Did Zariski really define the Zariski topology on the prime spectrum of a ring?
The question is not: “Did Zariski really define the Zariski topology?”
It is: “Did Zariski really define the Zariski topology on the prime spectrum of a ring?”
Here is the motivation. --- On page 80 of
Grothendieck, Alexander, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas. Publications Mathématiques de l'IHÉS, 4 (1960), p. 5-228,
availablable here, one reads:
L'introduction de cette topologie en géométrie algébrique est due à Zariski. Aussi est-elle souvent appelée la « topologie de Zariski » de $X$.
The introduction of this topology in algebraic geometry is due to Zariski. This is why it is often called the “Zariski topology“ of $X$.
The set $X$ is of course the prime spectrum of a ring.
EDIT. Theo Buehler has just posted a question inspired by Qiaochu's answer below. The title of Theo’s question is: Was Grothendieck familiar with Stone's work on Boolean algebras?
Solution 1:
Johnstone's Stone spaces contains the following historical note at the end of Chapter V:
The Zariski spectrum is really a misnomer; it would be better to call it the Jacobson-Zariski-Grothendieck spectrum, except that the latter is too much of a mouthful. As explained in the Introduction, it was Zariski [1952] who introduced a topology on an arbitrary algebraic variety, by taking its algebraic subsets as closed sets; then Grothendieck [1960], exploiting the correspondence between points of an affine variety and maximal ideals of its coordinate ring, transferred Zariski's topology to the set of prime ideals of an arbitrary commutative ring. However, essentially the same topology had been introduced much earlier by Jacobson [1945] under the name 'hull-kernel topology', and extensively studied by various authors (e.g. [Arens and Kaplansky 1948], [Kaplansky 1950], [Gillman 1957], [Kohls 1957]) in a line of develoment which remained separate from the Zariski-Grothendieck one, even for some years after the publication of [Grothendieck and Dieudonné 1960] and [Bourbaki 1961a].
The relevant part of the introduction is also an interesting read:
The other area [besides category theory] where one searches in vain for the influence of Stone's Theorem is in algebraic geometry, with the rise of the 'Zariski topology'. It was sometime in the late forties (see [Zariski 1952]) that O. Zariski realized how one might define a topology on any abstract algebraic variety, by taking its algebraic subsets as closed sets; the precise date is difficult to determine, since Zariski himself does not seem to have attached much importance to the idea. (There is no mention of the Zariski topology in the first edition of Weil's book [1946] on algebraic geometry, although it plays a central role in the second edition [1962].) It was not until the work of Serre [1955] that the Zariski topology became an important tool in the application of topological methods (in this case, sheaf cohomology) to abstract algebraic geometry. There is an obvious similarity between the topologies introduced by Zariski and Stone, and indeed Dieudonné [1974] asserts that Zariski was influenced by Stone's work; but there seems to be no acknowledgement of this influence in Zariski's own papers.
The refoundation of algebraic geometry using schemes in place of varieties, begun by Grothendieck [1959, 1960] in the late fifties, brought the Zariski and Stone topologies even closer together; indeed, the latter is just the special case of the former applied to the spectrum of a Boolean ring. But again, one will not find any reference to Stone in the work of Grothendieck, even though his use of the word 'spectrum' is an obvious echo of [Stone 1940], and Grothendieck, with his background in functional analysis, must have been familiar with Stone's work in that field. Again, when the Zariski topology made its first appearance in a book on commutative algebra, as opposed to algebraic geometry, [Bourbaki 1961a], there was no mention of Stone's name. (The Zariski topology does not occur in [Zariski and Samuel 1958].)
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.
- Zariski 1952: Zariski, Oscar. The fundamental ideas of abstract algebraic geometry. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, 77–89. Amer. Math. Soc., Providence, R. I., 1952. MR0045412
- Grothendieck 1960: Grothendieck, Alexander The cohomology theory of abstract algebraic varieties. 1960 Proc. Internat. Congress Math. (Edinburgh, 1958) pp. 103–118 Cambridge Univ. Press, New York. MR0130879
- Jacobson 1945: Jacobson, N. A topology for the set of primitive ideals in an arbitrary ring. Proc. Nat. Acad. Sci. U. S. A. 31, (1945). 333–338. MR0013138
- Arens and Kaplansky 1948: Arens, Richard F.; Kaplansky, Irving Topological representation of algebras. Trans. Amer. Math. Soc. 63, (1948). 457–481. MR0025453
- Kaplansky 1950: Kaplansky, Irving Topological representation of algebras. II. Trans. Amer. Math. Soc. 68, (1950). 62–75. MR0032612
- Gillman 1957: Gillman, L. Rings with Hausdorff structure space. Fund. Math. 45 (1957), 1–16. MR0092773
- Kohls 1957: Kohls, C. W. The space of prime ideals of a ring. Fund. Math. 45 1957 17–27.
- Grothendieck and Dieudonné: Grothendieck, A. Éléments de géométrie algébrique. I. Le langage des schémas. (French) Inst. Hautes Études Sci. Publ. Math. No. 4 1960 228 pp. 14.05
- Bourbaki 1961a: MR0163908 (29 #1207) Grothendieck, A. Éléments de géométrie algébrique. I. Le langage des schémas. (French) Inst. Hautes Études Sci. Publ. Math. No. 4 1960 228 pp. MR163908
Solution 2:
Let's listen to the master:
Mais en 1945, Jacobson observe [172 c] que le procédé de définition d'une topologie, imaginé par Stone, peut en fait s'appliquer à tout anneau A (commutatif ou non) pourvu que l'on prenne comme ensemble d'idéaux non pas l'ensemble des idéaux maximaux, mais l'ensemble des idéaux « primitifs » bilatères (i.e. les idéaux bilatères $\mathfrak b$ tels que $\mathrm A/\mathfrak b$ soit un anneau primitif); pour un anneau commutatif, on retrouve bien entendu les idéaux maximaux. De son côté, Zariski, en 1944 [340 a], utilise une méthode analogue pour définir une topologie sur l'ensemble des places d'un corps de fonctions algébriques.
Toutefois, ces topologies restaient pour la plupart des algébristes de simples curiosités, en raison du fait qu'elles sont d'ordinaire non séparées, et qu'on éprouvait une répugnance assez compréhensible à travailler sur des objets aussi insolites. Cette méfiance ne fut dissipée que lorsque A. Weil montra, en 1952, que toute variété algébrique peut être munie de façon naturelle d'une topologie du type précédent et que cette topologie permet de définir, en parfaite analogie avec le cas des variétés différentielles ou analytiques, la notion d'espace fibré [330 e]; peu après, Serre eut l'idée d'étendre à ces variétés ainsi topologisées la théorie des faisceaux cohérents, grâce à laquelle la topologie rend dans le cas des variétés « abstraites » les mêmes services que la topologie usuelle lorsque le corps de base est $\mathbf C$, notamment en ce qui concerne l'application des méthodes de la Topologie algébrique [283 a et b].
Dès lors il était naturel d'utiliser ce langage géométrique dans toute l'Agèbre commutative. On s'est rapidement aperçu que la considération des idéaux maximaux est d'ordinaire insuffisante pour obtenir des énoncés commodes*, et que la notion adéquate est celle de l'ensemble des idéaux premiers de l'anneau, topologisé de la même manière. Avec l'introduction de la notion de spectre, on dispose maintenant d'un dictionnaire permettant d'exprimer tout théorème d'Algèbre commutative dans un langage géométrique très proche de celui de la Géométrie algébrique de l'époque Weil-Zariski; ce qui d'ailleurs a amené aussitôt à élargir considérablement le cadre de cette dernière, de sorte que l'Algèbre commutative n'en est plus guère de ce point de vue, que la partie la plus élémentaire [138 a].
With the footnote:
L'inconvénient de se borner au « spectre maximal » provient de ce que, si $\varphi : \mathrm A \to \mathrm B$ est un homomorphisme d'anneaux et $\mathfrak n$ un idéal maximal de B, $\varphi^{-1}(\mathfrak n)$ n'est pas nécessairement un idéal maximal de A, alors que pour tout idéal premier $\mathfrak p$ de B, $\varphi^{-1}(\mathfrak p)$ est un idéal premier de A. On ne peut donc en général associer à $\varphi$ de façon naturelle une application de l'ensemble des idéaux maximaux de B dans l'ensemble des idéaux maximaux de A.
And the relevant parts of the bibliography:
[138 a] A. GROTHENDIECK, Éléments de géométrie algébrique, I., Publ. Math. IHÉS n°4 (1960).
[172 c] N. JACOBSON, A topology for the set of primitive ideals in an arbitrary ring, Proc. Nat. Acad. Sci. U.S.A., t. XXXI (1945), p.333-338.
[283 a] J.-P. SERRE, Faisceaux algébriques cohérents, Ann. of Math., t. LXI (1955), p. 197-278.
[283 b] J.-P. SERRE, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, t. VI (1956), p. 1-42.
[330 e] A. WEIL, Fibre spaces in Algebraic Geometry (Notes by A. Wallace), Chicago Univ., 1952.
[340 a] O. ZARISKI, The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc., t. L (1944)- p.683-691.
(N. Bourbaki, Éléments d'histoire des mathématiques) I don't translate because it is very long and anyone interested in Algebraic Geometry (or called Pierre-Yves) should be able to read French.
I hope I'm not exceeding the limit of fair use, but if Mr. Bourbaki thinks otherwise, he can contact me.
Solution 3:
According to Kohls, it was Neal H. McCoy who first introduced the Zariski topology to the prime spectrum in a 1949 paper.