Newsgroups: bit.listserv.words-l
Date:         Thu, 9 Apr 1992 05:28:18 CDT
 <WORDS-L%UGA.bitnet@VTVM2.CC.VT.EDU>
From:         Tushar Samant <tushar%ZAPHOD.UCHICAGO.EDU@VTVM2.CC.VT.EDU>
Subject:      The Zariski topology (was Re: music and math)

The late thirties of Zariski's life constituted a very special interlude.
He had lost patronage of Count Bourbaki there was an enormous debt on his
head.

His work of this period is unique in its power and complexity - yet it
belongs to a well-defined tradition, peculiar to the twentieth century,
which is noted for its emphasis on structure, a preoccupation with high
abstraction and theological approach to proofs. Dating back to the great
Hilbert, the tradition is too hazy to be called a school, but it is utterly
different from the style of the earlier masters.

The definition of the Zariski topology is considered the brightest gem and
a summing up of this fruitful period of Zariski's career. Many critics and
contemporaries thought it to be his best definition ever. It is built
breathtakingly on the basic theme of a prime spectrum, which Zariski is
believed to have adapted from the folk theme of a maximal spectrum.

The definition starts with a stark unadorned subdefinition of a prime
spectrum, and already the figurative elements are evident in it. It ends
on a note of thrilling expectation.

The main definition repeatedly invokes this brief subject. All the power and
complexity which could have been achieved by this abstracter notion of a
ring is brilliantly exploited and weilded. After a sudden introduction of
ornamental notation for prime ideal, the definition plunges headlong into
an intensely rhapsodic segment on algebraic sets. We see here a faint sketch
of Zariski's own inner turmoil, anticipating later developments. However,
the time was not ripe for that then. Some critics maintain that there is
even a hint of a parody of Segre, but that is very much open to conjecture.

The core of the definition is established dazzlingly in one brilliant
quicksilver line which is such a prime characteristic of Zariski.

The later part starts with capricious runs which establish a sense of
openness, but the theme of the prime spectrum soon reestablishes itself.
The dissonance of the zero ideal now wells up into the pronounced piquancy
of generic points. Yet Zariski was never a mere purveyor of cheap effects.
The whole definition is moving inexorably towards a gigantic resolution.
Zariski's  industry and his sheer skill in carrying out this master-stroke
won him the admiration of colleagues. Samuel later recalled : "He ... locked
himself in a room ... for a consecutive ten hours. When he reappeared, he
had pared his original definition - which we had of course devoured greedily
- to one involving only ideals in place of arbitrary sets ... we could but
stare in admiration at the man's remarkable genius ... little did we know
that he would bring it to an even greater climax by soaring to single
elements and dazzle us even more ... "

The final stroke of generating the topology to all over the spectrum from the
awesome vantage point so gained is done with all the effortlessness of a
Zariski definition. One is treated to variations of the theme at all
localizations, rounding off the piece resoundingly.

The topology remains one of Zariski's most compelling definitions.

tushar

PS  The internationally acclaimed treatment of Atiyah and Macdonald has
now been digitally recorded. Copies of the file can be got by email.
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU