12 - Adjunctions, Equivalences and Cohomological Dimension

Summary

This chapter is quite varied. In Sections 12.2 and 12.3, there is a detailed look at the derived bifunctors RHom(-,-) and (-⊗L-). In Section 12.4 we study cohomological dimensions of functors. They are used in Section 12.5 to prove some theorems about triangulated functors, such as a sufficient condition for a morphism ζ : F → G of triangulated functors to be an isomorphism. In Sections 12.6 and 12.7 we study several adjunction formulas that involve the derived bifunctors RHom(-,-) and (-⊗L-). We define derived forward adjunction and derived backward adjunction. We prove that if A → B is a quasi-isomorphism of DG rings, then the restriction functor D(B) → D(A) is an equivalence. Resolutions of DG rings are important in several contexts. In Section 12.8 we prove that given a DG K-ring A, there exists a noncommutative semi-free DG ring resolution à → A over K. In Subsection 12.9 there is a theorem providing sufficient conditions for the derived tensor-evaluation morphism to be an isomorphism.In Subsection 12.10 we present some adjunction formulas that pertain only to weakly commutative DG rings.