Re: Sheaves and the empty set
- From: Kira Yamato <kirakun@xxxxxxxxxxxxx>
- Date: Fri, 18 May 2007 15:49:30 -0400
On 2007-05-18 07:23:01 -0400, Jose Capco <cliomseerg@xxxxxxxxxxxxxxxxxxxxxxxxx> said:
On May 18, 12:37 pm, Kira Yamato <kira...@xxxxxxxxxxxxx> wrote:On 2007-05-18 04:53:33 -0400, Jose Capco
<cliomse...@xxxxxxxxxxxxxxxxxxxxxxxxx> said:
Dear NG,
We know very well that if F is a sheaf of rings/groups, then
F(emptyset)=the zero ring/groups..
Suppose X is any topological space. For every open set U in X, define
F(U) = Z
where Z is the ring of integers. Define all restriction maps to be the
identity map on Z.
Is F a sheaf?
-
Like I said, F(emptyset) must be the zero ring if F were a sheaf.. so
this is not a sheaf!
Oh, is the requirement that
F(emptyset) = zero ring
be part of the definition for a sheaf? Otherwise, it seems that my example satisfies the definition for a sheaf.
About your exact sequence
F(U) --> \prod_i F(U_i) ==> \prod_i, \prod_j F(U_i \cap U_j)
isn't the emptyset a covering (of a single open set) of the emptyset itself? So, the products in the exact sequence are not empty products.
--
-kira
.
- Follow-Ups:
- Re: Sheaves and the empty set
- From: Jose Capco
- Re: Sheaves and the empty set
- References:
- Sheaves and the empty set
- From: Jose Capco
- Re: Sheaves and the empty set
- From: Kira Yamato
- Re: Sheaves and the empty set
- From: Jose Capco
- Sheaves and the empty set
- Prev by Date: Linear Least Squares. Linear Alg perspective
- Next by Date: Re: Are x and dx/dt independent?
- Previous by thread: Re: Sheaves and the empty set
- Next by thread: Re: Sheaves and the empty set
- Index(es):
Relevant Pages
|
