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Home , Bimodule, Noncommutative ring, Opposite ring

1. Commutative vs. Noncommutative Theory

Differential Graded Rings and Derived Categories of 1. Commutative vs. Noncommutative

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 . Department of 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

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1. Commutative vs. Theory 1. Commutative vs. Noncommutative Ring Theory

The homological definition of regularity makes sense when A is Now let us look at a noncommutative 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 p. The criterion for localizeabilty is the , and it most often fails. Let’s indroduce the A op , which is the same abelian 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 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

Some of the most important work on noncommutative rings since 1985 was by the Artin school of noncommutative , 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 Proj A associated to a 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.

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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-. If M and N are A-bimodules, then Hom A(M, N) is an A- 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-.

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We are interested in the right

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 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.

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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.

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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.

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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 A is the ring of differential operators of the D Mod en in ( A ). C := K[t1,..., tn]. It comes with two filtrations: Often the ring A comes equipped with a filtration that is not as nice. This can be improved: ◮ G The order filtration G, where gr 0 (A) = C. ◮ The Bernstein filtration F, where gr F(A) = K. Theorem 3.7. ([MS], [YZ3]) Assume A admits a filtration G, such that 0 G G G gr (A) is a finite over its center Z(gr (A)) , and Z(gr (A)) is When C is any smooth K-ring (namely Spec A is smooth affine finitely generated as K-ring. variety), the ring of differential operators A := DC only comes with the Then A admits a Bernstein filtration F. order filtration G. Still, by Theorem 3.7, A has (noncanonical) Bernstein filtrations. Amnon Yekutieli (BGU) Derived Categories of Bimodules 15 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 16 / 39 4. Examples and Applications of Dualizing Complexes 4. Examples and Applications of Dualizing Complexes

By the Van den Bergh Existence Theorem, this implies that A = DC has Theorem 4.1. ([VdB2], [Ye3]) a rigid dualizing complex RA. The rigid dualizing complex R of A := U(g) is isomorphic to ∼ A It turns out that RA = A[n], where n = dim C. Namely RA is the trivial n bimodule A, sitting in cohomological degree −n. !A ⊗K V (g)[n]. In modern terminology (post 2005), this says that A is a Calabi-Yau ring n of dimension n. Explanation: A ⊗K V (g) is the bimodule A twisted by the character n V (g); and RA is this bimodule sitting in cohomological degree −n. K Lie . Suppose g is a over , of finite rank n. The theorem says that A is a twisted Calabi-Yau ring of dimension n. The universal enveloping algebra A := U(g) is equipped with a Note that the character n(g) is trivial when g is either semisimple or F V canonical Bernstein filtration F, and gr (A) is a commutative nilpotent; but it is nontrivial for many solvable Lie algebras. polynomial ring in n variables. Van den Bergh [VdB2] proved the theorem in the semisimple case. He Van den Bergh Existence Theorem tells us that A has a rigid dualizing used it to deduce what is now called Van den Bergh Duality for complex RA. Hochschild (co)homology.

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4. Examples and Applications of Dualizing Complexes 5. The Arithmetic Setup

The general case was done in [Ye3], and the proof relies on the rigid trace (Theorem 3.6). We applied it to the Borel subalgebra of a semisimple Lie algebra, to deduce facts on Verma modules. 5. The Arithmetic Setup

More Applications. Now we drop the assumption that K is a field . It is still a commutative ring, and A is a noetherian central K-ring. ◮ Derived Morita theory, and the derived Picard group DPic K(A), We refer to this as the arithmetic setup . which is an important noncommutative invariant. See [Ri], [Ye2], [RZ], [MY]. As we shall see, the difficulty lies in the fact that A might fail to be flat over ◮ Structure of a Hopf algebra A, noetherian but not finite over K. K. Here K-linear duality is replaced by the duality RHom (−, R ) A A The next two examples should convince us that rigid dualizing and the rigid trace Tr K : K → R . See [LWZ], [BZ], [RRZ] and A/ A complexes in the arithmetic setup are interesting. their references. ◮ Noncommutative homological identities, see [YZ2] and [JZ].

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Example 5.2. Here K := Zp, the p-adic . b Let G be a compact p-adic analytic group, and let A be the Iwasawa Example 5.1. In (commutative) algebraic geometry, rigid dualizing algebra Z complexes allow us to expand the range of Grothendieck duality. A := lim p[G/N], ← b For the first time, we can talk about global Grothendieck duality for where N ranges over the open normal subgroups of G. See [AB] and DM stacks and proper maps between them. [AW]. Since it is important to include arithmetic schemes and stacks in the Even though A itself is flat over K, if we look at a 2-sided ideal I of A, framework, we do not want to assume that our geometric objects are the B := A/I need not be flat over K. defined over a base field. We would like to have a theory of rigid dualizing complexes over A and its quotient rings, with rigid traces between them . This should give us a better understanding of the structure and representations of A.

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5. The Arithmetic Setup 5. The Arithmetic Setup

Now to the theory. Recall that the enveloping ring of A is Aen = A ⊗K Aop . When the ring A is commutative, the problem of K-injective resolutions that was discussed in the previous slide does not arise. Trying to mimic Definition 3.1 naively, a dualizing complex over A should be a complex This is because in the commutative theory of dualizing complexes, we R ∈ D(Mod Aen ). only work with central A-bimodules, so all the action takes place in D(Mod A). However, when we try to construct the complexes However, when we try to define rigid dualizing complexes, the bimodule en resolution problem resurfaces, even when the ring A is commutative. RHom A(R, R), RHom Aop (R, R) ∈ D(Mod A ) (I won’t go into details.) that appear in Definition 3.1, we run into a technical problem. Can we overcome this difficulty? We need a resolution R → I, where I is a complex of Aen -modules that is K-injective over A and over Aop . Here is where DG rings enter the picture. But when A is not flat over K, we do not know how to find such resolutions!

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A homomorphism of DG rings u : A˜ → B˜ is a graded ring homomorphism that respects the differentials. 6. DG Rings In cohomology, we get a graded ring homomorphism

H(u) : H (A˜) → H(B˜). Definition 6.1. A (nonpositive) differential graded ring is a graded ring

i A˜ = M A˜ , Definition 6.3. A left DG A-module is left graded A-module i≤0 ˜ ˜ i together with a differential d of degree 1, that satisfies the graded M = M M , Z Leibniz rule i∈ together with a differential d of degree 1, that satisfies the graded (6.2) d (a · b) = d(a) · b + ( −1)i · a · d(b) Leibniz rule, like (6.2). for a ∈ A˜ i and b ∈ A˜ j. The DG A˜-modules form a category, denoted by DGMod A˜.

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6. DG Rings 6. DG Rings

Definition 6.4. Let A be a central K-ring. Any ring A can be viewed as a DG ring concentrated in degree 0. A K-flat DG ring resolution of A (relative to K) consists of a DG ring A˜, As in the case of C(Mod A), we can invert quasi-isomorphisms in K ˜ DGMod A˜, and thus we obtain the derived category together with a factorization → A → A of the structural homomorphism K → A, such that: D(A˜) := D(DGMod A˜). ◮ The DG ring homomorphism K → A˜ is central, and A˜ is K-flat as a DG K-module. DGMod ˜ There are enough K-injective and K-flat resolutions in A, and ◮ The DG ring homomorphism A˜ → A is a quasi-isomorphism. ( ˜ ˜ ) ˜ ⊗L ˜ therefore the derived functors RHom A˜ M, N and M A˜ N exist. If A˜ → B˜ is a DG ring quasi-isomorphism, then the restriction functor It is known that K-flat DG ring resolutions exist. K D(B˜) → D(A˜) These resolutions are usually polynomial rings over in infinitely many noncommuting variables. is an equivalence. If A is commutative, then there are also commutative K-flat DG ring resolutions.

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≈ There are canonical equivalences D(A) −→ D(A˜) and ≈ D(Aop ) −→ D(A˜ op ). Thus we do not change the derived categories of 7. Derived Categories of Bimodules modules. We can now provide a good generalization of Definition 3.1: As before, K is a commutative base ring, and A is a noetherian central K-ring. Definition 7.1. A DG module R˜ ∈ D(A˜ en ) is called a dualizing complex Let A˜ be some K-flat DG ring resolution of A, and consider DG ring over A if these three conditions hold:

en op i ˜ op A˜ := A˜ ⊗K A˜ . (i) The cohomology bimodules H (R) are finite over A and over A . (ii) R has finite injective dimension over A˜ and over A˜ op . D ˜ en It has a derived category (A ). This is our candidate for the derived (iii) The canonical morphisms category of A-bimodules.

A˜ → RHom (R˜, R˜) and A˜ → RHom op (R˜, R˜) All the resolutions we need are available in D(A˜ en ). Therefore the A˜ A˜ package of standard derived functors associated to A (see slide 10) in D(A˜ en ) are isomorphisms. exists.

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7. Derived Categories of Bimodules 7. Derived Categories of Bimodules

Likewise we can extend Definition 3.3: This problem is settled by the next new result.

Definition 7.2. Let R˜ be a dualizing complex over A. Theorem 7.4. ([Ye5]) Let K be a commutative ring, and let A be a central K-ring. 1. A rigidifying isomorphism for R˜ is an isomorphism Suppose A˜ 0 and A˜ 1 are K-flat DG ring resolutions of A relative to K. ˜ ≃ ˜ ˜ L ˜ ρ : R −→ RHom A˜ en (A, R ⊗K R) There is a canonical equivalence of triangulated categories

in D(A˜ en ). Φ D ˜ en D ˜ en : (A0 ) → (A1 )

2. The pair (R˜, ρ) is called a rigid dualizing complex over A relative to K. that respects the package of standard derived functors.

The theorem tells us that the notion of dualizing complex does not Problem 7.3. There are many K-flat DG ring resolutions A˜ of A. ˜ ˜ D ˜ en depend on the resolution A chosen: a DG module R ∈ (A0 ) is Φ ˜ D ˜ en Do we get different answers (e.g. to the question of existence of rigid dualizing iff the DG module (R) ∈ (A1 ) is dualizing. dualizing complex) when choosing different resolutions? Similarly for rigidity.

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This justifies the following definition. 8. On the Proof of Theorem 7.4 Definition 7.5. The derived category of K-central A-bimodules is the category D(A˜ en ), where A˜ is any K-flat DG ring resolution of A over K. Here is some extra material (not in the notes). K Most of the results mentioned earlier on dualizing complexes, when I know two ways to prove this theorem, and they complement each is a field, appear to hold in the arithmetic setup. other. The big challenge for us is: Consider two K-flat resolutions A˜ 0 → A and A˜ 1 → A. Problem 7.6. Try to extend the Van den Bergh Existence Theorem to the We have enveloping DG rings arithmetic setup. ˜ en ˜ ˜ op Ai = Ai ⊗ Ai This is the subject of an ongoing project with Rishi Vyas. for i = 0, 1. – END – ?? The unadorned is over K.

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8. On the Proof of Theorem 7.4 8. On the Proof of Theorem 7.4 Look at the following DG ring: ˜ en ˜ en op ˜ ˜ op ˜ op ˜ A1 ⊗ (A0 ) = A1 ⊗ A1 ⊗ A0 ⊗ A0. This construction has an obvious pseudofunctorial property: if A˜ 2 is There is a well defined object yet another resolution of A, then there is a canonical isomorphism L op D ˜ en ˜ en op Φ Φ ∼ Φ P := A ⊗ A ∈ !A1 ⊗ (A0 ) . 1,2 ◦ 0,1 = 0,2 of equivalences As always, we must keep track of the four actions involved; but let’s D(A˜ en ) → D(A˜ en ). leave this aside. 0 2 It turns out that P is a tilting DG module . However, it is not easy to see why these equivalences respect the The functor package of standard derived functors. Φ L 0,1 (M) := P ⊗ ˜ en M A0 is the equivalence Φ D ˜ en D ˜ en 0,1 : (A0 ) → (A1 ) that we want. Amnon Yekutieli (BGU) Derived Categories of Bimodules 35 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 36 / 39 8. On the Proof of Theorem 7.4 8. On the Proof of Theorem 7.4 Passing to enveloping DG rings, we get a diagram of For this we turn to the second approach. quasi-isomorphisms ˜ Let Asf → A be a semi-free DG ring resolution. This means that when ˜ en Asf forgetting the differential, A˜ sf is a noncommutative polynomial ring K over is nonpositive graded variables. ~ ˜ en ˜ en Then we can find homomorphisms of DG rings (the dashed arrows) A0 A1 that fit into a commutative diagram

˜ en and thus a diagram of equivalences Asf D ˜ en (Asf ) Φ 9 e Φ ~ 0,sf 1,sf A˜ 0 A˜ 1 D(A˜ en ) D(A˜ en ) !  } 0 1 A These equivalences can be easily shown to respect the package of standard derived functors. Amnon Yekutieli (BGU) Derived Categories of Bimodules 37 / 39 Amnon Yekutieli (BGU) Derived Categories of Bimodules 38 / 39

8. On the Proof of Theorem 7.4 8. On the Proof of Theorem 7.4

[AB] K. Ardakov and K. A. Brown, Ring-Theoretic Properties of Iwasawa Algebras: A Survey, Documenta Mathematica, Extra Volume Coates (2006) 7-33. Finally, the diagram of equivalences [AW] K. Ardakov and S. Wadsley, On irreducible representations of D ˜ en compact p-adic analytic groups, Annals of Mathematics, (Asf ) Φ 9 e Φ 0,sf 1,sf Volume 178 (2013), 453-557. [AZ] M. Artin and J.J. Zhang, Noncommutative Projective Schemes, D ˜ en / D ˜ en (A0 ) Φ (A1 ) 0,1 Advances in Mathematics, Volume 109, Issue 2, December 1994, Pages 228-287. is commutative up to a canonical isomorphism. [BZ] K.A. Brown and J.J. Zhang, Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras, Journal of Algebra 320 (2008), 1814-1850. – END – [JZ] P. Jorgensen and J.J. Zhang, Gourmet’s Guide to Gorensteinness, Advances in Mathematics 151, 313-345 (2000).

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[Ke] B. Keller, Deriving DG categories, Ann. Sci. Ecole Norm. Sup. [RRZ] M. Reyes, D. Rogalski and J.J. Zhang, Skew Calabi-Yau 27 , (1994) 63-102. algebras and homological identities, Advances in Mathematics 264 (2014), 308-354. [LWZ] D.-M. Lu, Q.-S. Wu and J.J. Zhang, Homological Integral of Hopf Algebras, Transactions of the American Mathematical [RZ] R. Rouquier and A. Zimmermann, Picard groups for derived Society Volume 359, Number 10, October 2007, Pages module categories, Proc. London Math. Soc. 87 (2003), 197-225. 4945-4975. [SV] J.T. Stafford and M. Van den Bergh, Noncommutative curves [MY] J.-I. Miyachi and Amnon Yekutieli, Derived Picard Groups of and noncommutative surfaces, Bull. Amer. Math. Soc. 38 Finite Dimensional Hereditary Algebras, Compositio (2001), 171-216. Mathematica 129 (2001), 341-368. [VdB1] M. Van den Bergh, Existence theorems for dualizing [MS] J. C. McConnell and J. T. Stafford, Gelfand-Kirillov dimension complexes over non-commutative graded and filtered ring, J. and associated graded modules, J. Algebra 125 (1989), 197-214. Algebra 195 (1997), no. 2, 662-679. [Ri] J. Rickard, Morita theory for derived categories, J. London [VdB2] M. Van den Bergh, A Relation Between Math. Soc. 39 (1989), 436-456. and Cohomology for Gorenstein Rings, Proceedings of the [RD] R. Hartshorne, “Residues and Duality,” Lecture Notes in American Mathematical Society, Volume 126, Number 5, May Math. 20 , Springer-Verlag, Berlin, 1966. 1998, Pages 1345-1348

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8. On the Proof of Theorem 7.4 8. On the Proof of Theorem 7.4

[Ye1] A. Yekutieli, Dualizing Complexes over Noncommutative [YZ2] A. Yekutieli and J.J. Zhang, Rings with Auslander Dualizing Graded Algebras, J. Algebra 153 no. 1 (1992), 41-84. Complexes, J. Algebra 213 (1999), 1-51. [Ye2] A. Yekutieli, Dualizing complexes, and the [YZ3] A. Yekutieli and J.J. Zhang, Dualizing Complexes and derived Picard group of a ring, J. London Math. Soc. 60 (1999), Perverse Modules over Differential Algebras, Compositio 723-746. Mathematica 141 (2005), 620-654. [Ye3] A. Yekutieli, The rigid dualizing complex of a universal [YZ4] A. Yekutieli and J.J. Zhang, Dualizing Complexes and enveloping algebra, Journal of Pure and Applied Algebra 150 Perverse Sheaves on Noncommutative Ringed Schemes, (2000) 85-93. Selecta Math. 12 (2006), 137-177. [Ye4] A. Yekutieli, Rigid Dualizing Complexes via Differential Graded Algebras (Survey), in “Triangulated Categories”, LMS Lecture Note Series 375 , 2010. [Ye5] A. Yekutieli, Derived Categories of Bimodules, in preparation. [YZ1] A. Yekutieli and J.J. Zhang, Serre Duality for Noncommutative Projective Schemes, Proc. AMS 125 (1997), 697-707.

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