Closure properties of flat modules

Observation 8.5.14. Let \(R\) be a connective associative ring spectrum. Condition (3) of Theorem 8.5.13 immediately gives the following facts:

(a)

The subcategory of \(\LMod _R\) spanned by the flat \(R\)-modules is closed under coproducts, retracts, and filtered colimits.

(b)

Every free module is flat. Combining this with (a) and Proposition 8.5.6, it follows that every projective module is flat.

(c)

If \(A\) is an ordinary associative ring, then a connective left \(HA\)-module \(N\) is flat if and only if \(N\) is discrete and \(\pi _0 N\) is a flat \(A\)-module in the classical sense, i.e., satisfies condition (6) of Theorem 8.5.13.