dumb latex question: is there a good way to overset arrows in latex. I've been using xrightarrow but it looks like trash

I use xymatrix (for commutative diagrams in general)

xypic

that's not quite what I want it for. I want it in the text, or as a string of composable arrows in a displayed "equation." tikzcd (what I use) tends to make the arrows too long for this usage

but you can use xymatrix also outside of diagrams. You can regulate the length as well

What's the difference between a sheaf of modules over a sheaf of rings and a sheaf with values in a module category?

I know the former has a more structure but when people say a sheaf of modules I keep thinking about the second

@PeterNelson In AMS-TeX there's a command called \overset, which could be what you're looking for. tex.wikidot.com/snippets:overset-and-underset

iirc, overset doesn't change the length of the arrow if the label is long, so also looks like trash

If $A$ is a dg-algebra over a ring $k$, and let $P_A$ be the dg-category of perfect $A$-modules. What is the cleanest way to prove that the derived categories of dg-$A\otimes A^{op}$-modules and dg-$P_A\otimes P_A^{op}$-modules are equivalent?

@Sarah Well, if the sheaf of rings is the constant sheaf then there's no difference I believe. Of course, a quasicoherent sheaf on a scheme is a whole other kind of beast

@Sarah In the sheaf of modules over sheaf of rings definition the ring over which the modules are defined varies with the open subset: it is the set of sections of the ring sheaf over this subset. In the sheaf in module category definition the base ring is assumed roughly the same for all subsets, i.e. it corresponds to a constant sheaf of rings as Denis says above (note that over e.g. a disjoint union of two subsets we have pairs of modules, which are a module over a pair of base rings).

Say a functor is a localization if it is equivalent to a category of fractions construction $C\to C[S^{-1}]$.

It seems to be well known that any adjoint to a fully faithful functor (left or right) must be a localization.

However, it also seems to be the case that the complementary fact holds: any adjoint to a localization (left or right) must be fully faithful.

Is there a reference for this?

Also, the above with $\infty$ stuck on.