turkmath.org

Tensor products are important and ubiquitous objects in mathematics, and they are used in many application areas, including physics and engineering. In 1961 Auslander initiated the study of torsion in tensor products of modules. In particular Auslander proved that (Lichtenbaum (1966) in the ramified case), if R is a regular local ring and the tensor product of two nonzero finitely generated R-modules M and N is torsion-free, then M and N are both torsion-free. In 1994 Huneke and Wiegand proved a partial extension of the aforementioned result of Auslander, and obtained the following which is known as the second rigidity theorem: if R is a local hypersurface ring and the tensor product of two nonzero finitely generated R-modules M and N is reflexive, where N has rank (for example, N has finite projective dimension), then M is reflexive. The theme of the talk is the following question which stems from Huneke and Wiegand (2007): If R, M, and N are as in the second rigidity theorem, then are M and N both reflexive? I will discuss an example  which gives a negative answer to the question. If there is time, I also plan to mention some works that yield affirmative answer to the question under certain additional conditions.