Dear Kevin,
This is more or less an amplification of Tyler's comment. You shouldn't take it too seriously, since I am certainly talking outside my area of expertise, but maybe it will be helpful.
My understanding is that homotopy theorists are extremely (perhaps primarily) interested in torsion phenomena. (After all,
homotopy groups are often non-trivial but finite.) TMF, for example, involves quite subtle torsion phenomena. Coupled with Tyler's remark that homotopy theorists have no fear of $E_{infty}$ rings, and so are (a) happy to identify them
with dg-algebras in char. zero, and (b) don't feel any psychological need to fall back on
the crutch of dg-algebras, this makes me suspect that your assumption (1) is likely to be wrong. (I share your motivation (2), but this is a psychological weakness of algebraists that
homotopy theorists seem to have overcome!)
In particular, one of Lurie's achievements is (I believe) constructing equivariant versions of TMF,
which (as I understand it) involves (among other things) studying deformations of $p$-divisible groups of derived elliptic curves. It seems hard to do this kind of thing
without having a theory that can cope with torsion phenomena.
Also, when Lurie thinks about elliptic cohomology, he surely includes under this umbrella TMF and its associated torsion phenomena. (So your (3) may not include all the aspects
of elliptic cohomology that Lurie's theory is aimed at encompassing.)
No comments:
Post a Comment