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?
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This is called Künneth formula and depends on the ring over which you are tensoring your modules, see e.g. https://www.encyclopediaofmath.org/index.php/K%C3%BCnneth_formula