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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

1 Answer 1

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

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