If p is a homogeneous polynomial in C[x_1,...,x_n,y], then homog(dehomog(p)) = p / y^k, where k is taken largest possible. (Namely, it's taken to be the smallest power occurring among the terms of p.)
If J is a homogeneous ideal in C[x_1,..,x_n,y], then the map dehomog: J -> dehomog(J) is onto.
If I is an ideal in C[x_1,..,x_n], then the map homog: I -> homog(I) is not onto; its image is the homogeneous polynomials in homog(I) that are not multiples of y.
Feel free to use these without proving them (though they're easy).
