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We begin by proving that if K is a triangulated category and S ⊆ K is a denominator set of cohomological origin, then the localized category KS is triangulated and the localization functor Q : K → KS is triangulated. In the case of the triangulated category K(A,M) and the set of quasi-isomorphisms S(A,M) in it, we get the derived category D(A,M) := K(A,M)S(A, M) and the triangulated localization functor Q : K(A,M) → D(A,M). We look at the full subcategories of K(A, M) corresponding to boundedness conditions and the corresponding derived categories. We prove that the obvious functor M → D(M) is fully faithful. The section ends with a study of the triangulated structure of the opposite derived category D(A,M)op .
A good understanding of DG (differential graded) algebra is essential in our approach to derived categories. By DG algebra, we mean DG rings, DG modules, DG categories and DG functors. The first section is on cohomologically graded rings and modules, with a discussion of the monoidal braiding (i.e. the Koszul sign rule). After that we study DG rings, DG modules and operations on them. We go on to discuss DG categories, DG functors between them and morphisms between DG functors. We recall the DG category C(M) of complexes in an abelian category M. A new feature we introduce is the DG category C(A,M) of DG A-modules in M, where A is a DG ring and M is an abelian category. This includes as special cases the category C(M) mentioned above, and the category C(A) of DG A-modules over a DG ring A. Another new feature is the distinction between the DG category C(A,M) and its strict subcategory Cstr(A,M), whose morphisms are the degree 0 cocycles, and it is an abelian category.
The focus of this chapter is on balanced NC dualizing complexes (DC). Let A be a noetherian connected NC graded ring over the base field K, with enveloping ring Aen = A ⊗K Aop. A complex R ∈ D(Aen,gr) is called a graded NC DC if its cohomology is bounded and finite both sides; it has finite graded injective dimension on both sides; and it has NC derived Morita property (see abstract of Chapter 13) on both sides. A balanced NC DC over A is a pair (R,β), where R is a graded NC DC over A with symmetric derived m-torsion, and β : RΓm(R) → A* is an isomorphism in D(Aen,gr). A balanced DC (R,β) is unique up to a unique isomorphism, and it satisfies the NC Graded Local Duality Theorem. We prove that A has a balanced DC iff A satisfies the χ condition and has finite local cohomological dimension. If A is an Artin--Schelter (AS) regular graded ring, then it has a balanced DC R = A(φ,-l)[n], a twist of the bimodule A by an automorphism φ and integers -l and n.
We define K-injective and K-projective DG modules in K(A,M) and also K-flat DG modules in K(A). These constitute full triangulated subcategories of K(A,M), and we refer to them as resolving subcategories. The category K(A,M)inj of K-injectives in K(A,M) plays the role of the category J in the abstract of Chapter 8, and the category of K-projectives K(A,M)prj plays the role of the category P there. The K-flat DG modules are resolving for the tensor functor. Furthermore, we prove that the functors Q : K(A,M)inj → D(A,M) and Q : K(A,M)prj → D(A,M) are fully faithful.
Let A be a noetherian commutative ring. A complex R ∈ D(A) is called a dualizing complex (DC) if it has bounded finitely generated cohomology, finite injective dimension, and the derived Morita property, which says that the derived homothety morphism : A → RHomA(R, R) in D(A) is an isomorphism. We prove uniqueness of DCs and existence when A is essentially finite type over a regular noetherian ring. A residue complex is a DC that consists of injective modules of the correct multiplicity in each degree. There is a stronger uniqueness property for residue complexes. To understand residue complexes, we review the Matlis classification of injective A-modules. In the last two sections we talk about Van den Bergh rigidity. We prove that if A has a rigid DC R, then it is unique up to a unique rigid isomorphism. Existence of a rigid DC is harder to prove, and we just give a reference to it. Rigid residue complexes always exist, and they are unique in a very strong sense. We end this chapter with remarks that explain how rigid residue complexes allow a new approach to residues and duality on schemes and Deligne--Mumford stacks.
In this chapter we recall the material on abelian categories and additive functors that is needed for the book. There is one new topic here: our sheaf tricks, which are a “cheap substitute” for the Freyd--Mitchell Theorem.
In this section we take a close look at localization of categories. Let K be an abstract category (i.e. without any extra structure), and let S ⊆ K be a multiplicatively closed set of morphisms. The localization of K w.r.t. S is a category KS equipped with a functor Q : K → KS, such that the morphisms in Q(S) are invertible and the functor Q is universal for this property. We give a detailed proof of the fundamental theorem on localization: Q is an Ore localization iff S is a denominator set. We then prove that the localization KS of a linear category K at a denominator set S is a linear category too, and the localization functor Q : K → KS is linear.
This chapter is devoted to DG and triangulated bifunctors. We prove that a DG bifunctor between categories of DG modules induces a triangulated bifunctor on the homotopy categories. Then we define left and right derived bifunctors, and prove their uniqueness and existence, under suitable conditions. Bifunctors that are contravariant in one or two of the arguments are also studied.
We talk about the translation (or shift or suspension) functor and standard triangles in the DG category C(A,M). The translation T(M) of a DG module M is the usual one. A calculation shows that T is a DG functor from C(A,M) to itself. We introduce the degree -1 morphism tM : M → T(M), called the little t operator, which facilitates many calculations.
A morphism φ : M → N in Cstr(A,M) gives rise to the standard Cone(φ) = N ⊕ T(M) , whose differential is a matrix involving the degree 1 morphism φ ◦ (tM)-1. The standard cone sits inside the standard triangle associated to φ. A DG functor F : C(A,M) → C(B,N) gives rise to a T-additive functor F : Cstr(A,M) → Cstr(B,N), and it sends standard triangles in Cstr(A,M) to standard triangles in Cstr(B,N).
This chapter, as well as Chapters 16 and 17, are on algebraically graded rings, which is our name for graded rings that have lower indices and do not involve the Koszul sign rule (in contrast with the cohomologically graded rings that underlie DG rings). Simply put, these are the usual graded rings that one encounters in textbooks on algebra. With few exceptions, the base ring K in the four final chapters of the book is a field. Let A be an algebraically graded ring. The category of algebraically graded A-modules is M(A,gr). Its morphisms are the A-linear homomorphisms of degree 0. We talk about finiteness in the algebraically graded context and about various kinds of homological properties, such as graded-injectivity. Special emphasis is given to connected graded rings. The category of complexes with entries in the abelian category M(A,gr) is the DG category C(A,gr) := C(M(A,gr)). Its objects are bigraded, by cohomological degree and algebraic degree. The strict subcategory of C(A,gr) is Cstr(A,gr). The derived category is the triangulated category D(A,gr) := D(M(A,gr)). We present the algebraically graded variants of K-injective resolutions and the relevant derived functors.
Here we talk about derived functors. To make the definitions precise, we introduce 2-categorical notation. Suppose K and E are abstract categories, F : K → E is a functor and S ⊆ K is a multiplicatively closed set of morphisms. In this context we define the left and right derived functors of F w.r.t. S. These derived functors RF, LF : KS → E have universal properties, making each unique up to a unique isomorphism. Then we provide a general existence theorem for right and left abstract derived functors, in terms of the existence of suitable resolving subcategories J, P ⊆ K, respectively. In Section 8.4 we specialize to triangulated derived functors. Here K and E are triangulated categories, F : K → E is a triangulated functor and S ⊆ K is a multiplicatively closed set of cohomological origin. The right and left derived functors RF, LF : KS → E are defined like in the abstract setting, and their uniqueness is also proved the same way. Existence requires resolving subcategories P and J that are full triangulated subcategories of K. The chapter is concluded with a discussion of contravariant triangulated derived functors.
Let A be a connected graded ring over the base field K, with augmentation ideal m. In this chapter we study derived m-torsion, both for complexes of graded A-modules and for complexes of graded bimodules. The graded bimodule A* := HomK(A,K) is graded-injective and m-torsion on both sides. One of the main results is on the representability of the right derived m-torsion functor RΓm. Under quite general conditions the functor RΓm is isomorphic to the left derived tensor functor P⊗LA(-), where P := RΓm(A). We also prove the NC MGM Equivalence in the connected graded context and a theorem on symmetric derived m-torsion. The χ condition of Artin and Zhang is introduced in Section 16.5. We study how this condition interacts with symmetric derived m-torsion.
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 Ã → 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.
This chapter is a brief review of standard material on categories and functors, including limits and the Yoneda Lemmas. A reader who is familiar with this material can skip this section, yet we recommend looking at our notational conventions, which are spelled out in Conventions 1.2.4 and 1.2.5.
The introduction is detailed, containing generalities on the subject, a motivating discussion (Grothendieck Duality for finitely generated abelian groups), a synopsis outlining the content of each chapter, prerequisites, author’s credo and acknowledgments.
In this chapter we prove existence of K-injective, K-projective and K-flat resolutions in the category of DG modules K(A, M), under certain conditions on the abelian category M and certain boundedness conditions. Our proofs are explicit and quite detailed.