From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: Etale cohomology book recommendation needed
Date: Fri, 24 Mar 2000 07:53:11 GMT
Newsgroups: sci.math
Summary: [missing]

In article <38DADBCA.7DAC292F@sympatico.ca>,
  t@sympatico.ca wrote:
> Hi,
>
> my eventual goal is to learn etale cohomology to a level that would
> allow me to read Deligne's two IHES papers on the Weil conjectures.  I
> would like a preliminary "gentle" introduction.  The books I am
> considering are:
>
> Etale Cohomology by Milne
> Etale Cohomology by Tamme
> Etale Cohomology and the Weil Conjectures by Freitag and Kiel.
> and of course SGA
>
> The third book attempts to carry the reader to the conjectures' proofs,
> but I have heard that it is not sufficient.
>
> Any other suggestions for preparatory reading before tackling Deligne
> would also be welcome.

You might look at Milne's lecture notes on etale cohomology,
avaliable at

http://www.math.lsa.umich.edu/~jmilne/

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


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==============================================================================

From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: continuous-valued logic
Date: 27 Aug 2000 17:11:14 GMT
Newsgroups: sci.math
Summary: [missing]

[...]
|>Consider a circle C with its natural topology and the category of
|>espaces etales over C.
|>A sentence       is an open subset of C
|>A formula f(x,y) is sous espace etale of the fibred product X x Y
|>                                                              C
|>Conjunction      is set intersection
|>Disjunction      is espace etale associated with the union

In article <39a7c97a.340345587@nntp.sprynet.com>,
ullrich@math.okstate.edu (David C. Ullrich) writes:
|	Well, I don't know what any of that means, so I'm
|not sure whether it's interesting or not. If you wanted to
|explain what it meant that would be ok.

Well, "espace" is French for "space", and "etale" is a word we've
borrowed from the French for mathematical purposes, meaning "flat".
An etale space Y over a space X is a bit like a covering space of X;
each point of Y has to have a neighborhood which maps homeomorphically
onto a neighborhood of its image in X. In the case of a circle, an
etale space over it would have as connected components either open
intervals or circles, with continuous mappings to the circle (possibly
wrapping around the circle a number of times).

In case you want to navigate the literature, it would help you to know
that etale spaces are another way of treating sheaves over a space. To
each sheaf there's an associated etale space and to each etale space
there's an associated sheaf, with isomorphic things of the one kind
corresponding to isomorphic things of the other. Sheaves are used to
describe spaces of functions defined over open subsets of a space.

The fact being alluded to is that the category of etale spaces (of
sheaves) over a space X (where X has at least one point, I guess) is
a topos. A topos is a kind of category which in many key respects
resembles the category of sets.

For example, if X is just a single point, then the etale spaces over
X are just sets (with the discrete topology on them), so we get the
category of sets itself. If X is two points, then we get the category
of pairs of sets, which in a number of ways works like the category
of sets. Each standard set-theoretical operation just applies
separately to each component. If X has a more interesting topology on
it, we get a category of objects which one can think of as being like
sets which vary with a parameter given by a point moving in X. If X is
a circle, we can think of them as being sets where the elements may
appear and disappear as we move around the circle, or may continue to
exist clear around the circle, but come back to a different point if
we follow them all the way around. The set theoretical operations can
be applied roughly as though we were applying them to each point
separately.

The upshot is that if someone refers to a "gizmo over X" where a
gizmo is some kind of known mathematical object, and X is a
topological space, we have a (usually oversophisticated) way of
attempting a charitable interpretation of what they are saying by
assuming for argument's sake that they meant a gizmo object in the
category of sheaves over X, which often is the same as a "sheaf of
gizmos on X". In this case, though, that's not even as good an guess
as yours was-- what would have been the point of mentioning sheaves
of Banach spaces over a square in the context of the discussion?

Keith Ramsay
==============================================================================

From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: continuous-valued logic
Date: 30 Aug 2000 02:36:07 GMT
Newsgroups: sci.math

>In the case of a circle, an
>etale space over it would have as connected components either open
>intervals or circles, with continuous mappings to the circle (possibly
>wrapping around the circle a number of times).

An alert reader has pointed out that this is just a small sampling
of the connected etale spaces over the circle. For instance, if we
take a disjoint union of two copies of (0,1) identified on (0,1/2) and
map the resulting space to the circle in the obvious way, we get
another etale space over the circle.

Keith Ramsay