Let \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\subseteq \mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{\mathbb{Z}})\) 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 \[\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\in \mathrm{CAlg}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]).\]
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2