Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

Let X   be a Stein manifold, and let Y⊂XY⊂X be a closed complex submanifold. Denote by O(X)O(X) the algebra of holomorphic functions on X  . We show that the weak (i.e., flat) homological dimension of O(Y)O(Y) as a Fréchet O(X)O(X)-module equals the codimension of Y in X. In the case where X and Y   are of Liouville type, the same formula is proved for the projective homological dimension of O(Y)O(Y) over O(X)O(X). On the other hand, we show that if X is of Liouville type and Y   is hyperconvex, then the projective homological dimension of O(Y)O(Y) over O(X)O(X) equals the dimension of X.