derived smooth geometry
For usual (commutative) rings, Grothendieck introduced the notion of a prime spectrum. In order to accomodate not only polynomials but also formal power series, it is convenient to consider completions and topological rings. Order nilpotent elements in an ordinary ring compare to completions as truncations of general power series and geometrically represent certain -th infinitesimal neighborhood. Completions represent certain pro-objects in the category of rings. Adic completion corresponds to have all infinitesimal neighborhoods at once.
A formal spectrum is a generalization of prime spectrum to adic noetherian rings, therefore containing information on all infinitesimal neighborhoods, corresponding to the ideal of completion.
Assume is a commutative ring and is an ideal, such that its powers make a fundamental system of neighborhoods of zero of a complete Hausdorff topology (we say that is an separated complete ring in the -adic topology).
Regarding that all affine schemes for have the same underlying topological space because nilpotents in do not affect the underlying reduced scheme, so does . With our assumptions on -adic topology, in fact contains all closed points of and any open subset of containing is the whole of . The structure sheaf has the ring of sections where the limit is taken in the category of topological rings, and have a discrete topology. For example, the ring of global sections is .
A formal spectrum is an example of a formal scheme. Formal schemes in general form certain subcategory of the category of ind-schemes.
The formal spectrum of a separated complete topological -adic ring depends just on the underlying topology on and not on a choice of the ideal generating this topology.
standard references are EGA, Hartshorne