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 define by 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 ar...