If the projective dimension of an $R$-module $M$ is finite, then can we say that projective dimension of tensor product $M\otimes M$ (as an $R\otimes R$-module) is finite?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
Here's how it works:
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
This is called Künneth formula and depends on the ring over which you are tensoring your modules, see e.g. http://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula