Examples motivating t-flatness

Remark 6.5.3 (Motivation). The three main examples we have in mind for this abstract setup are:

(1)

The tensor product \(\Mod _R \times \Mod _R \xrightarrow {\otimes _R} \Mod _R\) for a commutative ring spectrum \(R\).

(2)

More generally the relative tensor product \(\RMod _R \times \LMod _R \xrightarrow {\otimes _R} \Sp \) for an associative ring spectrum \(R\).

(3)

The derived tensor product \(\D ^-(\Aa ) \times \D ^-(\Aa ) \xrightarrow {\otimes ^{\bL }} \D ^-(\Aa )\) for a symmetric monoidal abelian category \(\Aa \) with enough projectives (constructed in Section 6.6).

We work in this generality for two reasons. First, it makes clear that the notion of flatness does not depend on the intricate coherences involved in the definition of \(\Mod _R\) and its tensor product, discussed in Part II of this book. Second, it provides a single formulation that covers all relevant examples.