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?

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