A discrete \(Z\)-graded \(k\)-module \(M\) is flat if \(\oplus_{z \in Z} M_z\) is flat19 as a \(k\)-module. We let \(\mathrm{mod}_k^{Z, \mathrm{flat}}\) denote the full subcategory of the ordinary category of discrete \(Z\)-graded \(k\)-modules \(\mathrm{mod}_{k}^Z\) on the flat modules. A (not necessarily commutative) discrete \(Z\)-graded \(k\)-algebra \(A\) is flat if it is flat as a \(Z\)-graded \(k\)-module.
4.5 Morita categories of discrete flat algebras[008U]
For the rest of this section, we focus on the case where \(\mathbb{K}= Hk\) for an ordinary commutative ring \(k\) and where \(\mathcal Z\) is a discrete commutative monoid \(Z\). Our goal of this subsection is to restrict our Morita categories from corollary 4.4.5 to certain symmetric monoidal full subcategories which only contain discrete (i.e. ordinary) \(k\)-algebras and whose hom-categories are given by ordinary categories of discrete graded bimodules, or their derived variants.
In particular, the ordinary category of discrete \(Z\)-graded \(k\)-algebras is \(\mathrm{Alg}(\mathrm{mod}_k^{Z, \mathrm{flat}})\).
The following two observations and example are crucial when connecting back to section 2.
An ordinary \(Z\)-graded \(k\)-module \(M\) is flat if and only if it is degreewise flat, i.e. each graded component \(M_z\) is a flat \(k\)-module for all \(z\in Z\). This follows from the fact that flatness is preserved under infinite coproducts and retracts.
Free modules are flat. In particular, if \(k\) is a field, all \(Z\)-graded \(k\)-vector spaces are flat, and if \(k\) is an ordinary commutative ring and \(n\geq 0\), the polynomial algebra \(k[x_1,\ldots, x_n]\) is flat, and hence it is also flat if considered as a \(\mathbb{Z}\)-graded \(k\)-algebra with generators \(x_i\) in some degree \(n_i \in \mathbb{Z}\).
As flatness is closed under tensor products, \(\mathrm{mod}_k^{Z, \mathrm{flat}}\) is a symmetric monoidal full subcategory of \(\mathrm{mod}_{k}^Z\). On the other hand, \(\mathrm{mod}_k^{Z, \mathrm{flat}}\) is also a symmetric monoidal full subcategory of \(\mathrm{Mod}_{Hk}^{ \geq 0, Z} \hookrightarrow \mathrm{Mod}_{Hk}^Z\): under the equivalence \(\mathrm{Mod}_{Hk}^Z \simeq \mathcal D(\mathrm{mod}_{k}^Z)\) by example 3.5.15. The tensor product in \(\mathcal D(\mathrm{mod}_{k}^Z)\) is given by Day convolution of derived tensor products, which reduces to the Day convolution of ordinary tensor products on flat modules. In particular, tensor products of discrete flat \(Z\)-graded \(k\)-algebras are also discrete and flat.
For flat \(Z\)-graded \(k\)-algebras \(A\) and \(B\), we let \({}_A\mathrm{grbmod}_B\coloneqq{}_A\mathrm{BMod}_B(\mathrm{mod}_{k}^Z)\) denote the abelian \(1\)-category of ordinary graded \(A\)–\(B\) bimodules. Let \({}_A\mathrm{grbmod}_B^{\mathrm{gr-cp}}\) denote its full subcategory on those bimodules that are graded-compact-projective, see definition 3.6.8, as right \(B\)-modules.
Let \(\mathcal D({}_A\mathrm{grbmod}_B)^{\mathrm{gr-perf}}\) denote the full subcategory of the derived \(\infty\)-category \(\mathcal D({}_A\mathrm{grbmod}_B)\) on those objects that are graded-perfect, see remark 3.6.10 and preceeding definition, as derived right \(B\)-modules.
Most of the constructions in section 6 will build on the following \(\infty\)-categories.
Let \(k\) be an ordinary commutative ring and \(Z\) a discrete commutative monoid. We define \[\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\subseteq \mathrm{Morita}^{\mathrm{cp}}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\] to be the full \(\mathrm{add}_{Hk}^{BZ}\)-enriched subcategory on the discrete flat \(Z\)-graded \(k\)-algebras.
Similarly, we define \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\subseteq \mathrm{Morita}^{\mathrm{c}}(\mathrm{Mod}_{Hk}^{Z})\] to be the full \(\mathrm{st}_{Hk}^{BZ}\)-enriched subcategory on the discrete flat \(Z\)-graded \(k\)-algebras.
The following justifies the terminology ’Morita categories’, see also example 6.0.2.
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. ◻
The hom-categories \({}_A\mathrm{grbmod}_B^{\mathrm{gr-cp}}\) of \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) are ordinary \(1\)-categories, hence \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) is a (2,2)-category. On the other hand, \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\) is a genuine \((\infty,2)\)-category with non-trivial higher morphisms.
By definition, \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\) is a full symmetric monoidal subcategory of \(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}_{\mathrm{Mod}_{Hk}^{Z}}\). Thus, it comes equipped with a symmetric monoidal fully faithful \(\mathrm{st}_{Hk}^{BZ}\)-enriched functor \[\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\hookrightarrow \mathrm{Pr}^{\mathrm{L},\mathrm{c}}_{\mathrm{Mod}_{Hk}^{Z}} ~\stackrel{(-)^{\mathrm{c}}}{\simeq} ~\mathrm{st}_{Hk}^{BZ}.\]
Explicitly, the functor in observation 4.5.10 sends a flat \(Z\)-graded \(k\)-algebra \(A\) to the stable \(\infty\)-category \[\mathrm{RMod}_{HA}\left(\mathrm{Mod}_{Hk}^{BZ}\right)^{\mathrm{c}}~~ \stackrel{\mathrm{Prop.}~\href{/tag/006U}{3.6.9}}{\simeq} ~~\mathcal D(\mathrm{grmod}_A)^{\mathrm{gr-perf}}\] of graded-perfect right \(A\)-modules, with \(Z\)-action given by grading shift.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2