?? functor taking "generalized toric variety" to ordinary toric variety ?? ... ???? ... analogous to functor taking affine vsp to its translation vector space .... ?????......

?? comm monoid under (k,mult) ... ???.... cokernel functor ??? ...

?? but ... what happens under such functor to morphism taking main torus to non-main ??? ..... ????

?? hmmm ... (k,mult) vs gl(1,k) here .... ???? ..... "killing off 0" (in sense of setting it to 1 .... ???!!??? ...) as "dangerous" .... ??? .....