Remark 15.89.19. The equivalences of Proposition 15.89.15, Theorem 15.89.17, and Proposition 15.89.18 preserve properties of modules. For example if $M$ corresponds to $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ then $M$ is finite, or finitely presented, or flat, or projective over $R$ if and only if $M'$ and $M_ i$ have the corresponding property over $S$ and $R_{f_ i}$. This follows from the fact that $R \to S \times \prod R_{f_ i}$ is faithfully flat and descend and ascent of these properties along faithfully flat maps, see Algebra, Lemma 10.83.2 and Theorem 10.95.6. These functors also preserve the $\otimes $-structures on either side. Thus, it defines equivalences of various categories built out of the pair $(\text{Mod}_ R, \otimes )$, such as the category of algebras.

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