so let's try taking some nice simple examples of graded commutative algebras with maximal ideals corresponding to mildly (or something...) stacky points, and then try to understand the symmetric monoidal algebroid of fp graded modules of the quotient algebra by the square of the ideal. or something like that.

for example let's try "the orbit stack of gl(1) acting on k^1". so we have one generating grade g, and one generating element x in grade g. and let's consider the ideal generated by x, and the square of that ideal. and so it seems like this is an example that we've thought about before, where the graded modules end up being z-graded chain complexes, with the usual convolutional tensor product but without the usual sign-flip in the symmetry maps. and this is weirdly suggestive in a number of ways but we're not really sure what to make of it yet...

let's try another example. one generating grade g, generators x in grade 2 and y in grade 3. maximal ideal generated by x. a graded module of the quotient algebra is ...

hmm, so wait, that's not quite i wanted this last example to be... i don't want to just set x to zero; i also want to invertibilize y... or something like that... in which case a graded module now seems to be... a triple of vector spaces... with the tensor product being essentially the usual convolutional tensor product of "co-modules" of z/3, with the usual symmetry... and now if we instead mod out by the square of the ideal, while still invertibilizing y, then the graded modules now seem to be "chain complexes of period 3" ... again with the expected tensor product but the slightly unexpected symmetry.

hmm...