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Differential Graded Rings and Derived Categories of Bimodules 1 1. Commutative vs. Noncommutative Ring Theory Differential Graded Rings and Derived Categories of Bimodules 1. Commutative vs. Noncommutative Ring Theory In the theory of commutative rings, one of the important tools is Amnon Yekutieli localization at prime ideals. Example 1.1. Let A be noetherian commutative ring. Department of Mathematics Ben Gurion University Recall that A is called regular if all its local rings Ap are regular local email: [email protected] rings. Notes available at Namely http://www.math.bgu.ac.il/~amyekut/lectures 2 dim Ap = rank Ap/pp (pp/pp). updated 1 Feb 2016 Amnon Yekutieli (BGU) Derived Categories of Bimodules 1 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 2 / 39 1. Commutative vs. Noncommutative Ring Theory 1. Commutative vs. Noncommutative Ring Theory The homological definition of regularity makes sense when A is Now let us look at a noncommutative noetherian ring A. noncommutative, with a small modification: we have to take care of left modules and right modules. It is very rare that A can be localized at a prime ideal p. The criterion for localizeabilty is the Ore condition, and it most often fails. Let’s indroduce the opposite ring A op , which is the same abelian group as A, but with reversed multiplication. Furthermore, there is no good notion of “geometry” associated to A. There do exist some definitions of “spectrum of A”, but they are all The category of left A-modules is Mod A, and the category of right ad-hoc, and are only useful in special situations. A-modules is Mod Aop . Fortunately, homological methods can sometimes replace geometry in We say that A is regular if there is a number n ∈ N, such that noncommutative ring theory. i i ′ ′ Ext A(M, N) = 0 and Ext Aop (M , N ) = 0 It is known that the local definition of regularity in the commutative case (Example 1.1) has an equivalent global homological definition. for all i > n, all M, N ∈ Mod A, and all M′, N′ ∈ Mod Aop . The smallest such n is called the global cohomological dimension of A . Amnon Yekutieli (BGU) Derived Categories of Bimodules 3 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 4 / 39 1. Commutative vs. Noncommutative Ring Theory 2. Derived Categories of Bimodules: over a Base Field Some of the most important work on noncommutative rings since 1985 was by the Artin school of noncommutative algebraic geometry, with the participation of Schelter, Tate, Van den Bergh, Zhang, Stafford and 2. Derived Categories of Bimodules: over a Base Field others. They considered regular graded rings over a field K. Derived categories greatly increase the scope of the homological approach to noncommutative ring theory. The main achievement was the classification of regular graded rings of dimension 3. Until Section 4, we work in this setting: K is a base field, and A is a noetherian central K-ring. This classification used the noncommutative projective scheme Proj A associated to a graded ring A, that was introduced in [AZ]. The reason we need K to be a field is technical: it insures the existence of various kinds of resolutions. I will mention some instances as I go This noncommutative projective geometry is a way to translate along. homological properties of the graded ring A to “geometric” properties of the “scheme” Proj A. Though “technical”, this condition is crucial. Removing it is the motivation of our current research project! See the survey paper [SV] for an exposition. Amnon Yekutieli (BGU) Derived Categories of Bimodules 5 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 6 / 39 2. Derived Categories of Bimodules: over a Base Field 2. Derived Categories of Bimodules: over a Base Field Recall that the category of left A-modules is Mod A. The category of complexes is C(Mod A). I already mentioned the opposite ring Aop . The derived category D(Mod A) has the same objects as C(Mod A). The enveloping ring of A is en op There is a localization functor A := A ⊗K A . Q : C(Mod A) → D(Mod A), Left modules over Aen are the same as A-bimodules. which is the identity on objects, and it inverts quasi-isomorphisms. If M and N are A-bimodules, then Hom A(M, N) is an A-bimodule too. Thus any morphism Thus we obtain a functor φ : M → N Mod en op Mod en Mod en in D(Mod A) can be written (not uniquely!) as a fraction Hom A(−, −) : ( A ) × A → A . −1 φ = Q(φ0) · Q(φ1) , Notice that “op” also designates the opposite category, taking care of contravariance in the first argument. where φi are homomorphisms of complexes, and φ1 is a quasi-isomorphism. Amnon Yekutieli (BGU) Derived Categories of Bimodules 7 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 8 / 39 2. Derived Categories of Bimodules: over a Base Field 2. Derived Categories of Bimodules: over a Base Field We are interested in the right derived functor en op en en By changing the roles of A and Aop we get another functor RHom A(−, −) : D(Mod A ) × D(Mod A ) → D(Mod A ). en op en en Hom Aop (−, −) : (Mod A ) × Mod A → Mod A . It is constructed using K-injective resolutions . These are a generalization of the injective resolutions in the classical sense. This Hom functor can be right derived too, in the same way as above. Given complexes M, N ∈ D(Mod Aen ), we choose a K-injective There are also tensor functors M ⊗ N and N ⊗ M for bimodules. resolution N → I over Aen . Such a resolution exists even if N is A A L L unbounded. These have left derived functors M ⊗A N and N ⊗A M, that are constructed using K-flat resolutions . These are a generalization of the Now we use the fact that K is a field: the ring homomorphism K → Aop is flat resolutions in the classical sense. flat; therefore A → Aen is flat; and therefore I is also K-injective over A. We shall refer to these right derived Hom functors, and these left It follows that the complex derived tensor functors, as the package of standard derived functors en associated to A. RHom A(M, N) := Hom A(M, I) ∈ D(Mod A ) is well-defined, up to a canonical isomorphism. Amnon Yekutieli (BGU) Derived Categories of Bimodules 9 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 10 / 39 3. Dualizing Complexes 3. Dualizing Complexes 3. Dualizing Complexes D Mod D Mod op Let us denote by f( A) and f( A ) the categories of complexes with finite cohomology modules. The following definition is taken from [Ye1]. It is a variation of the commutative definition by Grothendieck in [RD]. Recall that K is a The next result, which was proved by Grothendieck in [RD] for field, A is a noetherian central K-ring, and Aen = A ⊗K Aop . commutative rings, explains the name “dualizing”. Definition 3.1. A complex R ∈ D(Mod Aen ) is called a dualizing Theorem 3.2. ([Ye1]) Let R be a dualizing complex over A. The functor complex if these three conditions hold: D Mod op D Mod op RHom A(−, R) : f( A) → f( A ) (i) The cohomology bimodules H i(R) are finite over A and over Aop . is an equivalence, with quasi-inverse RHom Aop (−, R). (ii) R has finite injective dimension over A and over Aop . (iii) The canonical morphisms There are usually many nonisomorphic dualizing complexes over A. However, some of them are “better” than others. A → RHom A(R, R) and A → RHom Aop (R, R) in D(Mod Aen ) are isomorphisms. Amnon Yekutieli (BGU) Derived Categories of Bimodules 11 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 12 / 39 3. Dualizing Complexes 3. Dualizing Complexes Here is a definition from [VdB1]. By a filtration F of A we mean an ascending exhaustive filtration {Fj(A)}j≥0, that respects multiplication. Definition 3.3. Let R be a dualizing complex over A. Such a filtration gives rise to a graded central K-ring 1. A rigidifying isomorphism for R is an isomorphism F F gr (A) = M gr j (A). ≃ j≥0 ρ : R −→ RHom Aen (A, R ⊗K R) F Bernstein filtration F(A) D Mod en Let us call the filtration a if gr is commutative, in ( A ). F K K connected (i.e. gr 0 (A) = ), and finitely generated as -ring. 2. The pair (R, ρ) is called a rigid dualizing complex over A relative to Theorem 3.5. (Van den Bergh Existence Theorem, [VdB1]) K. If A admits a Bernstein filtration, then it has a rigid dualizing complex. Theorem 3.4. ([VdB1], [YZ2]) If A has a rigid dualizing complex (R, ρ), This existence theorem is extremely powerful. then it is unique, up to a unique isomorphism. The idea of its proof also yields the next functoriality result. Amnon Yekutieli (BGU) Derived Categories of Bimodules 13 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 14 / 39 3. Dualizing Complexes 4. Examples and Applications of Dualizing Complexes Theorem 3.6. ([YZ2]) Let A → B be a surjective ring homomorphism. Assume A admits a Bernstein filtration. 4. Examples and Applications of Dualizing Complexes 1. The rigid dualizing complexes RA and RB exist. 2. There is a unique rigid trace morphism Rings of Differential Operators. Here we assume K has characteristic 0. Tr : R → R B/A B A The n-th Weyl algebra A is the ring of differential operators of the D Mod en in ( A ). polynomial ring C := K[t1,..., tn].
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