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.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2