Timeline for left adjoint to restriction functor

6 events

when toggle format	what		by	license	comment
May 25, 2012 at 15:25	vote	accept	gholam
May 14, 2012 at 9:30	comment	added	Martin Brandenburg		Yes, but this is a special case. When $X \setminus U$ is a disjoint union of opens, then it is open ...
May 14, 2012 at 9:24	comment	added	Yosemite Sam		does the decomposition of QC(X) you described for clopens work for an infinite number of them? (I'm thinking of things like the Hilbert scheme which decomposes according to topological data of the parameterised subscheme, that's a countable disjoint union)
May 13, 2012 at 9:05	comment	added	Martin Brandenburg		Well it is true when $U$ is a clopen subset of $X$, because then $\mathrm{Qcoh}(X) = \mathrm{Qcoh}(U) \times \mathrm{Qcoh}(X \setminus U)$. But otherwise probably it's only true in pathological cases. This is already visible in the affine case, say $X=\mathrm{Spec}(A)$ and $U=D(f)$ for some $f \in A$. When does $M \mapsto M_f$ preserve infinite products? Almost never.
May 13, 2012 at 8:30	comment	added	Yosemite Sam		can we add some assumptions on U or X to make this true? (or is U=X the only case when this works?)
May 13, 2012 at 7:45	history	answered	Martin Brandenburg	CC BY-SA 3.0