Homology of I-adic towers
  Let R be a commutative ring with unit and let I be an ideal generated by a regular sequence. Then it is known that the natural sequences   0-> Tor_*^R(R/I,I^s)-> Tor_*^R(R/I,I^s/I^{s+1})-> Tor_{*-1}^R(R/I,I^{s+1})-> 0 are short exact sequences of graded free R/I-modules, for any s>=0. The aim of this paper is to give a proof which accounts for the structural simplicity of the statement. It relies on a minimum of technicalities and exposes the phenomenon in a transparent way as a consequence of the regularity assumption. The ideas discussed here are used in math.AT/0411409 to obtain a better qualitative understanding of I-adic towers in algebraic topology. 
Samuel Wuethrich