Krull & Paris
The Category-Cafe ran an interesting post The history of n-categories claiming that “mathematicians’ histories are largely ‘Royal-road-to-me’ accounts”
To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently, while the mathematicians tend to tell a story where we learn how the present has emerged out of the past, giving the impression that things were always going to turn out not very dissimilarly to the way they have, even if in retrospect the course was quite tortuous.Over the last weeks I’ve been writing up the notes of a course on ‘Elementary Algebraic Geometry’ that I’ll be teaching this year in Bach3. These notes are split into three historical periods more or less corresponding to major conceptual leaps in the subject : (1890-1920) ideals in polynomial rings (1920-1950) intrinsic definitions using the coordinate ring (1950-1970) scheme theory. Whereas it is clear to take Hilbert&Noether as the leading figures of the first period and Serre&Grothendieck as those of the last, the situation for the middle period is less clear to me. At first I went for the widely accepted story, as for example phrased by Miles Reid in the Final Comments to his Undergraduate Algebraic Geometry course.
… rigorous foundations for algebraic geometry were laid in the 1920s and 1930s by van der Waerden, Zariski and Weil (van der Waerden’s contribution is often suppressed, apparently because a number of mathematicians of the immediate post-war period, including some of the leading algebraic geometers, considered him a Nazi collaborator).But then I read The Rising Sea: Grothendieck on simplicity and generality I by Colin McLarty and stumbled upon the following paragraph
From Emmy Noether’s viewpoint, then, it was natural to look at prime ideals instead of classical and generic points‚Äîor, as we would more likely say today, to identify points with prime ideals. Her associate Wolfgang Krull did this. He gave a lecture in Paris before the Second World War on algebraic geometry taking all prime ideals as points, and using a Zariski topology (for which see any current textbook on algebraic geometry). He did this over any ring, not only polynomial rings like C[x, y]. The generality was obvious from the Noether viewpoint, since all the properties needed for the definition are common to all rings. The expert audience laughed at him and he abandoned the idea.The story seems to be due to Jurgen Neukirch’s ‘Erinnerungen an Wolfgang Krull’ published in ‘Wolfgang Krull : Gesammelte Abhandlungen’ (P. Ribenboim, editor) but as our library does not have this book I would welcome any additional information such as : when did Krull give this talk in Paris? what was its precise content? did he introduce the prime spectrum in it? and related to this : when and where did Zariski introduce ‘his’ topology? Answers anyone?