Given a cotorsion pair
(
A
,
B
)
(\mathcal {A},\mathcal {B})
in an abelian category
C
\mathcal {C}
with enough
A
\mathcal {A}
objects and enough
B
\mathcal {B}
objects, we define two cotorsion pairs in the category
C
h
(
C
)
\mathbf {Ch(\mathcal {C})}
of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when
(
A
,
B
)
(\mathcal {A},\mathcal {B})
is hereditary. We then show that both of these induced cotorsion pairs are complete when
(
A
,
B
)
(\mathcal {A},\mathcal {B})
is the “flat” cotorsion pair of
R
R
-modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new “flat” model category structure on
C
h
(
R
)
\mathbf {Ch}(R)
. In the last section we use the theory of model categories to show that we can define
Ext
R
n
(
M
,
N
)
\operatorname {Ext}^n_R(M,N)
using a flat resolution of
M
M
and a cotorsion coresolution of
N
N
.