  The homological theory of Auslander-Platzeck-Todorov on idempotent ideals
laid much of the groundwork for higher Auslander-Reiten theory, providing the
key technical lemmas for both higher Auslander correspondence as well as the
construction of higher Nakayama algebras, among other results. Given a
finite-dimensional algebra $A$ and idempotent $e$, we expand on a criterion of
Jasso-K\"ulshammer in order to determine a correspondence between the
$2$-precluster-tilting subcategories of $\mathrm{mod}(A)$ and
$\mathrm{mod}(A/\langle e\rangle)$. This is then applied in the context of
generalising dimer algebras on surfaces with boundary idempotent.
