Alternative Construction of Sheaf from Sheaf on a Base

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In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too.



Suppose $mathscr F$ is a sheaf on bases of a topological space $X$. For an open set $Usubset X$ where $U$ is open, define



$barmathscr F(U)=lim_Bsubset Umathscr F(B)$, where the $B$s are basic open sets.



This seems to satisfy all the sheaf conditions since limits commute with products and kernels and hence the equalizer diagram for sheaves is satisfied. But I must have overlooked something since I haven't seen this anywhere else. Could you tell me if I'm right?




algebraic-geometry sheaf-theory




share|cite|improve this question















  • 1




    What kind of limit is that?
    – Lord Shark the Unknown
    Jul 28 at 9:02










  • An inverse limit.
    – Jehu314
    Jul 28 at 9:17










  • An inverse limit? Over what ordered set/category?
    – Lord Shark the Unknown
    Jul 28 at 9:18










  • The category of the Basic open sets and the restriction maps.
    – Jehu314
    Jul 28 at 9:21










  • I got this idea from the fact that for any sheaf $mathscr F$, $mathscr F(U)=lim_U_isubset U mathscr F(U_i)$ for any open covering $U_i$ of $U$
    – Jehu314
    Jul 28 at 9:26















up vote
2
down vote

favorite












In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too.



Suppose $mathscr F$ is a sheaf on bases of a topological space $X$. For an open set $Usubset X$ where $U$ is open, define



$barmathscr F(U)=lim_Bsubset Umathscr F(B)$, where the $B$s are basic open sets.



This seems to satisfy all the sheaf conditions since limits commute with products and kernels and hence the equalizer diagram for sheaves is satisfied. But I must have overlooked something since I haven't seen this anywhere else. Could you tell me if I'm right?




algebraic-geometry sheaf-theory




share|cite|improve this question















  • 1




    What kind of limit is that?
    – Lord Shark the Unknown
    Jul 28 at 9:02










  • An inverse limit.
    – Jehu314
    Jul 28 at 9:17










  • An inverse limit? Over what ordered set/category?
    – Lord Shark the Unknown
    Jul 28 at 9:18










  • The category of the Basic open sets and the restriction maps.
    – Jehu314
    Jul 28 at 9:21










  • I got this idea from the fact that for any sheaf $mathscr F$, $mathscr F(U)=lim_U_isubset U mathscr F(U_i)$ for any open covering $U_i$ of $U$
    – Jehu314
    Jul 28 at 9:26













up vote
2
down vote

favorite









up vote
2
down vote

favorite











In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too.



Suppose $mathscr F$ is a sheaf on bases of a topological space $X$. For an open set $Usubset X$ where $U$ is open, define



$barmathscr F(U)=lim_Bsubset Umathscr F(B)$, where the $B$s are basic open sets.



This seems to satisfy all the sheaf conditions since limits commute with products and kernels and hence the equalizer diagram for sheaves is satisfied. But I must have overlooked something since I haven't seen this anywhere else. Could you tell me if I'm right?




algebraic-geometry sheaf-theory




share|cite|improve this question











In Vakil's Notes and Mumford's 'Algebraic Geometry II', one can find the usual recovery of a sheaf from the data on the base using stalks. I was wondering if this construction would work too.



Suppose $mathscr F$ is a sheaf on bases of a topological space $X$. For an open set $Usubset X$ where $U$ is open, define



$barmathscr F(U)=lim_Bsubset Umathscr F(B)$, where the $B$s are basic open sets.



This seems to satisfy all the sheaf conditions since limits commute with products and kernels and hence the equalizer diagram for sheaves is satisfied. But I must have overlooked something since I haven't seen this anywhere else. Could you tell me if I'm right?





algebraic-geometry sheaf-theory





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 28 at 8:03









Jehu314

356




356







  • 1




    What kind of limit is that?
    – Lord Shark the Unknown
    Jul 28 at 9:02










  • An inverse limit.
    – Jehu314
    Jul 28 at 9:17










  • An inverse limit? Over what ordered set/category?
    – Lord Shark the Unknown
    Jul 28 at 9:18










  • The category of the Basic open sets and the restriction maps.
    – Jehu314
    Jul 28 at 9:21










  • I got this idea from the fact that for any sheaf $mathscr F$, $mathscr F(U)=lim_U_isubset U mathscr F(U_i)$ for any open covering $U_i$ of $U$
    – Jehu314
    Jul 28 at 9:26













  • 1




    What kind of limit is that?
    – Lord Shark the Unknown
    Jul 28 at 9:02










  • An inverse limit.
    – Jehu314
    Jul 28 at 9:17










  • An inverse limit? Over what ordered set/category?
    – Lord Shark the Unknown
    Jul 28 at 9:18










  • The category of the Basic open sets and the restriction maps.
    – Jehu314
    Jul 28 at 9:21










  • I got this idea from the fact that for any sheaf $mathscr F$, $mathscr F(U)=lim_U_isubset U mathscr F(U_i)$ for any open covering $U_i$ of $U$
    – Jehu314
    Jul 28 at 9:26








1




1




What kind of limit is that?
– Lord Shark the Unknown
Jul 28 at 9:02




What kind of limit is that?
– Lord Shark the Unknown
Jul 28 at 9:02












An inverse limit.
– Jehu314
Jul 28 at 9:17




An inverse limit.
– Jehu314
Jul 28 at 9:17












An inverse limit? Over what ordered set/category?
– Lord Shark the Unknown
Jul 28 at 9:18




An inverse limit? Over what ordered set/category?
– Lord Shark the Unknown
Jul 28 at 9:18












The category of the Basic open sets and the restriction maps.
– Jehu314
Jul 28 at 9:21




The category of the Basic open sets and the restriction maps.
– Jehu314
Jul 28 at 9:21












I got this idea from the fact that for any sheaf $mathscr F$, $mathscr F(U)=lim_U_isubset U mathscr F(U_i)$ for any open covering $U_i$ of $U$
– Jehu314
Jul 28 at 9:26





I got this idea from the fact that for any sheaf $mathscr F$, $mathscr F(U)=lim_U_isubset U mathscr F(U_i)$ for any open covering $U_i$ of $U$
– Jehu314
Jul 28 at 9:26
















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