Let $M$ be a smooth manifold. 


I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on $M$). Sadly the best results I could find showed at best that pseudo-differential form a sheaf of left $C^{\infty}$-modules without treating the question of whether the multiplication is well defined nevermind associative. So my question is rather simple:



> 
> Do pseudo-differential operators form a sheaf of (associative?) $\mathbb{C}$-algebras on $M$?
>  If not, what fails?
> 
> 
> 


I'm pretty sure that if one considers pseudo-differential operators modulo smothing operators than these do form a sheaf of algebras (However a precise reference here would be welcome). As for the entire space of pseudo-differential operators this seems rather non-trivial to me (I'm actually not entirely sure whether this should be true or not). I apologize if this is too elementary for this site.



Do pseudo-differential operators form a sheaf of algebras?
Pseudodifferential operators are operators whose Schwartz kernel
is smooth outside of the diagonal and is conormal with respect to the diagonal
(differentiating finitely many times with respect to smooth vector
fields tangent to the diagonal produces a distribution in a certain Besov space).


From this definition one immediately sees that pseudodifferential operators do form a presheaf of abelian groups, albeit in a nonobvious way: a Schwartz kernel on M×M
can be restricted to U×U.
But there is no way this could be a sheaf: knowing restrictions to U×U
and V×V such that U and V cover M does not tell what happens on M×M
outside of U×U and V×V.


Sheafification produces germ equivalence classes of conormal distributions
near the diagonal.


This is a larger sheaf than equivalence class of pseudodifferential
operators modulo smoothing operators (which do form a sheaf for trivial reasons).
Indeed, all germs of smooth functions near the diagonal
produce different elements in the sheafification,
but become equal once one mods out smoothing operators.


The above talks about sheaves of abelian groups.
Pseudodifferential operators cannot always be composed
(and accordingly do not form presheaves of algebras);
one needs conditions like properness of support.


However, the sheafification and the equivalence classes modulo smoothing operators both do form sheaves of algebras.


