Lemma 15.56.1. Let R \to S be a flat ring map. If I^\bullet is a K-injective complex of S-modules, then I^\bullet is K-injective as a complex of R-modules.
15.56 Derived categories of modules
In this section we put some generalities concerning the derived category of modules over a ring.
Let A be a ring. The category of A-modules is denoted \text{Mod}_ A. We will use the symbol K(A) to denote the homotopy category of complexes of A-modules, i.e., we set K(A) = K(\text{Mod}_ A) as a category, see Derived Categories, Section 13.8. The bounded versions are K^+(A), K^-(A), and K^ b(A). We view K(A) as a triangulated category as in Derived Categories, Section 13.10. The derived category of A, denoted D(A), is the category obtained from K(A) by inverting quasi-isomorphisms, i.e., we set D(A) = D(\text{Mod}_ A), see Derived Categories, Section 13.111. The bounded versions are D^+(A), D^-(A), and D^ b(A).
Let A be a ring. The category of A-modules has products and products are exact. The category of A-modules has enough injectives by Lemma 15.55.9. Hence every complex of A-modules is quasi-isomorphic to a K-injective complex (Derived Categories, Lemma 13.34.6). It follows that D(A) has countable products (Derived Categories, Lemma 13.34.2) and in fact arbitrary products (Injectives, Lemma 19.13.4). This implies that every inverse system of objects of D(A) has a derived limit (well defined up to isomorphism), see Derived Categories, Section 13.34.
Proof. This is true because \mathop{\mathrm{Hom}}\nolimits _{K(R)}(M^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(S)}(M^\bullet \otimes _ R S, I^\bullet ) by Algebra, Lemma 10.14.3 and the fact that tensoring with S is exact. \square
Lemma 15.56.2. Let R \to S be an epimorphism of rings. Let I^\bullet be a complex of S-modules. If I^\bullet is K-injective as a complex of R-modules, then I^\bullet is a K-injective complex of S-modules.
Proof. This is true because \mathop{\mathrm{Hom}}\nolimits _{K(R)}(N^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(S)}(N^\bullet , I^\bullet ) for any complex of S-modules N^\bullet , see Algebra, Lemma 10.107.14. \square
Lemma 15.56.3. Let A \to B be a ring map. If I^\bullet is a K-injective complex of A-modules, then \mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet ) is a K-injective complex of B-modules.
Proof. This is true because \mathop{\mathrm{Hom}}\nolimits _{K(B)}(N^\bullet , \mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{K(A)}(N^\bullet , I^\bullet ) by Algebra, Lemma 10.14.4. \square
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