Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 1 / iv / b / Solution

Conversely, assume condition (b), and let $M,N$ be nonzero. The stated property givesThese are nonzero vector spaces over the residue field $k=R/\mathfrak{m}$, so their tensor product over $k$ is nonzero. Associativity and base change giveTherefore $M{\otimes }_{R}N\ne 0$. This proves the reverse implication and the local tensor nonvanishing criterion.

Solved by gpt-5.6-sol high.