Characterizations of projective modules

Observation 8.5.3. The characterizations of t-projective objects from Proposition 6.4.11 specialize to the following equivalent conditions for a connective left \(R\)-module \(P\):

(1)

\(P\) is projective;

(2)

For every connective left \(R\)-module \(Q\) and every \(i > 0\), we have \(\Ext ^i_R(P,Q) = 0\);

(3)

For every connective left \(R\)-module \(Q\), we have \(\Ext ^1_R(P,Q) = 0\);

(4)

Given an exact sequence \(N' \to N \to N''\) of connective left \(R\)-modules, the induced map \(\Ext ^0_R(P,N) \to \Ext ^0_R(P,N'')\) is surjective.

Since the t-structure on \(\LMod _R\) is left complete (Corollary 8.2.5), these are further equivalent to:

(5)

For every \(Q \in \LMod _R^{\heartsuit }\) and every \(i > 0\), we have \(\Ext ^i_R(P,Q) = 0\).