Is there any research relating varieties with filters instead of radical ideals? For example, Suppose we have a variety V in C^n, now consider the fixed filter consisting of all algebraic sets (zeroes of polynomials) which contain this variety. Now instead of considering the spectrum of a ring, perhaps we can look at the set of all these fixed filters, and give it a natural topology (I believe initial topology should work).
It is well known that for a compact space X, the set of all ultrafilters on X corresponds to the set of points in X. However, for a locally compact space Y (Like R), this isn't true. Instead, we need to look at the Stone-Cech Compactification of this space to give us all of our Ultrafilters.
Is there any useful analogy with this? For example, can an affine scheme be defined by the corresponding filters (with initial topology) of the prime ideals of some commutative ring? And if so, is this of any use?