Re: Sheaves of T-algebras

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

Dear Mike,

In that case you want to look at John Kennison's 1976 paper
"Integral domain type representations in sheaves and other
topoi", Math. Z. 151, 35-56, and my own "Rings, fields and
spectra" in J. Algebra 49 (1977), 238-260. There's a summary
in section V 3.11 of "Stone Spaces".

Peter Johnstone

On Mon, 24 Sep 2012, Michael Barr wrote:

> Well, I actually know quite a bit more about it.  First off, each point in
> the base has a local neighborhood whose sections are algebras and the
> restrictions are homomorphisms.  I should have mentioned that all ops and
> partial ops are finitary, but of increasing arity.  We know the answer is
> positive; I am just unhappy with the proof.
>
> The situation is that the stalks are integral domains.  Now integral
> domains are not models of an LE theory.  But they are models of a theory
> with an infinitude of LE operations (all finitary, but of increasing
> arity) and I want the global sections to be
> models of that theory (which turns out to characterize the rings in the
> limit closure of the domains).  The space is the set of primes of some
> arbitrary semi-prime (no nilpotents) commutative ring with the co-Zariski
> topology.
>
> Mike
>
> On Mon, 24 Sep 2012, Steve Vickers wrote:
>
>> Dear Mike,
>>
>> Once you have that a sheaf of T-algebras is equivalent to a T-algebra in
>> the category of sheaves, it's obvious: because the global sections
>> functor preserves finite (in fact all) limits.
>>
>> But the first step is not so trivial, and in fact is not true unless you
>> take care over "the stalks are T-algebras". For example, if the base
>> space is Sierpinksi then a sheaf is just a function, with the domain and
>> codomain as the stalks (over the bottom and top points respectively).
>> You might put algebra structures on the stalks, but it won't be an
>> algebra in the sheaf category unless the function is a homomorphism.
>>
>> You will get further issues if the base space is non-spatial, so there
>> aren't enough global points to provide enough stalks.
>>
>> What statement of the result did you have in mind?
>>
>> One sufficient condition that gets round those problems is for the
>> stalks and their T-algebra structure to be defined geometrically.
>>
>> Regards,
>>
>> Steve.
>>
>>
>>
>> Michael Barr wrote:
>>> It must be known that if T is a left exact theory (same as nearly
>>> equational theory) and you make a sheaf of T-algebras (meaning the stalks
>>> are T-algebras, then the set of global sections is also a T-algebra.  Can
>>> anyone give me reference?
>>>
>>> Michael
>>>
>>
>
>


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