Affine schemes and structure sheaves

For a ring A, the points of Spec A are its prime ideals, with closed sets consisting of primes containing a specified ideal. The distinguished open D(f) consists of the primes avoiding f. These opens form a basis, and their pairwise intersections are D(fg). On D(f), the function f must be invertible, so its ring of regular sections is the localization A_f. Restricting to a smaller distinguished open uses the corresponding universal localization map.

This prescription first defines the structure sheaf on a basis, where the gluing identities can be checked algebraically, and then extends it to arbitrary opens. One must not assign A_f to an arbitrary open merely because it resembles a punctured region in a picture. General opens may need several distinguished charts, and their sections are compatible tuples of localized elements. In particular, different denominators are allowed on different members of a cover as long as they define the same section on overlaps.

A ring homomorphism A→B sends a prime of B to its inverse image in A. This gives the continuous map Spec B→Spec A and, after localization, the required local maps on stalks. Conversely, every affine-scheme morphism arises uniquely in this way. Taking global sections recovers A from Spec A, including its nilpotents and zero divisors. Thus affine schemes and rings form contravariantly equivalent categories, not covariantly equivalent categories.

A map on the underlying topological spectra alone is insufficient. It may admit several different sheaf pullbacks, or no suitable pullback, and it cannot specify what happens to infinitesimal functions. For example, maps out of a ring containing a square-zero element must choose where that element goes, subject to its relation. Two such choices can induce the same map on every prime ideal while defining distinct scheme morphisms. The structure sheaf is therefore essential morphism data rather than a decorative addition to a space.

Every prime ideal contains every nilpotent element. Consequently the quotient by the nilradical has the same underlying prime spectrum, with the same Zariski topology. The reduced scheme removes those nilpotent functions from the sheaf. A nonreduced scheme can therefore be topologically indistinguishable from a reduced one while carrying different local rings and different infinitesimal maps. Saying that a nilpotent vanishes at all points is a statement about residue-field values, not a statement that it is the zero section.

The dual numbers illustrate this distinction on a single underlying point, while adjoining an ordinary coordinate spreads that infinitesimal direction along a line. A first-order map into the thickened object can detect a square-zero parameter that an ordinary field-valued point cannot. Such information later becomes important in tangent spaces and deformation theory. Before counting points or declaring two geometric objects equal, compare the local rings, the sheaf sections, and the chosen base field as well as the topology.