Talk:Spectrum of a ring

Can someone please elaborate on what the object K[T] is? The article mentions that a vector space, equipped with a linear operator on it, can be viewed as a module over K[T], but then subsequently only mentions the ring K[T]. If K[T] is just another name for the polynomial ring K[x], then how are the spectral properties of a particular operator T supposed to manifest in the spectrum of K[x]? Or is K[T] instead supposed to be something like "the polynomial ring mod out by the minimal polynomial of T"? — Preceding unsigned comment added by 129.97.226.227 (talk) 12:38, 10 June 2013 (UTC)

You should think of K[T] as living inside GL(V). Inside GL(V), there is a field K (determined by diagonal matrices) and an element T (given to us). The subring of GL(V) generated by K and T is K[T].

This kind of notation turns up in other places. For instance, it's analogous to the notation Q(i) for the field of rational numbers together with a square root of −1 (which is a complex number—note that we've implicitly chosen a copy of C containing Q).

General facts imply that K[T] is isomorphic to K[x] modulo the minimal polynomial of T. Indeed by the universal property of K[x], there is a homomorphism K[x] → K[T] that sends x to T. The map is obviously surjective, and the kernel is generated by the minimal polynomial of T by the definition of the minimal polynomial.

By the way, on Wikipedia talk pages it's conventional to write new comments at the bottom. Ozob (talk) 14:04, 10 June 2013 (UTC)

Header added. —Nils von Barth (nbarth) (talk) 07:48, 7 December 2009 (UTC)

The functor F is co-representable by B, as over the category of algebras it is covariant. Fourier-Deligne Transgirl (talk) 03:16, 7 December 2023 (UTC)

I removed the following example:

A special but quite typical case of an affine scheme is obtained as follows. Take a field K and n variables, x₁,...,x_n. Given m polynomials, p₁,...,p_m in these variables over K, there is a functor F from the category of commutative K-algebras to sets characterized by F(A)={(x₁,...,x_n) in Aⁿ|p₁=...=p_m=0}. Then F is represented by Spec(B) where B is the quotient of K[x₁,...,x_n] by the ideal I generated by the p_j.

Reasons:

• This uses functors, but the article hasn't mentioned the functor connection yet.
• The technical term "represented" is not explained. The functor F is not represented by Spec(B), but by B.
• A more useful example would describe Spec(B) in detail.

AxelBoldt 16:07, 18 Jan 2004 (UTC)

IMO, this article contains too much general rubbish. Just focus on the connections to geometry. Schemes are a generalisation of this setting and reside in a seperate article. agrosquid

See the relevant manual of style entry: External link vs. External links. It is permissible (and arguably more correct) to leave off plural in the case of a single link. - Gauge 06:59, 25 February 2006 (UTC)

Summarize Timeline Fact Check

As expressed at the moment, ${\displaystyle \Gamma (U,{\mathcal {O}}_{X})}$ is a limit of objects in the category of rings. But a limit of objects is always a product. And the product in the category of rings is just a Cartesian product of rings. Is this actually correct? There's no citation. What are the restriction maps here? --Svennik (talk) 17:14, 26 December 2023 (UTC)

Should this not be a colimit of morphisms: In other words, should we not define ${\displaystyle \Gamma \left(\bigcup _{i\in I}D_{fi},{\mathcal {O}}_{X}\right)=\operatorname {colim} _{i\in I}\phi _{i}}$ where each ${\displaystyle \phi _{i}:R\rightarrow R_{f}}$ is the localisation map? This should correspond to repeatedly localising. which is what I intuitively would've expected.--Svennik (talk) 17:23, 26 December 2023 (UTC)

This is a direct limit. I have fixed notation and added a link to dirct limit. D.Lazard (talk) 18:42, 26 December 2023 (UTC)

It's not a direct limit because the basis ${\displaystyle (D_{f})_{f}}$ of ${\displaystyle U}$ doesn't form a directed set. It's likely to be the colimit I described above. I think this needs to be given in more detail. --Svennik (talk) 19:03, 26 December 2023 (UTC)

In fact, this is a limit (category theory) with respect to a preordered set that is not directed. I have edited the article accordingly. D.Lazard (talk) 12:10, 27 December 2023 (UTC)

By definition, Spec(R) is T₁ if and only if every prime ideal is maximal, which is the statement that R has Krull dimension 0.

In general topology, a T₁ spectral space is called a Stone space, and such space is T₄ (normal), but not necessarily T₅ (completely normal). 2001:861:4285:F310:14F2:AD65:112F:79E1 (talk) 18:33, 9 September 2025 (UTC)

Summarize Timeline Top Qs Fact Check

I have just added a section § Affine sheaves. This section duplicates a large part of section § Sheaves and schemes, which is a mess. I'll soon clean up accordingly this section (probably by removing most of it) and the following ones.

Unfortunately, I use to do a lot of typos, and, when reading again the new section, I missed probably many of them. Also , feel free to complete the section if I forgot some important facts or explanations. (ping Alsosaid1987) D.Lazard (talk) 15:58, 23 June 2026 (UTC)

Thank you D.Lazard for this new, more pedagogical presentation. I can also proofread for typos when I have the time. The categorical approach that you take is clean and illuminating for those who have a good intuitive understanding of the language. However, I do feel that giving the compatible collection of germs definition (that I attempted to present) is useful for a more concrete and classical feel for those who may not have the background for this viewpoint. After all, this is the definition given by Hartshorne's, Mumford's classics, as well as more recent elementary presentations by Ueno and lecture notes by Gathmann. The special case of what a regular function is for R being an integral domain that a previous editor added is useful for the same reason. I would therefore advocate that you consider reincorporating the alternative definition and concrete examples of O_X(U) in special cases into your new exposition to help with intuition for beginners.

This is merely my view as a dabbler in mathematics (and complete newcomer to algebraic geometry), and so I respect your expert judgment and defer to your opinions on this. Alsosaid1987 (talk) 20:44, 23 June 2026 (UTC)

I hasten to add that having a connection with regular functions of algebraic varieties is a strong motivation for the much more abstract ring of regular functions of an affine scheme, which appears rather late in this version. For the beginner it's really not at all obvious that one should consider prime ideals to be points and ring elements to be functions without this motivation. This connection should be made clearer earlier in the article, perhaps where O_X(D_f) is introduced. Alsosaid1987 (talk) 20:57, 23 June 2026 (UTC)

One must not confuse help for intuitive understanding and mathematical definition. This was one of the problems of the previous version, which was also incorrect in interpreting as "function" something whose codomain depends on input. Moreover, interpreting the elements of ⁠${\displaystyle {\mathcal {O}}_{X}(U)}$⁠ as "functions" may be helpful for analysts, but is certainly confusing for algebraists. We must take care of both.

I introduced § Affine algebraic varieties essentially for providing an example where ⁠${\displaystyle {\mathcal {O}}_{X}(U)}$⁠ may effectively be interpreted as a ring of functions. In reality, in this case, ⁠${\displaystyle {\mathcal {O}}_{X}(U)}$⁠ is a ring of functions only up to several canonicl isomorphisms (isomorphism between polynomials and polynomial functions and isomorphism between ⁠${\displaystyle \mathbb {C} }$⁠ and the residue field at a maximal ideal). I am not fully satisfied with my version of the section because I did not find a way to make these isomorphisms explicit while remaining readable and reasonably short.

For explaining better the interpretation as "functions", I intended to write a section § Analogy with rings of functions. I think that this is the best way for helping people with knowledge in analysis and for explaining why one commonly uses phrases such as "restriction to a smaller open set of a section over a larger open set". By the way, it would certainly improve the article to add a definition of these terms to the definition of sheafs and ringed spaces, with a note explained that the terminology, originated from sheaves of functions, is generalized to the case where ⁠${\displaystyle {\mathcal {O}}_{X}(U)}$⁠ is not a ring of functions. D.Lazard (talk) 11:03, 24 June 2026 (UTC)

Of course, in the general case, the "function" maps to a different field for every point that depends on the point, and it is also not determined by its values, so there are several reasons that should be pointed out for why it is not an actual function. Nonetheless, regarding the structure sheaf as a collection of permissible functions on given open sets is important intuition, even if the objects, in full generality, cannot be regarded as true functions anymore. (The definition of a section as a sequence or collection of germs retains the intuitive idea well.) Personally, as a recent learner, it took a while for me to understand the construction based on sheafification, so I appreciate that it is a challenge to explain well within the space limitations of an encyclopedia entry, but I think it is worth pointing out this alternative, more concrete perspective of "what" O_X(U) actually contains.

As examples, definition 12.16 from https://agag-gathmann.math.rptu.de/class/alggeom-2021/alggeom-2021-c12.pdf and https://math.stackexchange.com/a/4182759 are, in my opinion, relatively clear and suitable for someone who might come across the idea for the first time.

Anyway, I overall support your efforts on the points you state above, and I await more improvements to the article from you! Alsosaid1987 (talk) 15:15, 24 June 2026 (UTC)

Summarize Timeline Fact Check

The new section § Historical motivation is useful, but sets various questions:

• Part of its duplicates § Functional analysis perspective. I suggests to remove the latter section.
• My opinion is that it should be moved down in the article: WP is not a textbook, and, generally, readers of this article are primarily interested to learn what is the subject than to learn his history.
• It should be clear that the spectrum of an operator is a special case of the spectrum of a ring. I edited the section for that, but more must be done.
• It remains unclear who introduced the term "spectrum" for the set of the prime or maximal ideals. It could be Grothendieck in his work on topological vector spaces, but this has to be checked.
• Also, who considered first the Zariski topology on a spectrum. Probably Zariski, since he knew the bijection between the points of a variety and the maximal ideals of the ring of its regular functions (Hilbert's Nullstellensatz). Again this has to be checked.
• Before the introduction of scheme theory by Grothendieck, the spectrum of a ring was its maximal spectrum. I ignore whether someone considered the prime spectrum before.
• "The prime spectrum of a commutative ring also has a separate origin in commutative algebra and algebraic geometry": This seems to contradicts the beginning of the section, saying that the term comes from operator theory. It is true that prime and maximal ideals were studied in algebraic geometry independently of operator theory. But the use of "spectrum" in this context does not come from nothing.

In any case the section contains too much technical material for remaining the first section. D.Lazard (talk) 15:17, 25 June 2026 (UTC)

I think the functional analysis section should be removed. The goal of the historical motivation section is to be less technical than the old first section, and to justify the term "spectrum", which otherwise appears somewhat unconnected from the rest of mathematics (especially the spectrum from linear algebra, with which most readers will be familiar). According to Bourbaki, the word spectrum in ring theory comes from the functional-analytic side. The term itself comes from operator theory, first via Hilbert, then Stone's representation theorem, and later Gelfand. Jacobson in 1945 observed that Stone's topology can be applied to any commutative ring. Meanwhile, Zariski in 1944 defined a topology on the set of places of a function field (later identified with prime ideals). In 1952, Weil showed how to develop the Zariski topology on an algebraic variety. Serre's early 1958 work on sheaves is also invoked, and Serre seems to have been the first to introduce the "prime spectrum" (that I could find), but it seems clear that Grothendieck's EGA had the final synthesis. Regarding the apparent condradiction, these are parallel developments, and the section frames them that way. The Zariski topology, for example, was not formulated as a topology on a spectrum; that still awaited final synthesis from Weil, Serre, and ultimately Grothendieck. Sławomir Biały (talk) 18:18, 25 June 2026 (UTC)