We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring
Γ
e
≔
(
1
−
e
)
R
(
1
−
e
)
\Gamma _e ≔(1-e)R(1-e)
of a semiperfect noetherian basic ring
R
R
by a construction inside
m
o
d
R
\mathsf {mod}\,R
. This is then applied to investigate homological properties of idempotent subrings
Γ
e
\Gamma _e
under the assumption of
R
/
⟨
1
−
e
⟩
R/\langle 1-e\rangle
being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module
S
e
≔
e
R
/
rad
e
R
S_e ≔eR /\operatorname {rad}eR
with
Ext
R
1
(
S
e
,
S
e
)
=
0
\operatorname {Ext}_R^1(S_e,S_e) = 0
is self-orthogonal, that is
Ext
R
k
(
S
e
,
S
e
)
\operatorname {Ext}^k_R(S_e,S_e)
vanishes for all
k
≥
1
k \geq 1
, whenever
gldim
R
\operatorname {gldim}R
and
pdim
e
R
(
1
−
e
)
Γ
e
\operatorname {pdim}eR(1-e)_{\Gamma _e}
are finite. Indeed, a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose
e
∈
R
e \in R
is an idempotent such that all idempotent subrings
Γ
\Gamma
sandwiched between
Γ
e
\Gamma _e
and
R
R
, that is
Γ
e
⊆
Γ
⊆
R
\Gamma _e \subseteq \Gamma \subseteq R
, have finite global dimension. Then the simple summands of
S
e
S_e
can be numbered
S
1
,
…
,
S
n
S_1, \dots , S_n
such that
Ext
R
k
(
S
i
,
S
j
)
=
0
\operatorname {Ext}_R^k(S_i, S_j) = 0
for
1
≤
j
≤
i
≤
n
1 \leq j \leq i \leq n
and all
k
>
0
k > 0
.