A topological space is irreducible if and only if it is not empty and any two nonempty open subsets have nontrivial intersection. Equivalently, if and only if the space is not empty, and cannot be written as a union of two proper closed subsets.
(To see the equivalence: suppose any two nonempty open subsets have nontrival intersection, and let $C_1$ and $C_2$ be closed subsets such that $C_1\cup C_2 = X$. Let $O_1=X-C_1$, $O_2=X-C_2$; since $O_1\cap O_2 = (X-C_1)\cap(X-C_2) = X-(C_1\cup C_2) = \emptyset$, either $O_1$ is empty or $O_2$ is empty, hence either $C_1 =X$ or $C_2=X$. Conversely, suppose that $X$ is not the union of two proper closed subsets, and let $O_1$ and $O_2$ be nonempty open sets. Then $C_1=X-O_1$ and $C_2=X-O_2$ are proper closed subsets, so $\emptyset\neq X-(C_1\cup C_2) = (X-C_1)\cap(X-C_2) = O_1\cap O_2$.)
For a ring $R$, $\mathrm{Spec}(R)$ as a set consists of all prime ideals of $R$, and the closed sets are the sets of the form $V(E)$, where $E\subseteq R$ is a subset, and $V(E) = \{\mathfrak{p}\in\mathrm{Spec}(R) \mid E\subseteq \mathfrak{p}\}$.
So, suppose $R$ is a domain. Is $\mathrm{Spec}(R)$ empty? Hint: Remember that if $S$ is a commutative ring with identity, then $\mathfrak{I}$ is a prime ideal if and only if $S/\mathfrak{I}$ is an integral domain.
Now suppose that $E$ and $F$ are subsets of $R$ such that $\mathrm{Spec}(R) = V(E) \cup V(F)$. That means: every prime ideal of $R$ contains either $E$ or $F$ (or both).
In particular, there is a very special prime ideal that must contain either $E$ or $F$, which tells you that $E$ or $F$ is actually equal to blank. And therefore, either $V(E)$ or $V(F)$ is blah. Which means that $\mathrm{Spec}(R)$ is blankety-blank.
For an encore, try Exercise I.19 of Atiyah-MacDonald, which says that in general, $\mathrm{Spec}(A)$ is irreducible if and only if the nilradical of $A$ is a prime ideal (and notice that your proposition is just one implication of a special case of this).
[...] Grothendieck, with his background in functional analysis, must have been familiar with Stone's work in that field.
cited by Qiaochu reveals a gap in the knowledge of Grothendieck's work (which is of course nothing to be blamed for, given its vastness and ramifications in so many fields) but it also misses a link to one of Grothendieck's beautiful theorems in functional analysis. I'd like to make the point that
not only was Grothendieck well aware of that work, he even exploited it!
The crucial reference is his article Une caractérisation vectorielle–métrique des espaces $L^{1}$. While Grothendieck does not cite Stone, he relies crucially on Nachbin's paper A theorem of the Hahn–Banach type for linear transformations, improves on that paper and exploits the main results in a serious way. This is only possible if you understand these results profoundly. Nachbin in turn cites three works of Stone, see references 4,5,6 below.
To prevent a possible criticism of that argument, let me note that Nachbin and Grothendieck knew each other well. Grothendieck spent some time in Brazil and even planned to write a book on topological vector spaces with Nachbin:
Grothendieck planned to write a book on topological vector spaces with Leopoldo Nachbin, who was in Rio de Janeiro, but the book never materialized. However, Grothendieck taught a course in São Paulo on topological vector spaces and wrote up the notes, which were subsequently published by the university.
see Allyn Jackson's Notices article on Grothendieck, part 1, p.1044. It could seem conceivable that the results were orally transmitted and that Grothendieck never looked at that paper. However, to such an objection I'd respond: you have to take into account that Grothendieck asserted the main results of that work earlier but he later discovered that his arguments had a serious gap. Given that he had to work rather hard to fix it, it seems quite implausible to me that he never took a closer look at Nachbin's work and missed the work of Stone cited in there entirely.
Secondly Grothendieck refers to two works of Kakutani (references 8 and 9) below, which in turn refer to Stones works 4 and 5 and rely on it, too.
In conclusion I think it is safe to say that Grothendieck not only "must have been" but actually "was" aware of Stone's work (Edit: As Didier Piau pointed out in a comment this is still inconclusive, but please see the update at the end of this post).
I'm now parting from the historical ramblings and make an attempt at a description of Grothendieck's result in a way that I hope is appealing to algebraists. For the actual work, please consult the MathSciNet review and the paper itself.
Further References: It don't know of many self-contained expositions of the results I mention below. The classification of "injective" Banach spaces (called $P_1$-spaces in the classical literature) can be found in Day's booklet Normed Linear Spaces as well as in H. Elton Lacey's The isometric theory of classical Banach spaces. For a categorical view on Banach spaces please consult the Book Banach modules and functors on categories of Banach spaces by Cigler–Losert–Michor. I take the liberty and refer to Part 2 (Chapter IV) of my thesis for further facts, references and details.
Recall that Grothendieck's thesis was called Produits tensoriels topologiques et espaces nucléaires which is a masterpiece of its own and even mentioning its ramifications to the theory of distributions would lead us too far astray—the first line of the MathSciNet review reads: Le grand nombre de résultats importants contenus dans ce mémoire (Thèse), rend difficile d'en mettre en évidence les lignes essentielles, même si l'on a recours au résumé des résultats publié auparavant [...]. Incidentally, one characteristic of that work is that quite often the statement of theorems span more than a page while the proofs only take a couple of lines or are dismissed with a mere "Preuve: évidente."
The simplest instance of a topological tensor product is due to Murray (and von Neumann)'s student Schatten and is called the projective tensor product $\otimes$ of Banach spaces (usually denoted $\hat{\otimes}$ or $\hat{\otimes}_{\pi}$). Given two Banach spaces, the space of bounded linear operators equipped with the operator norm yields an (internal) Hom-bifunctor and by design the projective tensor product is its left adjoint. From that point of view this is the most algebraically sane tensor product of Banach spaces to look at. (Yes, it is part of a monoidal closed structure etc etc)
Let us call a Banach space flat if $F \otimes {-}$ preserves short exact sequences of Banach spaces (here short exact means isometric inclusion of subspace + corresponding quotient map—we're working in the category of Banach spaces and linear contractions).
A Banach space $I$ will be called injective if every contractive map $E \to I$ from a subspace $E \leq F$ extends to a contractive map $F \to I$.
Note: The scalar field is of course flat (it is the tensor unit) and it is injective (that's equivalent to Hahn–Banach).
Using this language we have:
Theorem (Grothendieck). A Banach space $F$ is flat if and only if $F$ is isomorphic to a space $L^1(\Omega)$ where $\Omega$ is some measure space.
Now how does Grothendieck prove that? In his thesis he observed that $L^1(\Omega)$ is flat. This is not very hard to prove, as from an analytic perspective the projective tensor product $L^1(\Omega) \otimes E$ really is the space of Bochner integrable functions $L^1(\Omega, E)$ and from that identification (due to Grothendieck as well, as far as I know) the exactness of $L^1(\Omega) \otimes {-}$ is rather obvious .
The other direction needed much more insight. I won't elaborate on precisely what Grothendieck did, but here's what it comes down to morally, from the skewed perspective I adopted in this part of the answer so far:
- "Lambek's theorem": A Banach space is flat if and only if its dual space $F^{\ast}$ is injective (rather trivial, see see also Lam Theorem 4.9, p.125).
- A Banach space is injective if and only if it is isomorphic to a space of the form $C(K)$ with $K$ Stonean (compact Hausdorff and extremally disconnected).
- Grothendieck's theorem can now be phrased as: An injective Banach space is a dual space if and only if it is of the form $L^{\infty}(\Omega)$; its pre-dual $L^1(\Omega)$ is unique (contrary to general Banach spaces).
The theorem in 2. (due to various efforts by Akilov-Goodner-Kelley-Nachbin) is seriously deep. A feeling for this can be obtained by noting that the easiest example of an infinite Stonean space is $K = \beta \mathbb{N}$, the Stone–Čech compactification of the natural numbers. Suffice it to say that the space $K$ arises from the geometry of the unit ball of an injective Banach space, which is strongly related to lattice theory and Boolean algebras, hence the link. Note also that Stonean spaces are called thus because they arose from Stones study of spectra of Boolean algebras.
Note that by the Riesz-Kakutani representation theorem the space $C(K)^\ast$ is the space of measures on $K$. Now given a flat Banach space $F$ it embeds into $F^{\ast\ast} = C(K)^{\ast}$ and Grothendieck needed to recover it from the fact that he knew only that $C(K)$ is a dual space. He did that by identifying the space $F$ with the normal measures on $K$ and this does involve a deep understanding of the Stonean spaces.
Added: I forgot to mention one further punchline: Grothendieck also proves and makes use of the fact that a $C(K)$ space which is a dual space can be realized as a von Neumann algebra (in the spatial definition, of course). In remarque 3 he states:
On peut se demander si le théorème 2 se généralise à toute $C^{\ast}$-algèbre (non nécessairement abélienne comme dans notre énoncé): Si une telle algèbre $C$ est isomorphe (avec sa norme) au dual d'un sous-espace $L$ de $C$, est-il vrai que $C$ est isomorphe à une algèbre de von Neumann, et que $L$ est exactement l'ensemble des formes normales sur $C$? [...] La seule difficulté est dans la question si $L$ est engendré par sa partie positive, le raisonnement donné dans le cas commutatif ne vaut pas tel quel. Il semble probable cependant que la technique des $C^{\ast}$-algèbres jointe au théorème de Banach-Dieudonné doive permettre de donner une réponse affirmative à notre question.
Of course, this is nothing but giving the outline of the proof of the celebrated Sakai theorem that characterizes von Neumann algebras as precisely those $C^{\ast}$-algebras which are dual spaces as a Banach space. Those acquainted with Sakai's theorem will of course recognize that this is exactly how Sakai proceeded and indeed the "seule difficulté" in the proof... but I digressed enough. Let me stop by remarking that Sakai's paper was submitted while Grothendieck proofread his paper and I'll finish by quoting him:
Ajouté pendant la correction des épreuves. Monsieur Lowdenslager m'a fait observer les faits suivants. La généralisation du théorème 2 conjecturée dans la remarque 3 a été prouvée récemment par Sakai, dans un papier qui sera publié dans le Pacific Journal of Mathematics. Le théorème de Nachbin cité au début de ce travail a été prouvé indépendamment par D. B. Goodner; la plus jolie preuve connue semble être celle de Kelley (Banach spaces with the extension property, Trans. Amer. Math. Soc, 72 (1952), 323-326), qui ne suppose pas que la boule unité admette un point extremal.
References:
- MR0075539:
Grothendieck, Alexandre,
Produits tensoriels topologiques et espaces nucléaires.
Mem. Amer. Math. Soc. (1955), no. 16, 140 pp.
- MR0076301:
Grothendieck, A.
Une caractérisation vectorielle-métrique des espaces $L^1$.
Canad. J. Math.7 (1955), 552–561.
- MR0032932: Leopoldo Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28-46.
- MR1501905: Stone, M. H.,
Applications of the theory of Boolean rings to general topology.
Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481.
- MR2023: Stone, M. H., A general theory of spectra. I.,
Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 280–283.
- MR4092: Stone, M. H., A general theory of spectra. II.
Proc. Nat. Acad. Sci. U. S. A. 27, (1941). 83–87.
- MR29091: M. H. Stone, Boundedness properties in function-lattices, Canadian Journal of Mathematics vol. 1 (1949) pp. 176-186.
- MR0004095: Kakutani, Shizuo. Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem.
Ann. of Math. (2) 42, (1941). 523--537.
- MR0005778: Kakutani, Shizuo. Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)
Ann. of Math. (2) 42, (1941). 994--1024.
- MR0084115: Sakai, Shôichirô,
A characterization of $W^{\ast}$-algebras. Pacific J. Math. 6 (1956), 763–773.
- MR0007568: Robert Schatten, On the direct product of Banach spaces, Trans. Amer. Math. Soc. 53 (1943), 195-217.
- T. Bühler, On the algebraic foundation of bounded cohomology, Mem. Amer. Math. Soc., posted on March 11, 2011,
PII S 0065-9266(2011)00618-0 (to appear in print). Also available here.
- MR533819: Cigler, Johann; Losert, Viktor; Michor, Peter
Banach modules and functors on categories of Banach spaces.
Lecture Notes in Pure and Applied Mathematics, 46. Marcel Dekker, Inc., New York, 1979. xv+282 pp.
- MR0344849: Day, Mahlon M. Normed linear spaces.
Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. Springer-Verlag, New York-Heidelberg, 1973. viii+211 pp.
- MR0493279:
Lacey, H. Elton. The isometric theory of classical Banach spaces.
Die Grundlehren der mathematischen Wissenschaften, Band 208. Springer-Verlag, New York-Heidelberg, 1974. x+270 pp.
- MR1653294: T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999.
- MR58866: A. Grothendieck, Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$., Canadian J. Math.5,(1953). 129–173.
Update:
In the paper Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$ we find Stone's paper Applications of the theory of Boolean rings to general topology listed as reference 14. It is mentioned only once as a mere reference for the Stone–Čech compactification at the very end of the proof of Proposition 6 (b):
Here's the bibliography of that paper:
I'd like to emphasize that my intention of this post was by no means to debunk anything Johnstone writes (and as a matter of fact: I didn't!): the excerpt provided by Qiaochu explicitly speaks of Grothendieck's work on Algebraic Geometry a bit before. I merely wanted to point out that there was an interesting link that may not be as widely known as it deserves to be.
For me the above settles the matter whether Grothendieck was aware of Stone's work. Whether he was familiar with it, particularly with the aspect of spectra, it is probably something he alone could answer, unless a more specific reference is provided. However, I hope I have succeeded in making the point that he was familiar enough with the contents that he could prove a beautiful theorem with it.
Best Answer
Johnstone's Stone spaces contains the following historical note at the end of Chapter V:
The relevant part of the introduction is also an interesting read:
I don't quite have the time to add in links to all those references... anyway, in short, it seems that Zariski only considered the maximal spectrum of varieties.