HOMOLOGY OF POWERS OF REGULAR IDEALS

WÜTHRICH, SAMUEL

Glasgow mathematical journal, Tome 46 (2004) no. 3, pp. 571-584

Résumé

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 $I^s$ and of $R/I^s$ for $s \geq 1$ and which generalise the Koszul resolution. We derive these from a certain multiplicative double complex ${\mathbf K}$. By means of a Cartan–Eilenberg spectral sequence we express ${\rm Tor}_*^R(R/I, R/I^s)$ and ${\rm Tor}_*^R(R/I, I^s)$ 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 ${\rm Tor}_*^R(R/I, I^s)\,{\otimes_R}\,{\rm Tor}_*^R(R/I, I^t)\,{\to}\,{\rm Tor}_*^R(R/I, I^{s+t})$. This paper is based on ideas by Andrew Baker; it is written in view of applications to algebraic topology.

Détail

DOI : 10.1017/S0017089504002009

Mots-clés : 13D02, 13D07, 55U15