Homology of I-adic towers

2004/11/09 by Wuethrich, Samuel
#18E30 (Secondary) #18G10 #18G15 (Primary) 18G25 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0411187

Abstract

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.

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