??so what about something about .... ????... ??"derivation at a pair of points" ... ??or something ... ???as something about .... ?????isomorphism class of short exact sequences .... ????or something ?????? ...... ?????....... ????....

????...

???something about .... ?"hall algebra" ... ???? .... ???

???something about "symmetric" vs "anti-symmetric" pattern here ???? .... ???or something ??? .... and so forth .... ????.....

??something about ... ext(m,m) ... ????..... ????something about "derivation" .... ?????? or something???? ...... .... ??????.....

hmmmm.....

"deformation" ....

"normal bundle" .... ????.....

??????.........

??so consider ... ???commutative k-algebra... ideal power filtration at a maximal (or something....) ideal .... corresponding to point ... ?? .... ????something about "maximal in maximal" ideal corresponding to tangent vector .... 2-stage fitration on quotient algebra ... ... associated graded... ???something about "fake tangent space" ... ??or something?? ... well, not quite, i guess, but ... ???.....

??hmm, something about the extension module as symmetric monoidal in the "tangent vector" case, but not the other case ... ?????or something ??? .... .... ??...

??so what about this concept of "derivation twisted by an automorphism" that baez mentioned?? ... hmmm... ??...

??so suppose that the automorphism is, for example, (x,y) |-> (-x,-y) ... ???

or also (x,y) |-> (-x,y) ???...

...and so forth ...

d(fg)=d(f)g(p)+f(q)d(g) ....

d(fg)=d(f)g(p)+f(-p)d(g) ... ??

d(f1)=d(f)+f(-p)d(1) ... ??

f = 1 ... ?? d(1) = d(1) + d(1) ... ??so d(1) = 0 ???

d("x^n" "x") = d("x^n") + "(-x)^n" d("x")

d("x^[n+1]") = d(x^[n]) + (-x)^n a ???

0
a
a - ax
a - ax + ax^2
a - ax + ax^2 - ax^3

???or something ??...

??hmmm... some idea that i had somewhere up above (and/or elsewhere...??...) seems confused now... backwards or something...

homomorphism from smooth functions on manifold m to upper-triangular 2x2 matrixes ... something about ... ??ext between skyscraper sheaves, or something ... ???...

??hmm, or_does_ it make some sense... ?? module of the algebra of smooth functions, with 2d underlying vector space ... ???and so forth, or something???...