12 - The Zariski Spectrum

Summary

The prime spectrum, or Zariski spectrum, of a commutative ringwas introduced in Section 2.5. There the purpose was to show early on that spectral spaces arise naturally in algebra. The way we defined Zariski spectra was the model for a general method by which spectral spaces can be attached to various algebraic structures in a functorial way, see Sections 2.5, 3.1, and 14.3 as well as the method from 7.2.12 for other constructions.

Now we return to rings and their Zariski spectra for a more detailed presentation. Our main objective is to show how spectral spaces can be helpful in the study of rings. It is essential to gain a thorough understanding of the connections between algebra and topology. This is a huge success story in commutative algebra and algebraic geometry, driven to a large extent by Grothendieck's work laying newfoundations for algebraic geometry. The Zariski spectrum introduces geometric intuition and geometric tools in ring theory. Any ring is viewed as a ring of functions on its Zariski spectrum. In view of the subject's breadth, we present only basic definitions and facts, but enough to give an impression of how and why spectral spaces are such an important tool in ring theory.

In ring theory the role of spectral spaces is different from their role in the theory of bounded distributive lattices. By Stone duality, Chapter 3, every spectral space is the spectrum of a unique bounded distributive lattice. In this respect the Zariski spectrum behaves very differently. While it is true that every spectral space is the prime spectrum of some ring (which is a famous result by Hochster, [Hoc69], and is discussed in Section 12.6), there does not exist anything like a duality between rings and spectral spaces. In fact, every spectral space is the prime spectrum of a ring in many different ways. Just note that all fields have the same Zariski spectrum, namely the one-point space. Thus it is not possible to reverse the construction of the Zariski spectrum and functorially associate a ring with every spectral space.We explore this topic in Section 12.6.

We start in Section 12.1 with the introduction of the Zariski spectrum.