An Ext-Tor duality theorem, cohomological dimension, and applications

We provide a duality theorem between Ext and Tor modules over a Cohen-Macaulay local ring possessing a canonical module, and use it to prove some freeness criteria for finite modules. The applications include a characterization of codimension three complete intersection ideals and progress on a long-held multi-conjecture of Vasconcelos. By a similar technique, we furnish another theorem which in addition makes use of the notion of cohomological dimension and is mainly of interest in dimension one; as an application, we show that the (still open) complete intersection case of the celebrated Huneke-Wiegand conjecture holds true provided that a single finiteness condition is satisfied.