$\DeclareMathOperator{\Ext}{Ext}$ Let $\mathcal{A}$ be an abelian category and $(\mathcal{D},\mathcal{E})$ be a cotorsion pair, i.e. classes of objects of $\mathcal{A}$, such that
- $D\in \mathcal{D}$ if and only if $\Ext^1(D,E)=0$ for all $E\in \mathcal{E}$ and
- $E\in \mathcal{E}$ if and only if $\Ext^1(D,E)=0$ for all $D\in \mathcal{D}$.
We call the pair hereditary if $\Ext^i(D,E)=0$ for all $i>0$ and $D\in \mathcal{D}$, $E\in \mathcal{E}$.
All cotorsion pairs I have yet encountered in nature are hereditary. Is there a simple example of a non-hereditary cotorsion pair I have been missing?
Let R be a domain. Consider the matlis cotorsion pair M= (SF,MS)，where $SF =\{M:Ext^1(Q,M)=0\}$. Q denotes the quotient field of R. By the book ‘’Approximation and endomorphism of algebra 06 ‘’Lemma 4.4.13, M is hereditary iff the base ring is Matlis domain (pd $Q$≤1).