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Results tagged with sheaf-theory
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user 75
For questions about sheaves on a topological space.
21
votes
Accepted
Are subfunctors of left exact functors also left exact?
Here's a counterexample with additive functors on abelian categories. If $A$ is an abelian group, let $F(A)$ denote the subgroup of elements that are divisible by $2$. It is easy to see that $F:Ab\t …
- 31.2k
17
votes
Why is the rank of a locally free sheaf well-defined?
Locally free means not just that $F(U)$ is a free $O_X(U)$-module for an open cover of $U$'s, but that as a sheaf the restriction $F|_U$ is isomorphic to a direct sum of copies of $O_X|_U$. In partic …
- 31.2k
8
votes
When does the sheaf direct image functor f_* have a right adjoint?
If $f_\ast$ has a right adjoint, it must preserve colimits and hence be right-exact. Thus a necessary condition is that the higher derived functors vanish. In particular, when everything is affine an …
- 31.2k
8
votes
Accepted
Exact sequence of groups to exact sequence of sheaves
The following is perhaps more of an extended comment than an answer. The sequence of sheaves is exact iff the quotient map $G\to H$ has a section over a neighborhood of every point (in fact, because …
- 31.2k
6
votes
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
This isn't entirely complete, but here are some results. Exercise III.4.11 in Hartshorne gives that whenever a sheaf is acyclic on any intersection of sets in a cover, Cech cohomology agrees with the …
- 31.2k