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All rings and algebra in this question are commutative and contains unity.

Suppose $M$ is an $A$ module and $A$ a $R$ algebra. If $pd_R(M) < \infty$, then will that imply $pd_A(M) < \infty$? In particular if $M = \frac{R[X_1, X_2, \ldots,X_n]_N}{I}$, $A = R[X_1, X_2, \ldots,X_n]_N$ where $R[X_1, X_2, \ldots,X_n]$ is polynomial algebra , $N$ a multiplicative closed set and $I$ an ideal, is the answer affirmative?

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    If $R$ is the coordinate ring of an affine algebraic variety over an algebraically closed field $k$, then the localizations $R_{\mathfrak{m}}$ are $k$-algebras. The projective dimension of $k$ over $k$ is zero. However, the projective dimension of $k$ over $R_{\mathfrak{m}}$ (with $\mathfrak{m}$ acting by zero) is finite if and only if the corresponding point of the variety is smooth.2011-02-24

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