turkmath.org

The tensor product of two nonzero finitely generated modules over a local ring (commutative and Noetherian), even when the modules in question are nice (e.g., torsion-free), is not typically well-behaved (e.g., has nonzero torsion). Hence the assumption that there is a nonzero torsion-free tensor product of modules influences the structure of the modules in the tensor product, as well as that of the ring in question. This connection over regular local rings made its first appearance in the seminal paper of Maurice Auslander, Modules over unramified regular local rings (Illinois J. Math. 5, 1961). Subsequently Craig Huneke and Roger Wiegand extended and studied Auslander’s results over hypersurface rings in their prominent paper, Tensor products of modules and the rigidity of Tor (Math. Ann. 299, no. 3, 1994), and did pioneering work on the study of torsion in tensor products of modules.

Studies in this direction by Auslander, Huneke and Wiegand, and other researchers, yielded several questions, which still remain open, on how nonzero torsion arises in tensor products of modules over certain local rings. One such example is a long-standing conjecture of Huneke and Wiegand, which is concerned with the torsion submodule of tensor products of modules with their algebraic duals. Over Gorenstein rings, an affirmative answer to this conjecture would imply an affirmative answer to a celebrated conjecture of Maurice Auslander and Idun Reiten on the vanishing of Ext for maximal Cohen-Macaulay modules, a conjecture which was initially proposed over finite dimensional algebras in the 1970s.

In this talk I plan to partly survey results on the existence of torsion in tensor products. In particular I plan to discuss some of the recent work done that corroborate the aforementioned conjecture of Huneke and Wiegand.

Link to the seminar is:

https://teams.microsoft.com/l/meetup-join/19%3a0ae91f7b86a24e8fa1d818a6f74b22a1%40thread.tacv2/1670068534385?context=%7b%22Tid%22%3a%22066690f2-a8a6-4889-852e-124371dcbd6f%22%2c%22Oid%22%3a%22a343f6fd-86f8-4abe-95cf-3c7f4ad5f0ca%22%7d