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Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...

Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable. Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials. Let also $\widehat{R}$ be the ring of ...

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...