Homology of powers of regular ideals

Original · EN

For a commutative ring R with an ideal I, generated by a finite regular sequence, we construct differential graded algebras which provide R-free resolutions of Iˢ and of R/Iˢ for s>0 and which generalise the Koszul resolution. We derive these from a certain multiplicative double complex. By means of a Cartan-Eilenberg spectral sequence we express Tor*ʳ(R/I,R/Iˢ) and Tor*ʳ(R/I, Iˢ) in terms of exact sequences and find that they are free as R/I-modules. Except for R/I, their product structure turns out to be trivial; instead, we consider an exterior product. The paper is based on ideas by Andrew Baker; it is written in view of applications to algebraic topology.