TITLE: My first question - on Affine Schemes in Algebraic Geometry
QUESTION [11 upvotes]: If R is a commutative ring (with unit) then we have an affine scheme Spec(R) which is an object of the category of ringed topological spaces. Is there any way of characterising this object relative to the category of ringed topological spaces? The underlying space of an affine scheme is compact and the structure sheaf is a ring, but these statements hardly go any way towards characterising an affine scheme. I am not looking for an answer that is necessarily strictly tied to the structure of the category of ringed topological spaces - just something that is topological and/or about the algebraic structure of the structure sheaf.
A non-answer is: 'An affine scheme is a ringed topological space of the form SpecR for some cummutative ring R.' 
Thanks for any pointers, Christopher

REPLY [3 votes]: Maybe you will be interested in the thesis of Mel Hochster. He characterizes the image of Spec in Top. The article based on it is called Prime Ideal Structures in Commutative Rings and appeared in the Transactions of the AMS in 1969.