Flat modules form a symmetric monoidal subcategory

Observation 8.5.16. Every free \(R\)-module is flat (Observation 8.5.14), so in particular the monoidal unit \(R\) is flat. Combined with Lemma 8.5.15, this shows that the full subcategory \(\Mod _R^{\flat } \subseteq \Mod _{R,\geq 0}\) spanned by the flat \(R\)-modules contains the unit and is closed under the tensor product. It therefore inherits a symmetric monoidal structure by the Part II result Lemma 14.5.2.