We construct what we call a "good" projective resolution of Weyl modules for the general linear group over an arbitrary commutative ring.
From this, we obtain a resolution of the co-Specht modules for the symmetric group by permutation modules.
These resolutions allow us to generalize  in the corresponding Grothendieck groups the classical determinantal identity known for the representation theory of the symmetric group over fields of characteristic zero.
This is joint work with Ivan Yudin.