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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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regarded as a module over $R\otimes R$ if $M$ is an $R$-module or do you have extra properties? – Julian Kuelshammer Aug 29 '12 at 15:20
yes, regarded as a module over R⊗R if M is an R -module – Necati Olgun Aug 29 '12 at 15:23
Is it true generally? If M is projective or flat module it is true. – Necati Olgun Aug 29 '12 at 15:25

This is called Künneth formula and depends on the ring over which you are tensoring your modules, see e.g.

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