etale ... descent ... geometric morphism ...

consider for example projective modules over z[1/3] .... or something like that ...

suppose that we have a morphism of symmetric monoidal finitely co-complete algebroids from the algebroid of representations of a finite abelian group g to the algebroid of z[1/3]-modules...

this was supposed to help us in thinking about the relationship between "galois stacks" and algebroids of coherent sheaves over a base scheme ... ??or something like that... ??and maybe it's actually working, sort of?? ...

let's consider for example z[6^[1/2]], or something like that...

no wait... let's consider some cyclotomic extension of z... well in fact we could have used z[6^[1/2]], but that example i tend to use as a "base", since it's got a nice explicit non-trivial line bundle over it... or something like that...

so perhaps try thinking about the "20" cyclotomic extension of z, for example ...
and then we have some idea of where this is ramified... "at 20", or something like that, because 20 is the discriminant, or something like that... ??though do we have to worry about ramification at archimedean primes here?? or something like that.... ???... ??maybe to avoid such trouble we should focus on some "real" sub-extension, or something like that...

hmm, in a slightly different direction... (or maybe really more or less the same direction, but pushing the idea further... to the point where it should probably really break if it's ever going to...) suppose that we consider morphisms of symmetric monoidal finitely cocomplete algebroids (and equivalences between such morphisms...) from the algebroid of representations of for example a finite abelian group to the algbebroid of vector spaces over, for example, the algebraic numbers, or the complex numbers, or something...

i'm still confused here... "vector space over the field of algebraic numbers" sounds so boring... whereas "representation of the etale fundamental group of spec(q)" (or something like that...) sounds interesting ... ???....

??am i making some level slip here?? something about endo-equivalences of the identity functor on the algebroid of modules of a commutative ring .... ???or something????...... ??what about models of the theory here???? ..... ????...... ??maybe we're dealing with theories that 'lack classical models" ?? .... ??how does this relate to some ideas that we were thinking about recently about certain "galois stacks" corresponding to "pure property theories" ?? ??or something like that???.....