using proposition 3.6.9 and observation 4.5.5 we obtain the following equivalences for discrete flat \(Z\)-graded \(k\)-algebras \(A\) and \(B\) \[\begin{aligned} {}_A\mathrm{BMod}_B(\mathrm{Mod}_{Hk}^{Z}) & \simeq \mathrm{RMod}_{A^{\mathrm{op}}\otimes^{L, \mathrm{gr}} B}(\mathrm{Mod}_{Hk}^Z) \simeq \mathrm{RMod}_{A^{\mathrm{op}}\otimes^{\mathrm{gr}} B}(\mathrm{Mod}_{Hk}^Z) \\& \simeq \mathcal D(\mathrm{grmod}_{A^{\mathrm{op}}\otimes^{\mathrm{gr}} B}) \simeq \mathcal D({}_A\mathrm{grbmod}_B). \end{aligned}\] Recalling notation 4.4.2 for the full subcategories \({}_{A}\mathrm{BMod}^{\mathrm{cp}}_B(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) and \({}_{A}\mathrm{BMod}^{\mathrm{c}}_B(\mathrm{Mod}_{Hk}^{Z})\) on those bimodules which are compact-projective, resp. compact as right \(B\)-modules, the above equivalence restricts to an equivalence between subcategories (see notation 4.5.6) \[{}_A\mathrm{BMod}^{\mathrm{cp}}_{B}(\mathrm{Mod}_{Hk}^{\geq 0, Z}) \simeq {}_A\mathrm{grbmod}_B^{\mathrm{gr-cp}} \quad \mathrm{and} \quad {}_A\mathrm{BMod}^{\mathrm{c}}_{B}(\mathrm{Mod}_{Hk}^{Z}) \simeq \mathcal D(_A\mathrm{grbmod}_B)^{\mathrm{gr-perf}}.\] Using these observations, corollary 4.5.8 follow directly from corollary 4.4.5. ◻