• DOI:10.1090/S0002-9947-1969-0251026-X
• Corpus ID: 122783870

Prime ideal structure in commutative rings

@article{Hochster1969PrimeIS,
  title={Prime ideal structure in commutative rings},
  author={Melvin Hochster},
  journal={Transactions of the American Mathematical Society},
  year={1969},
  volume={142},
  pages={43-60}
}

• M. Hochster
• Published 1 August 1969
• Mathematics
• Transactions of the American Mathematical Society

0. Introduction. Let ' be the category of commutative rings with unit, and regard Spec (as in [1]) as a contravariant functor from ' to g$7 the category of topological spaces and continuous maps. The spaces of the form Spec A, A a ring (ring always means object of '), are well known to have many special properties. It is worthwhile to discover whether these well-known properties of the spaces characterize them, since we then will know the limitations of the topological approach. As a by-product of this study, remarkable facts about the structure of the prime ideals in a commutative ring come to light. E.g. given any ring A there is a ring whose prime ideals have precisely the reverse order of the primes of A. In the same vein, on the topological level there is a complete duality between localization and taking residue class rings (see ?8, Proposition 8). We call a space spectral if it is To and quasi-compact, the quasi-compact open subsets are closed under finite intersection and form an open basis, and every nonempty irreducible closed subset has a generic point. Spec A, A a ring, is well known to be spectral, and in fact these properties do characterize the spaces in the image of Spec. However, there are much more enlightening characterizations. For example, the last property on the list is a very special case of a much more general property of spectral spaces-best described as the compactness of a new topology on the spectral space which is derived from the original topology. Again, the spectral spaces are precisely the projective limits of finite To spaces. Proving that these properties characterize the image of Spec involves constructing a large number of rings. This construction (??3-7) is very intricate, but is practically choice-frep and turns out to be fairly functorial. Moreover, we can fix any field k and get the constructed rings to be k-algebras. To be more precise about the functorial aspects, we define a continuous map of spectral spaces to be spectral if inverse images of quasi-compact open sets are quasi-compact. Let Y be the subcategory of $-consisting of spectral spaces and spectral maps. Then it is well known that every space and map in the image of Spec is in Y and it will turn out that up to isomorphism, every space and map in Y is in the image of Spec. In this sense, we may say that Y is precisely the image of Spec. We say that Spec is invertible on a subcategory I of Y if there is a (contravariant) functor F: q -W' whose composition with Spec is isomorphic with the inclusion