1 Spec as a set
For any subset of , we define the variety of to be the set
It is enough to restrict attention to subsets of which are ideals, since, for any subset of , we have where is the ideal generated by . In fact, even more is true: where denotes the radical of the ideal .
2 Spec as a topological space
for any ideals , of , establish that this collection does constitute a topology on . This topology is called the Zariski topology in light of its relationship to the Zariski topology on an algebraic variety (see Section 4 below). Note that a point is closed if and only if is a maximal ideal.
A distinguished open set of is defined to be an open set of the form
for any element . The collection of distinguished open sets forms a topological basis for the open sets of . In fact, we have
The topological space has the following additional properties:
For , let denote the localization of at the prime ideal . Then the topological spaces and are naturally homeomorphic, via the correspondence sending a prime ideal of contained in to the induced prime ideal in .
3 Spec as a sheaf
for each open subset . The presheaf satisfies the following properties:
4 Relationship to algebraic varieties
is sometimes called an affine scheme because of the close relationship between affine varieties in and the of their corresponding coordinate rings. In fact, the correspondence between the two is an equivalence of categories, although a complete statement of this equivalence requires the notion of morphisms of schemes and will not be given here. Nevertheless, we explain what we can of this correspondence below.
Let be a field and write as usual for the vector space . Recall that an affine variety in is the set of common zeros of some prime ideal . The coordinate ring of is defined to be the ring , and there is an embedding given by
The function is not a homeomorphism, because it is not a bijection (its image is contained inside the set of maximal ideals of ). However, the map does define an order preserving bijection between the open sets of and the open sets of in the Zariski topology. This isomorphism between these two lattices of open sets can be used to equate the sheaf with the structure sheaf of the variety , showing that the two objects are identical in every respect except for the minor detail of having more points than .
The additional points of are valuable in many situations and a systematic study of them leads to the general notion of schemes. As just one example, the classical Bezout’s theorem is only valid for algebraically closed fields, but admits a scheme–theoretic generalization which holds over non–algebraically closed fields as well. We will not attempt to explain the theory of schemes in detail, instead referring the interested reader to the references below.
Remark. The spectrum of a ring may be generalized to the case when is not commutative, as long as contains the multiplicative identity. For a ring with , the , like above, is the set of all proper prime ideals of . This definition is used to develop the noncommutative version of Hilbert’s Nullstellensatz.
- 1 Robin Hartshorne, Algebraic Geometry, Springer–Verlag New York, Inc., 1977 (GTM 52).
- 2 David Mumford, The Red Book of Varieties and Schemes, Second Expanded Edition, Springer–Verlag, 1999 (LNM 1358).
- 3 Louis H. Rowen, Ring Theory, Vol. 1, Academic Press, New York, 1988.
|Date of creation||2013-03-22 12:38:07|
|Last modified on||2013-03-22 12:38:07|
|Last modified by||CWoo (3771)|
|Defines||distinguished open set|