Re: Sheaves and the empty set
- From: Jose Capco <cliomseerg@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: 19 May 2007 01:29:41 -0700
On May 19, 12:47 am, Kira Yamato <kira...@xxxxxxxxxxxxx> wrote:
But the definition calls for any *arbitrary* open covering, not just a
particular open covering as you have chosen with the product set with
the empty indexing set. So you should take the maximum covering which
would include the emptyset, hence the covering would be nonempty.
If I understand correctly by what you mean by the maximum covering,
you see this only works when your covering cannot be the empty
covering. If your coverings cannot be the empty covering there is a
canonical map from the product of those section sheaves of one
covering to a finer covering
i.e. if {V_i}_I is a finer covering than {U_j}_J
you have the following canonical maps
F(U) --> \prod_J F(U_j) ---> \prod_I F(V_i)
and then you have your exact sequence because the maps F(U) -->
\prod_I F(V_i) is the composition of the above
But if the empty covering can cover your open set, then the
composition is not the same (the composition in rings becomes the zero
map, whilst the canonical map
F(emptyset)--> \prod_I F(V_i) ... V_i=emptyset
becomes the identity map, or just the products of them.
Sincerely,
Jose Capco
--
-kira
.
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