;[1] in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings
is a compact space, but almost never Hausdorff: In fact, the maximal ideals in
denotes the inverse limit with respect to the natural ring homomorphisms
Any ringed space isomorphic to one of this form is called an affine scheme.
As above, this construction extends to a presheaf on all open subsets of
, that is, a prime ideal, then the stalk of the structure sheaf at
Note that this agrees with the notion of a regular function in algebraic geometry.
is an algebraically closed field) that are defined as the common zeros of a set of polynomials in
is such an algebraic set, one considers the commutative ring
, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets).
By studying spectra of polynomial rings instead of algebraic sets with the Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes.
First, there is the notion of constructible topology: given a ring A, the subsets of
satisfy the axioms for closed sets in a topological space.
is clear from the context, then relative Spec may be denoted by
More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative
The relative spec is the correct tool for parameterizing the family of lines through the origin of
This example can be generalized to parameterize the family of lines through the origin of
From the perspective of representation theory, a prime ideal I corresponds to a module R/I, and the spectrum of a ring corresponds to irreducible cyclic representations of R, while more general subvarieties correspond to possibly reducible representations that need not be cyclic.
The connection to representation theory is clearer if one considers the polynomial ring
As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of
corresponds to choosing a basis for the vector space.
In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in n-space, by the Nullstellensatz (the maximal ideal generated by
Thus, points in n-space, thought of as the max spec of
The non-maximal ideals then correspond to infinite-dimensional representations.
Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain.
Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity.
For instance, for the 2×2 identity matrix has corresponding module: the 2×2 zero matrix has module showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module showing algebraic multiplicity 2 but geometric multiplicity 1.
of the algebra of scalars, indeed functorially so; this is the content of the Banach–Stone theorem.
Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum.
Generalizing to non-commutative C*-algebras yields noncommutative topology.