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At p. 107 of the book Cohen-Macaulay Rings by Bruns and Herzog, the authors write

"any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, simply because $R$ has finite injective dimension by definition."

Could someone please explain why this conclusion is true?

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Your question is Exercise 3.1.25 (the easy part) from the same book. Note that any free module of finite rank has finite injective dimension, and then look at a finite free resolution of the module (that is, decompose it in short exact sequences) starting from the left.

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Because a Noetherian ring $R$ has finite injective dimension if and only all R-modules with finite projective dimension also have finite injective dimension. See the proof of this on page $8$, and also the Theorem below there in http://www.math.hawaii.edu/~lee/homolog/Goren.pdf.

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