Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 ∪ T_2$ (by which I mean the smallest Grothendieck topology on C containing $T_1$ and $T_2$).

Assume that every $T_3$ cover has a refinement of the form $\stackrel{f_1}{\to} \stackrel{f_2}{\to}$ where $f_2$ is a $T_2$ cover and $f_1$ is a $T_1$ cover (this is true for example, if $T_1$ is the Nisnevich topology, and $T_2$ is the topology consisting of proper cdh covers; in this case $T_3$ is the cdh topology).

If $F$ is a $T_1$-sheaf, is its $T_2$-sheafification still a $T_1$-sheaf (and therefore a $T_3$-sheaf)?

I can show this is true if $F$ is a $T_2$ separated $T_1$ sheaf, but I want to remove this condition.

share|improve this question
3  
You should probably note that this is very similar to, but not identical to, mathoverflow.net/questions/8115/… –  Anon Nov 4 '11 at 18:21

1 Answer 1

How about this for a counter-example: We take the category

$$ A \stackrel{f_2}{\to} B \rightrightarrows C \stackrel{f_1}{\to} D $$

where there is a unique morphism from $A$ to $C$ and from $B$ to $D$. The only $T_1$ cover is $f_1$ and the only $T_2$ cover is $f_2$.

This category and these topologies satisfy my conditions.

Consider the $T_1$ sheaf $F$:

$F(A) = $ a singleton set $\underline{0} = ${$0$}.

$F(B) = $ a two element set $\underline{1} = ${$0, 1$}.

$F(C) = $ a singleton set $\underline{0} = ${$0$}.

$F(D) = \varnothing$ the empty set.

$$ \underline{0} \leftarrow \underline{1} \leftleftarrows \underline{0} \leftarrow \varnothing $$

The two morphisms $F(B) \leftleftarrows F(C)$ are the two distinct morphims $\underline{1} \leftleftarrows \underline{0}$. This is a $T_1$ sheaf.

Then the $T_2$ separated presheaf (which happens to already by a $T_2$ sheaf) associated to this $T_1$ sheaf is

$$ \underline{0} \leftarrow \underline{0} \leftleftarrows \underline{0} \leftarrow \varnothing $$

which is no longer a $T_1$ sheaf.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.