Étale cohomology

The second in an extremely irregular series of posts made on behalf of my father, who has spent much of his retirement so far doing very hard mathematics. What is attached here is the essay he wrote for the Part III of the Cambridge Mathematical Tripos, a one year taught course. The subject is the Étale cohomology.

Says my Dad: “I am afraid that I have been lured away from the translation of SGA 4.5 for some time by the attraction of working on Wolfgang Krull’s report on “Idealtheorie” from 1935 (again I am not aware of an English version anywhere) which is yet another important classic. However during a year at Cambridge I did write an essay as a very basic introduction to Étale Cohomology which was based on the first part of SGA 4.5. So with the usual imprecation of caveat lector, here it is as a temporising partial substitute should any other beginner be interested."

Here’s part of the introduction:

This essay has been written as part of the one year Certificate of Advanced Study in Mathematics (CASM) course at Cambridge University which coincides with Part III of the Mathematical Tripos. The starting point is, of necessity, roughly that reached in the lectures which in this particular year did not include much in the way of schemes and sheaves, nor, in the case of the author, much in the way of algebraic number theory.

Thus the frontiers of the subject can safely rest undisturbed by the contents of this essay. Rather it has been written with a reader in mind corresponding roughly to the author at the start of the enterprise. That is someone who is interested to find out what all the fuss was with the French algebraic geometers in the 1960s but is in need of some fairly elementary background to map out the abstractions involved and with any luck to avoid drowning in the “rising sea”.

And here’s the essay itself!