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1 DERIVED EQIVALENCE OF PPER TRIANGLAR DIFFERENTIAL GRADED ALGEBRA arxiv: v1 ath.ra 11 Nov 29 DANIEL MAYCOCK 1. Introduction The question of when two derived categories of rings are equivalent has been studied extensively. Morita theory answered the question of when two odule categories of rings are equivalent and a version of Morita theory for derived categories was developed by Rickard in 6 which ade use of the concept of tilting odules. This approach was applied by Ladkani in 5 to the situation of derived equivalences of upper triangular atrix rings. In this paper we extend the ain results fro 5 to the ore general case of upper triangular atrix differential graded algebras (henceforth referred to as DGAs). For this we will ake extensive use of the tool of recolleents and in particular the situation given by Jørgensen in 3. ection 2 sets out the notation used. We begin properly in section 3 by introducing the upper triangular atrix DGA Λ which has the for R M, where R and are DGAs and R M is a R--DG-biodule. R M There are also the left-dg-odules B and C which we will use throughout the paper. Next, by proving soe properties of B and C, we are able to use the ain result fro 3 to obtain the recolleent i j! D(R) i D(Λ) j D() i! j where i ( R R) B and j! ( ) C. We then conclude the section by presenting soe useful results obtained fro the recolleent which we will need in the next section. 1

2 2 DANIEL MAYCOCK In section 4 we turn our attention to the ain ai of the paper, generalizing the ain theore fro Ladkani to DGA s. To do so we follow a siilar idea as used in the proof of 5, Theore 4.5, by considering the DG-odule T Σi X j j Λ where X is copact and X D(R), where X denotes the sallest triangular subcategory containing X which is closed under the taking of coproducts. We begin with a stateent of Keller s theore which we will require to prove the following theore, our first attept at generalising 5, Theore 4.5. Theore. Let X be a DG R-odule such that R X is copact and R X D(R). Let R M be copact as a DG-R-odule. Let T Σi X j j Λ with E End Λ (P), where P is a K-projective resolution of T. Then E is an DGA with D(Λ) D(E op ). We then turn our attention to considering P, the K-projective resolution of T, and by doing so we are able to calculate its endoorphis DGA, which leads to our generalisation of the ain theore of Ladkani below. Theore. Let X be a DG R-odule such that R X is copact and R X D(R). Let R M be copact as an DG R-odule and let and V be K-projective resolutions of X and M respectively. Then for the upper triangular differential graded algebras Λ R M we have that D(Λ) D( Λ). and Λ HoR (V, ) Ho R (, ) op One specific advantage of considering the DGA case rather than the ring case is that with the DGA case we can do without a lot of constraints which are required in the ring case to ensure that the derived equivalence is between two triangular atrix rings. Finally in section 5 we conclude with a look at soe special cases. In the first we reconsider the original case in Ladkani, involving just rings and show that by aking the sae assuptions in our general theore we obtain the sae equivalence. We then briefly consider what happens in the special case where R X RR. In the final exaple, we require that our DGA s are over soe field k and that R is self dual, that is, Ho k (R, k) R in the derived category of DG-R-odules. This gives us the following result. Corollary. Let R be a self dual finite diensional DGA and be a DGA, both over a field k. Let R M be copact as a DG-R-odule.

3 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 3 Then Λ are derived equivalent. R M and Λ DM R 2. Notation and terinology In this we will fix the notation which we shall use throughout the paper; ore details on DGA s and DG-odules can be found in 4 and 1. Throughout this paper we will ake use of Differential Graded Algebras; these are always assued to be over soe coutative ground ring k unless stated otherwise. For any graded object r we will denote its degree by r. Note that we shall observe the Koszul sign convention so that whenever two graded eleents of degrees and n are interchanged we introduce a sign ( 1) n. For a DGA R we can define the opposite DGA, denoted R op, this is the sae as R except that the product is given by r.s ( 1) r s sr where. denotes ultiplication in R op. We will often identify DG-right-Rodules with DG-left-R op -odules. We shall often need to consider DG-odules with ore than one DGodule structure, for instance a DG-left-R-right--odule, denoted by R M. In these cases the different structures are required to be copatible, for the R M case this eans that the rule (r)s r(s) holds. For a DGA R we denote the category of all DG-left-R-odules by Mod R. We define the hootopy category of R, which we denote by K(R), as the category consisting of all DG-left-R-odules whose orphiss are the orphiss of DG-odules udololo hootopy. We define the derived category of R, denoted by D(R), fro K(R) by forally inverting the quasi-isoorphiss. Both K(R) and D(R) are triangulated categories. A ore detailed construction of the derived category and details of triangulated categories can be found in 2. ince we can identify DG-right-R-odules with DG-left-R op -odules we can also identify the derived category of DG-right-R-odules with D(R op ).

4 4 DANIEL MAYCOCK 3. A Recolleent ituation We begin by defining the differential graded algebras and DG-odules which we will be using throughout the paper. Definition 3.1. Throughout this paper, let R and be Differential Graded algebras with R M a DG-biodule which is quasi-isoorphic to R V where V is K-projective as a DG-left-R-odule and let Λ R M r Λ s denote the upper triangular atrix DGA with the differential R r M. s Reark 3.2. In the case where the base ring k is a field we always have that R M is quasi-isoorphic to soe R V with V K-projective as a DG-R-odule. 1 Definition 3.3. Let e R and e and define the 1 DG-left-Λ-odules R M B Λe R and C Λe ( ) R r and C has the dif- ( r where B has the differential ) ( ) ( ) ferential M s. s The first ai is to construct a recolleent involving the objects we have defined above. To do this we shall use 3, Theore 3.3 but before we can use this theore we first need the following Leas involving the DG-odules B and C. Definition 3.4. For X a full subcategory of a triangulated category T, we can define a full subcategory X {Y T Ho T (Σ l X, Y ) for all l} Lea 3.5. Λ B C in D(Λ) and hence both B and C are K- projective DG-left-Λ-odules which are copact in D(Λ).

5 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 5 Proof. Define Θ : Λ B C and Φ : B C Λ by ( ) ( ) r r Θ, s s and (( )) r Φ, s r. s It is obvious that Θ andφ are inverses of each other and it is straightforward to check that they are hooorphiss of DG-odules. o we have that Λ B C as DG-Λ-odules and hence also in D(Λ). Lea 3.6. B C as DG-odules and hence in D(Λ). Proof. Let C f B be a orphis of DG-odules. It suffices to show that f and since generates C we only need to show that 1 ( f. ( r Let f for soe r R. Then ( f f as required, hence f and so B C. ( ( r e. e f 1 Lea 3.7. B C in D(Λ). Proof. Let X B C, then Ho D(Λ) (Σ i B, X) and Ho D(Λ) (Σ i C, X) for each i. H i X H i Ho Λ (Λ, X) Ho K(Λ) (Λ, Σ i X) HoD(Λ) (Λ, Σ i X) Ho D(Λ) (B C, Σ i X) HoD(Λ) (B, Σ i X) Ho D(Λ) (C, Σ i X) for all i. Hence we have that X in D(Λ) and so B C.

6 6 DANIEL MAYCOCK We now have shown that B and C satisfy the conditions required to apply 3, Theore 3.3. However before we do so we prove the following lea about the endoorphis DGAs of B and C. Lea 3.8. Let F End Λ (B) and G End Λ (C), then F op R and G op as Differential Graded Algebras. 1 Proof. ince is a generator of B each eleent of End Λ (B) depends 1 entirely on where it sends. For each r R define the hooorphis f r as the eleent of F which sends to. 1 r We can now define φ : R op F by φ(r) f r. ince eleents of F 1 depend entirely on where they send this is obviously a bijection. It is also straightforward to show that φ is a hooorphis and so an isoorphis of DGAs. Now let g G. ince C is generated by we know that g depends 1 ( entirely on where it sends. Let g. 1 s ( ( ( However g e. g e g e, so s s ( and hence g. o for each s we can define the s hooorphis g s G as the eleent of G which sends to. 1 s Hence we can define a ap θ : op G sending s g s which is easily shown to be an isoorphis and so op G. By taking the above leas together with 3, Theore 3.3 we get the following recolleent:

7 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 7 i j! D(R) i D(Λ) j D(). i! j Five of the functors are given by j! ( ) Λ C L, i ( ) Λ B R L R, j ( ) RHo Λ ( Λ C, ), i! ( ) RHo Λ ( Λ B R, ), j ( ) RHo ( C Λ, ). Here C Λ RHo Λ( Λ C, Λ). In particular, i (R) B and j! () C. We shall now end this section with a nuber of results, involving the recolleent we have constructed, which we will find to be of great use in the next section. L Reark 3.9. The functor i ( ) Λ B R R X X to the DG-Λ-odule. sends a DG-R-odule Proposition 3.1. For C Λ RHo Λ( Λ C, Λ) we have that: (i) C Λ as DG-left--right-Λ-odules where has the differential ( s ) s. (ii) CΛ is a K-projective object over both and Λ. Proof. (i) First observe that C is generated by and that 1 RHo Λ ( Λ C, Λ) Ho Λ ( Λ C, Λ) since C is K-projective over Λ. ( Let θ Ho Λ (C, Λ) such that θ r Λ. s

8 8 DANIEL MAYCOCK However ( θ ( o θ ( ( r θ e. e.θ e s. s. s o for every s we can define θ s Ho Λ (C, Λ) to be the eleent which sends to. 1 s We can now use this to define an ap Θ : Ho Λ (C, Λ) given by Θ(θ s ) s. This ap is obviously a bijection and it is straightforward to check that it is an isoorphis of DG-left--right- Λ-odules. (ii) To see that C is K-projective over we observe that C as -odules. It reains to show now that C is also K-projective over Λ. We do this by showing that C is a direct suand of Λ as DG-right-Λ-odules. First observe that R M is a DG-right-Λ-odule with the differential r R r M. Now define Φ : Λ Λ R M such that Λ Λ ( ) r Φ ( r, s ). s It is clear to see that Φ is bijective and a hooorphis of DGodules. o C Λ Λ is a direct suand of Λ Λ and so is a K-projective DG-right-Λ-odule. Lea In the set up of the recolleent we have that: (i) j (Λ) in D(), (ii) j ( ) Λ C in D(Λ). Λ M Proof. (i) j (Λ) RHo Λ ( Λ C, Λ) C.

9 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 9 (ii) ince C is a K-projective -odule we have that j ( ) RHo ( C Λ, ) Ho ( C Λ, ) Ho ( Λ, ). Now observe that is generated by 1 and for all s define φ s Ho ( CΛ, ) to be the eleent which sends 1 to s. We can now define the ap Φ : Ho ( CΛ, ) Λ C/ Λ M given by M Φ(φ s ), where denotes the eleent + in s s s Λ C/ Λ M. This is obviously a bijection and it is easy to show that it is a hooorphis of DG-Λ-odules. 4. Derived Equivalences of pper Triangular DGA s We are now alost in the position where we can ake a start on what is the ain ai of the paper, to obtain a generalised version of 5, Theore 4.5 for upper triangular DGAs. In Theore 4.2, which is the first ajor step towards our goal, we obtain a dervived equivalence between D(Λ) and D(E ), where E is the endoorphis DGA of a K-projective resoultion of the DG-odule T Σi X j j Λ. We then follow this up by constructing a K-projective resoultion for T in proposition 4.6 which in turn is followed by the structure of the endoorphis DGA E in Proposition 4.1. The reainder of this section is involved in the details of coputing an quasi-isoorphiss between the DGA E and the upper triangular atrix DGA Λ HoR (V, ) Ho R (, ) op with the final result being Theore 4.15, the ain result of the paper which gives us a derived equivalnence between the upper triangular atrix DGAs Λ and Λ. We start however with a stateent of Keller s Theore which we use in the proof of Theore 4.2. Theore 4.1 (Keller s Theore). Let A be a DGA and let N be a K-projective DG A-odule which is copact in D(A) such that N D(A) and let H End A (N, N). Then D(A) D(H op ). Proof. ee 4. We are now able to ake our first attept at generalising 5, Theore 4.5 for DGAs. For this we follow a siilar ethod by introducing a DG-Λ-odule T Σi X j j Λ.

10 1 DANIEL MAYCOCK Theore 4.2. Let X be a DG R-odule such that R X is copact and R X D(R). Let R M be copact as a DG-R-odule. Let E End Λ (P), where P is a K-projective resolution of T Σi X j j Λ. Then E is an DGA with D(Λ) D(E op ). Proof. Our ai to to apply Kellers theore. To do this we need to show that T is copact and that T D(Λ). We begin with the copactness of T. ince T is a direct su it is sufficent to show that both its direct suands i X and j j Λ are copact. To show that i X is copact we first note that by adjointness Ho D(Λ) (i X, A k ) Ho D(R) (X, i! ( A k )) and that i! ( A k ) RHo Λ (B, A k ). Also, since B is copact it is not hard to show that i! ( A k ) i! (A k ) in D(Λ). We therefore have that Ho D(Λ) (i X, A k ) Ho D(R) (X, i! ( A k )) Ho D(R) (X, i! A k ) Ho D(R) (X, i! A k ) o i X is copact as required. Ho D(Λ) (i X, A k ). To show that j j Λ is copact we observe fro Lea 3.11 that j j Λ C We know that C is copact and since there is a distinguished triangle C C in D(Λ) it is sufficent to show that M. M M M is copact. M ince i M B L R M and both B and M are copact we have that Ho D(Λ) (B L R M, A k ) H RHo Λ (B L R M, A k ) H RHo R (M, RHo Λ (B, A k )) H RHo R (M, RHo Λ (B, A k )) H RHo R (M, RHo Λ (B, A k )) H RHo Λ (B L R M, A k )

11 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 11 Ho D(Λ) (i M, A k ). M o is copact and since C is also copact we have that j j Λ C M is copact, and so T Σi X j j Λ is copact. It reains to show that T D(Λ). For this it is sufficent to show that Λ T. ince Λ B C we only have to show that both B and C are in T. To show that B is contained in T we first observe that the functor i ( ) respects the operations of taking distinguished triangles, set indexed coproducts, quotients and suspensions. This gives us that i ( X ) i (X) for all X D(R). Hence B i R Ess.I(i ) i (D(R)) i X i X T. C To show that C T we first observe that M j j Λ T so M if we can show that T then C is in T. To show this we first observe that X D(R) so R M can be built fro X. ince i M preserves the possible constructions, we can build i M fro i X T. Hence we have that both B and C T and therefore that Λ T so T D(Λ). We are now in a position to apply Keller s Therore to get that D(Λ) D(E op ). Our ai now is to find a K-projective resolution of T in the above theore so that we can calculate E. For this we first need the following leas. Lea 4.3. Let be a K-projective resolution of a DG-R-odule X. X Then is a K-projective resolution of over Λ.

12 12 DANIEL MAYCOCK Proof. Let J be an exact DG-Λ-odule. Then ( ) Ho Λ, J HoΛ (B R, J) Ho R (, Ho Λ (B, J)). ince both and B are K-projective we have that this is exact and hence is K-projective. Lea 4.4. Let f : X Y be a orphis of K-projective DGodules over soe DGA R and let Z be the apping cone of f. Then Z is also K-projective. Reark 4.5. Fro definition of R M we have a quasi-isoorphis f RV R M where V is K-projective over R. Also for the DG-Rodule R X we can choose a quasi-isoorphis R g R X where is a K-projective resolution. We can now prove the following proposition about the structure of P, a K-projective resolution of T. Proposition 4.6. Let T Σi X j j Λ as defined in Theore 4.2. Then T has K-projective resolution P Σ W over Λ where W V is the apping cone of h i f M. Proof. By lea 4.3 we have that is a K-projective resolution of X V M i X and that is a K-projective resolution of over Λ. We now wish to find a K-projective resolution of j j Λ. To do this we first recall that j j Λ C M. We now consider the ap V h i f M

13 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 13 of DG-Λ-odules. This ebeds into the distinguished triangle V h i f M W. We can now use this to obtain the diagra M C C/ M V C W of distinguished triangles in D(Λ) so there exists a quasi-isoorphis W C/ M. By Lea 4.4 we also have that W is K-projective and hence a K- projective resolution of C/ M j j Λ. We now have K-projective resolutions for both direct suands of T and hence T has the K-projective resolution, P Σ W. Now that we have a K-projective resolution for T in Theore 4.2 we can try to calculate the endoorphis DGA E End Λ (T), but before we do so we need a few facts about W as defined in the above propostion. V Reark 4.7. W is the apping cone of h i f M so M W V Σ i.e any eleent w W is of the for w equipped with the diffential W f C V. s v. In addition W is

14 14 DANIEL MAYCOCK and the quasi-isorphis W C/ M in the proof of proposition 4.6 is give by s v. s Z Lea 4.8. W is isoorphic in K(Λ) to where Z is the apping cone of f. Furtherore Z is exact. Proof. ince Z is the apping cone of f we have a distinguished triangle of the for: V f M h Z k ΣV. We can use this to construct a distinguished triangle V h i f M h 1 Z k V Σ. Which in turn we can use to obtain the diagra of distinguished triangles: V h i f M h 1 Z k V Σ V h i f M W V Σ. Hence we have that there exists an isoorphis Z W. Finally since Z is the apping cone of a quasi-isoorphis it is exact. Lea 4.9. Let A and B be DG-R-odules. Then ( Ho R (A, B) A B Ho Λ, ) as coplexes of abelian groups.

15 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 15 Furtherore Ho R (A, A) Ho Λ ( A A, ) as DGA s. ( A B Proof. Define Θ : Ho R (A, B) Ho Λ, by Θ(φ) ) φ. It is easy to see that Θ is an isoorphis of coplexes of abelian groups. In addition in the case B A, Θ becoes an isoorpis of DGA s. The following proposition give the structure of E which by Theore 4.2 is dervived equivalent to the upper triangular atix DGA Λ. Proposition 4.1. In the setup of Theore 4.2, E HoR (, ) Ho Λ (W, Σ ) Ho Λ (Σ,W) Ho. Λ(W, W) Proof. ince P Σ W consists of a direct su we have that HoΛ (Σ E Ho Λ (P, P),Σ ) Ho Λ(W, Σ ) Ho Λ (Σ, W) Ho. Λ(W, W) Furtherore fro Lea 4.9 above we have that ( Ho Λ (Σ, Σ HoΛ, Ho ) ) R (, ). Our attention now is with obtaining a quasi-isoorphis between the entries of E and the corresponding entries Λ HoR (V, ) Ho R (, ) op which will allow us to construct a isoorpis between the two DGAs. ) Lea The coplex of abelian groups Ho Λ (Σ, W is exact.

16 16 DANIEL MAYCOCK Proof. For all i we have that ) ( H i Ho Λ (Σ, W H Ho Λ ), Σ i 1 W ) ( ), Σ i 1 W Z HoK(Λ), Σ i 1 ( HoK(Λ) ( ) Z However for θ Ho Λ, Σ i 1 for soe u, z Z and s, we have ( ( ) z u 1 u θ θ s ) 1 z z. s ( u z o s and so θ. ) ( u such that θ ) ( 1 u θ ) z s ) ( ) Z Hence Ho Λ (Σ, Σ i 1 Z HoΛ Σ, Σ i 1 and by Lea 4.9 this is isoorphic to Ho R (Σ, Σ i 1 Z). Taking this together with being K-projective and the exactness of Z gives us that ( ) Z ( Ho K(Λ), Σ i 1 HoK(Λ), Σ i 1 Z ) HoD(Λ) (, Σ i 1 Z ), ) ( ) Hence H i Ho Λ (Σ, W for all i and so HoΛ Σ, W is exact. Proposition There is a quasi-isoorphis of DGAs α : op Ho Λ (W, W). Proof. Define α : op Ho Λ (W, W) by α( s) g s l s ( ) ( ) s v v s g s ( 1) s s s and l s ( 1) s s s ( v +1). where

17 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 17 We first want to show that α is a hooorphis of Differential Graded Algebras. It is straightforward to check that α respects the operations of addition and ultiplication. o it reains to check that α is copatible with the differential. o ( ) HoΛ(W,W) (α( s)) Ho Λ(W,W) g s l s W g s ( 1) s g s l s W l s f C V g s ( 1) s g s l s l s C g s f l s V l s ( 1) s C g s ( 1) s g s C f f C V g s C f g s l s V l s ( 1) s f g s V l s + ( 1) s l s V. Considering each of the ters in this atrix seperately, starting with the upper left ter, we get: ) ( C g s ( 1) s g s C)( s ( ) ( ) s C ( 1) s s ( 1) s s s M () g s (s) ( 1) s s M ( s) ( 1) (s s) s ( 1) s ( s 1) M () s (s) s ( 1) s s M () s + ( 1) s ( s) (s) s + ( 1) s s ( 1) s s M () s ( s) (s) s ( 1) ( 1) s s s ( s) ( 1) s s ( 1) ( s) s ( s) ( s) s ( s) ( ) g ( s). s o C g s ( 1) s g s C g ( s).

18 18 DANIEL MAYCOCK By a siilar argueent we also have for the lower right entry that V l s + ( 1) s l s V l ( s). Finally for the upper right entry we have: ( ( ) f f v l s ( 1) s g s ) ( ) ( ) f v s f(v) ( 1) s ( v +1) ( 1) s g s f(v s) f(v) s ( 1) s ( v +1) ( 1) s ( 1) s v f(v) s f(v) s ( 1) s ( v +1) ( 1) s ( v +1). ubstituting these values back into the atrix gives us that: HoΛ(W,W) g (α( s)) ( s) α( l ( s)). ( s) Hence we have that α is a hooorphis of Differential Graded Algebras. It reains to show that it is also a quasi-isoorphis. ( ) Now let θ Ho Λ (W, C/ M ) such that θ s v. This s quasi-isoorphis gives us the hooorphis of coplexes of abelian groups Ho Λ (W, θ) : Ho Λ (W, W) Ho Λ (W, C/ M ). Define β Ho Λ (W, θ) α : op Ho Λ (W, C/ M ). Then β is a hooorphis of coplexes of abelian groups with ( ) ( β( s) s v θ ( 1) s s s s s ) v ( 1) s s. s s Now let ψ Ho Λ (W, C/ M ). Then for any M, s and ( ) v V we have that ψ s s v for soe s. Hence we have ( ) ( ) ( ) ψ s s v ψ s 1 s 1 v ψ s.

19 Furtherore DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 19 ψ ( ) ( ) s v ψ s ( ) ψ 1 s. o each eleent of Ho Λ (W, C/ M sends 1. ) depends entirely on where it We therefore have that for every s we have that β(s) is the eleent of Ho Λ (W, C/ M ) which sends 1 to. s ince eleents of Ho Λ (W, C/ M ) depend entirely on where they send 1 and since s s for all s, s with s s we have that β is a bijection and so a isoorphis of coplexes of abelian groups. Furtherore since W is K-projective and θ is a quasi-isoorphis we have that Ho Λ (W, θ) is a quasi-isoorphis and therefore since β is an isoorphis we have that α ust also be a quasi-isoorphis. Lea There exists a quasi-isoorphis Ψ : Ho R (V, ) Ho Λ (W, Σ of coplexes of abelian groups, such that ) Ψ(θ) ( ) s θ(v) v ( 1) θ. Proof. Consider the distinguished triangle V h i f M ι W π V Σ in K(Λ). f ince W is the apping cone of we have that π is given by ( ) π s v v. By applying the functor Ho Λ (, Σ we get a distinguished tri- ) angle

20 2 DANIEL MAYCOCK ( ( M Ho Λ, Σ Ho ) Λ W, Σ ) ( π V M Ho Λ (Σ, Σ Ho ) Λ Σ, Σ ) in K(Ab). ( ) M M Now let θ Ho Λ, Σ i. Then, since is generated by as a DG-Λ-odule, we have that θ depends entirely upon where it 1 sends. 1 ( u Let θ. Then u ( ( ) θ θ 1 1 so θ and hence Ho Λ ( M ( u θ, 1 1 ), Σ i for all i. Hence the distinguished triangle above shows that ( V π : Ho Λ (Σ, Σ Ho ) Λ W, Σ ) is a quasi-isoorphis. We can now use this along with the suspension Σ and the isoorphis Θ defined in the proof of Lea 4.9 to obtain the diagra ( Θ V Σ( ) Ho R (V, ) Ho Λ, ) V π Ho Λ (Σ, Σ ) Ho Λ (W, Σ ) ince each of the aps in the diagra is a quasi-isoorphis we can use the to define the quasi-isoorphis Ψ : Ho R (V, ) Ho Λ (W, Σ by the coposition ) Ψ π Σ( ) Θ.

21 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 21 Finally for θ Ho R (V, ) we have that o for s v W, we have that Ψ(θ) Ψ(θ) π Σ Θ(θ) ( ) θ π Σ ( ) θ π ( 1) θ θ ( 1) θ π. ( ) s θ v ( 1) θ π ( ) s v ) θ v θ(v) ( 1) θ ( 1) ( θ. Reark Fro the right DG--odule structure on V we have that Ho R (V, ) is a left DG--odule. In addition Ho R (V, ) is a HoR (V, ) left DG-Ho R (, )-odule. Hence we have that Ho R (, ) op is a DGA. We are now in a position to produce our ain Theore, a version of 5, Theore 4.5 for DGAs. Theore Let X be a DG R-odule such that R X is copact with R X D(R) and let R M be copact as a DG-R-odule. Then for the upper triangular differential graded algebras Λ R M we have that D(Λ) D( Λ). and Λ HoR (V, ) Ho R (, ) op

22 22 DANIEL MAYCOCK Proof. Fro Theore 4.2 and lea 4.1 we have that D(Λ) D(E op ) where HoR (, ) Ho E Λ (W, Σ ) Ho R (Σ, W) Ho. Λ(W, W) We therefore only need to show that there is a quasi-isoorphis of DGA s fro Λ op HoR (, ) Ho R (V, ) op to E. Fro proposition 4.12 we have that there exists a quasi-isoorphis α : op Ho Λ (W, W). Hence we can define the ap HoR (, ) Ho Φ : R (V, ) HoR (, ) Ho op Λ (W, Σ ) Ho R (Σ,W) Ho Λ(W, W) ( ) ( ) φ θ φ ( 1) by Φ θ Ψ(θ). s α(s) Here Ψ : Ho R (V, ) Ho Λ (W, Σ is the quasi-isoorphis ) fro Lea We now need to show that Φ is a orphis of DGA s. Both addition and copatibility with the differential follow fro the fact that α is a orphis of diffential graded algebras. o we only need to check ultiplication: φ θ Let. denote ultiplication in op. Let s Λ φ θ i and s Λ j ; then we have ( ) φ θ Φ s and ( ) φ θ Φ φ ( 1) s i Ψ(θ) φ ( 1) j Ψ(θ ) α(s) α(s ) φφ ( 1) j φψ(θ ) + ( 1) i Ψ(θ)α(s ) α(s)α(s ) φφ ( 1) j φψ(θ ) + ( 1) i Ψ(θ)α(s ) α(s.s ) ( ) φ θ φ θ Φ s s ( ) φφ φθ + θ.s Φ s.s φφ ( 1) (i+j) Ψ(φθ + ( 1) ij s θ) α(s.s ).

23 However DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 23 ( 1) (i+j) Ψ(φθ ) ( 1) (i+j) Ψ((φθ + ( 1) ij s θ)) ( ) s v ( ) s v + ( 1) (i+j) ( 1) ij Ψ(s θ) ( ) s v ( ) φθ ( 1) (i+j) ( 1) (i+j) (v) (s + ( 1) (i+j) ( 1) ij ( 1) (i+j) θ)(v) φ θ (v) θ(vs + ( 1) ij ( 1) j(i+(i+1)) ) φ θ (v) θ(vs + ( 1) j(i+1) ) ( ) ( ( 1) j φψ(θ ) s s ) v + ( 1) j(i+1) ( 1) i Ψ(θ) ss vs (( 1) j φψ(θ ) + ( 1) i Ψ(θ)α(s )) ( ) s v, so ( 1) j φψ(θ ) + ( 1) i Ψ(θ)α(s ) ( 1) (i+j) Ψ((φθ + ( 1) ij s θ)) and therefore ( ) ( ) ( ) φ θ φ θ Φ Φ φ θ φ θ s s Φ s s. We therefore have that Φ is a orphis of Diffential Graded Algebras. ) Furtherore since fro lea 4.11 we have that Ho Λ (Σ, W is ) exact and so the ap Ho Λ (Σ, W is a quasi isoorphis. Taking these together with the fact that α : op Ho Λ (W, W) is a quasi-isoorphis we have that Φ is a quasi-isoorphis. Hence E Λ op and so D(Λ) D(E op ) D( Λ).

24 24 DANIEL MAYCOCK 5. Exaples We shall now conclude with soe exaples. In the first exaple we will show that by taking R and to be k-algebras and aking the sae assuptions as in 5, we obtain what is in essence the sae result. Definition 5.1. An R-odule X is called rigid if Ext i R (X, X) for all i. Theore 5.2. Let R and be rings and R M a R--biodule such that R M is copact in D(R) and when R and are considered as DGAs then R M as a DG-biodule is quasi-isoorphic to R V which is a K-projective DG-R-odule. Let R X be a copact and rigid R- odule with X D(R) and Ext n R ( RM, R X) for all n. Then the triangular atrix rings Λ R M are derived equivalent. and Λ HoR (M, X) End R (X) op Proof. By considering the rings R and and odules M and X to be DGA s and DG-odules respectively we can apply Theore 4.15 to get that the DGA s R M and HoR (V, ) Ho R (, ) op are derived equivalent, where is a K-projective resolution of X. ince Ho R (, ) RHo R (X, X) we have that H i Ho R (, ) H i RHo R (X, X) Ext i R (X, X) for all i since X is rigid and H Ho R (, ) H RHo R (X, X) End R (X). iilarly since Ho R (V, ) RHo R (M, X) we have that H i Ho R (V, ) H i RHo R (M, X) Ext i R(M, X) for all i and H Ho R (V, ) H RHo R (M, X) Ho R (M, X).

25 DERIVED EQIVALENCE OF PPER TRIANGLAR DGA 25 Hence we have that H i HoR (V, ) Ho R (, ) op for all i and H HoR (V, ) Ho R (, ) op HoR (M, X) End R (X) op. HoR (M, X) We therefore have that the atrix ring End R (X) op is derived HoR (V, ) equivalent to the DGA Ho R (, ) op and so derived equivalent R M to the atrix ring. Our next exaple considers the special case obtained when we take RX R R. Corollary 5.3. Let R M be copact as a DG-R-odule. Then the triangular atrix DGAs R M Λ and Λ HoR (V, R) R where V is K-projective over R and is quasi-isoorphic to R M, are derived equivalent. For the next exaple we require the idea of the duality on ModR which we define next. Definition 5.4. Let R be a finite diensional DGA over a field k. Then we can define the duality on ModR by D : ModR Mod R op where D( ) Ho k (, k). The final exaple below considers the case where the DGA s R and are over soe field k and R is self dual in the sense of the above definition. Theore 5.5. Let R be a finite diensional and self dual in the sense that DR R in the derived category of DG-bi-R-odules and let R M be copact as a DG-R-odule. Then Λ R M and Λ DM R

26 26 DANIEL MAYCOCK are derived equivalent. Proof. Fro corollary 5.3 we have that R M HoR (V, R) and R are derived equivalent, where R V is quasi-isoorphic to R M and R V is K-projective. ince R is self dual we have that Ho R (V, R) Ho R (V, DR) Ho R (V, Ho k (R, k)) Hok (R R V, k) Ho k (V, k) DV. Furtherore, applying the functor D( ) to the quasi-isoorphis V M gives us the quasi-isoorphi DM DV. This in turn DM DV allows us to define a quasi-isoorphis, so R R DM DV and R R are derived equivalent and hence R M and are derived equivalent. References DM R 1 A. Frankild and P. Jørgensen, Gorenstein Differential Graded Algebras, Israel J. Math. 135 (23), R. Hartshorne, Residues and Duality, Lecture Notes in Math., Vol. 2, pringer, Berlin, 1966, Lecture notes of a seinar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. 3 P. Jørgensen, Recolleent for differential graded algebras, J. Algebra 299 (26), B. Keller, Deriving DG categories, Ann. ci. École Nor. up. (4) 27 (1994), Ladkani, Derived Equivalences of triangular atrix rings arising fro extensions of tilting odules, to appear in Algebr. Represent. Theory. 6 J. Rickard, Morita Theory for Derived Categories, J. London Math. oc. 39 (1989),