Flat resolvents, minimal injective resolutions and negative torsion functors

Abstract

The functors , an integer originated in algebraic topology but quickly became standard algebraic tools especially useful in commutative algebra. The values of these functors are computed using either a projective resolution of M or one of N, or the double complex that one gets by tensoring these two resolutions.

There has long been an obvious way to defineby using an injective resolution of N. However, it was not clear what sort of resolutions of .M could be used to compute these Tor's. There is now a method for getting such resolutions of M.. Most of the applications we then have result from the two different ways that can be used to compute the same object. Given the above, it is time to study these so-called negative torsion functors. For example, the existence of long exact sequences, commutativity and associativity questions and vanishing problems are all of interest.

The resolutions of M which are needed involve the notions of flat envelopes and pre-envelopes introduced by Edgar E. Enochs. These are defined in a manner which is categorically similar to the definitions of injective envelopes. Weinvestigate properties of these flat envelopes and pre-envelopes and apply our results to the negative torsion functors.

In case R is a two-sided noetherian ring and M is finitely generated, we get what we call complete resolutions of M. In case the ring is the group algebra IG with G a finite group and M=Zwith trivial group action, we get as a special case the complexes used to compute Tate homology. We give a natural generalization of Tate homology and also a new way to compute it.