Theorem 35.4.22. The following conditions are equivalent.
The morphism f is a descent morphism for modules.
The morphism f is an effective descent morphism for modules.
The morphism f is universally injective.
Theorem 35.4.22. The following conditions are equivalent.
The morphism f is a descent morphism for modules.
The morphism f is an effective descent morphism for modules.
The morphism f is universally injective.
Proof. It is clear that (b) implies (a). We now check that (a) implies (c). If f is not universally injective, we can find M \in \text{Mod}_ R such that the map 1_ M \otimes f: M \to M \otimes _ R S has nontrivial kernel N. The natural projection M \to M/N is not an isomorphism, but its image in DD_{S/R} is an isomorphism. Hence f^* is not fully faithful.
We finally check that (c) implies (b). By Lemmas 35.4.16 and 35.4.20, for (M, \theta ) \in DD_{S/R}, the natural map f^* f_*(M,\theta ) \to M is an isomorphism of S-modules. On the other hand, for M_0 \in \text{Mod}_ R, we may tensor (35.4.18.1) with M_0 over R to obtain an equalizer sequence, so M_0 \to f_* f^* M_0 is an isomorphism. Consequently, f_* and f^* are quasi-inverse functors, proving the claim. \square
Comments (2)
Comment #8763 by Dongryul Kim on
Comment #9308 by Stacks project on
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