DERIVED CATEGORIES AND THEIR APPLICATIONS

Transcription

1 REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 48, Número 3, 2007, Páinas 1 26 DERIVED CATEGORIES AND THEIR APPLICATIONS Abstract In this notes we start with the basic deinitions o derived cateories, derived unctors, tiltin complexes and stable equivalences o Morita type Our aim is to show via several examples that this is the best ramework to do homoloical alebra, We also exhibit their useulness or ettin new proos o well known results Finally we consider the Morita invariance o Hochschild cohomoloy and other derived unctors 1 Introduction Derived cateories were invented by A Grothendieck and his school in the early sixties The volume [Ve77] reproduces some notes o his pupil JLVerdier, datin rom 1963, which are the oriinal source on derived cateories (see also [Ve96]) Let A be an abelian cateory and C(A) the cateory o complexes in A, that is, objects are sequences o maps n 1 d n 1 n d n n+1 with d n d n 1 = 0 or all n Z, and a morphism : Y between complexes is a amily o morphisms n : n Y n in A such that d n Y n = n+1 d n The cohomoloy roups o the complex are by deinition H n () = Kerd n / Im d n 1 A morphism : Y induces a roup morphism H n () : H n () H n (Y ) in each deree We say that is a quasi-isomorphism i H n () is an isomorphism or all n Z The derived cateory D(A) o A is obtained rom C(A) by ormally invertin all quasi-isomorphisms Recall that the deinition o derived unctors in homoloical alebra is as ollows Assume that A has enouh projectives and let F : A Ab be a contravariant let exact unctor Then the riht derived unctor R n F : A Ab is deined in the ollowin way Let M be an object in A and let P 2 P 1 P 0 0 1

2 2 be a projective resolution o M, that is, P 2 P 1 P M 0 is a quasi-isomorphism Then R n F(M) is the roup homoloy in deree n o the complex 0 F(P 0 ) F(P 1 ) F(P 2 ) These are well deined unctors since the projective resolution o an object M is unique up to quasi-isomorphisms Moreover, or any short exact sequence 0 M 1 M2 M3 0 there is a lon exact sequence R n F(M 3 ) Rn F() R n F(M 2 ) Rn F() R n F(M 1 ) δn R n+1 F(M 3 ) where δ is the so called connectin morphism In act, when we are doin homoloical alebra, we are not dealin with objects and cohomoloy roups but with complexes up to quasi-isomorphisms and their cohomoloy roups Hence it is natural to work in D(A) instead o A or C(A) Any object in A can be identiied in D(A) by the complex concentrated in deree zero, which will be denoted by [0] Let 0 Y Z 0 be a short exact sequence in A Then 0 [0] Y [0] Z[0] 0 is a short exact sequence in C(A) Since 0 Y Z 0 is a quasi-isomorphism, we can replace Z[0] by the complex appearin in the irst line o the previous picture, hence the short exact sequence 0 [0] Y [0] Z[0] 0 can be identiied in D(A) by the sequence o complexes which is not an exact sequence Y = Y

3 DERIVED CATEGORIES AND THEIR APPLICATIONS 3 In this new cateory, the concept o short exact sequences (which are determined by two morphisms, ) will be replaced by that o distinuished trianles (determined by three morphisms,, h) I 0 Y Z 0 is a short exact sequence in A, then is a trianle in D(A) = Y = Y [0] Y [0] Z[0] [1] We will consider cohomoloical unctors deined in D(A), that applied to distinuished trianles will ive lon exact sequences with morphisms induced by,, h That is, the extra morphism we are considerin when deinin trianles is a morphism o complexes that induces the correspondin connectin morphism So, when considerin the derived cateory D(A) o A, we replace objects in A by complexes, and invert quasi-isomorphisms between complexes As we shall see, D(A) is not abelian i A is not semisimple The abelian structure o A, and o C(A), has to be replaced by a trianulated structure Given an alebra A over a commutative rin k, or simplicity, we will write D(A) or the derived cateory D(Mod A) Given two k-alebras A and B the natural question is: when are D(A) and D(B) equivalent cateories? (trianle equivalent?) O course, i A and B are Morita equivalent (that is, Mod-A and Mod-B are k- linearly equivalent) then D(A) and D(B) are equivalent We will see that there other equivalences and we shall present some examples 2 Trianulated cateories Let T be an additive cateory with an additive automorphism T : T T called the translation unctor We will write [n] and [n] or T n () and T n () respectively A trianle in T is a diaram o the orm Y Z h [1]

4 4 that will be denoted as (, Y, Z,,, h), and a morphism o trianles is a triple (α, β, γ) : (, Y, Z,,, h) (, Y, Z,,, h ) ormin a commutative diaram in T Y Z [1] α β γ α[1] Y Z h [1] A morphism o trianles is said to be an isomorphism i α, β, γ are isomorphisms in T DEFINITION 21 A structure o trianulated cateory on T is iven by a translation unctor T and a class o distinuished trianles veriyin the ollowin our axioms: TR1 a) (,, 0, id, 0, 0) is a distinuished trianle, or any object ; b) Every trianle isomorphic to a distinuished one is distinuished; c) Every morphism Y can be embedded in a distinuished trianle (, Y, Z,,, h) TR2 (Rotation) A trianle (, Y, Z,,, h) is distinuished i and only i (Y, Z, [1],, h, [1]) is distinuished TR3 (Morphisms) Every commutative diaram α Y β Z h h [1] Y Z h [1] whose rows are distinuished trianles can be completed to a morphism o trianles by a morphism Z γ Z TR4 (The octahedral axiom) Given Y and Y Z morphisms in A, and distinuished trianles (, Y,,,, s), (, Z, Y,, h, r), (Y, Z, Z,,, t), there exist morphisms u Y, Y v Z such that is a distinuished trianle and (, Y, Z, u, v, [1] t) Y s [1] u Z h Y r [1] v [1] Y Z Z t Y [1] h [1] u Y v Z w [1] is a commutative diaram

5 DERIVED CATEGORIES AND THEIR APPLICATIONS 5 Observe that all the rows in the previous diaram are distinuished trianles, and morphisms between rows determined the ollowin morphisms o trianles (, Y,,,, s) (Y, Z, Z,,, t) (id,,u) (, Z, Y,, h, r) (,h,id Z ) (, Y, Z, u, v, w) (,id Z,v) (Y, Z, Z,,, t) (s,tv,w) ([1], Y [1], [1], [1], [1], s[1]) The name o last axiom comes rom the act that it can be viewed as a picture o a octahedron, where our aces are distinuished trianles, and all the other aces are commutative Y Z u v Z Y DEFINITION 22 Let T be a trianulated cateory and A an abelian cateory An additive unctor H : T A is said to be a cohomoloical unctor i, or any distinuished trianle (, Y, Z,,, h) in T, we et an exact sequence in A H i () Hi () where H i denotes H T i = H( [i]) H i (Y ) Hi () H i (Z) REMARK 23 By TR2, i (, Y, Z,,, h) is a distinuished trianle then (Y, Z, [1],, h, [1]) is also distinuished Then, i H is a cohomoloical unctor, H i (Y ) Hi () H i (Z) Hi (h) is also exact So we et a lon exact sequence H i () Hi () H i (Y ) Hi () H i+1 () H i (Z) Hi (h) H i+1 () LEMMA 24 Let (, Y, Z,,, h) be a distinuished trianle in T Then i) = 0; ii) I u : U Y is such that u = 0, then there exists v : U such that u = v; iii) I s : Y W is such that s = 0, then there exists t : Z W such that s = t

6 6 Proo i) From TR1 we know that (,, 0, id, 0, 0) is a distinuished trianle, and rom TR3 we know that there exists a morphism o trianles id 0 [1] id s id [1] Y Z h and hence = 0 ii) From TR2 and TR3 we know that the diaram idu [1] U U 0 U[1] u Y Z h [1] can be completed to a morphism o trianles Hence, there exists U v such that v = u iii) It ollows as (ii) REMARK 25 Usin TR2 we also et that h = 0 and [1] h = 0 COROLLARY 26 I U is an object in a trianulated cateory T, then Hom T (U, ) : T Ab and Hom T (, U) : T op Ab are cohomoloical unctors COROLLARY 27 Any distinuished trianle is determined, up to isomorphisms, by one o its morphisms Proo From TR2, it suices to prove that the distinuished trianles (, Y, Z,,, h) and (, Y, Z,,, h ) are isomorphic By TR3, there exists a morphism o trianles Y Z [1] id id Y t id [1] Y Z h [1] I we apply the cohomoloical unctors Hom T (, Z) and Hom T (Z, ), by the 5- lemma we et that t : Hom T (Z, Z) Hom T (Z, Z), t : Hom T (Z, Z) Hom T (Z, Z ) are isomorphisms Then, t has let and riht inverse, and thereore, t is an isomorphism PROPOSITION 28 For any distinuished trianle (, Y, Z,,, h), the ollowin conditions are equivalent: (i) is a monomorphism; (ii) h = 0; (iii) there exists Z s Y such that s = id Z ; (iv) is an epimorphism; (v) there exists Y t such that t = id h

7 DERIVED CATEGORIES AND THEIR APPLICATIONS 7 Proo (iii) (iv) and (v) (i) are immediate (i) (ii) By TR2 and Lemma 24(i) we have that [1] h = 0 Since is a monomorphism, h = 0 (ii) (iii) I h = 0, by Corollary 26, Hom T (Z, Y ) Hom T (Z, Z) 0 Hom T (Z, [1]) is exact, so there exists s : Z Y such that s = id Z (iv) (v) By Lemma 24(i) we know that h = 0 Since is an epimorphism we have that h = 0, and by Corollary 26, Hom T (Y, ) Hom T (, ) 0 Hom T (Z[ 1], ) is exact, so there exists t : Y such that t = id COROLLARY 29 In any trianulated cateory any monomorphism splits and any epimorphism splits Moreover, is an isomorphism i and only i is a monomorphism and an epimorphism 3 Distinuished trianles in K(A) Let A be an abelian cateory, and let C(A) be the cateory o complexes in A The homotopy cateory K(A) is deined as ollows: the objects o K(A) are the objects in C(A), and morphisms in K(A) are the homotopy equivalence classes o morphisms in C(A) That is, Hom K(A) (, Y ) = Hom C(A) (, Y )/, where i there exists h n : n Y n 1 with d n 1 Y h n + h n+1 d n = n n So, the morphisms homotopic to zero in C(A) become the zero morphisms in K(A) and the homotopic equivalences become isomorphisms It can be checked that K(A) is well deined as a cateory, and moreover, it is an additive cateory, and the quotient C(A) K(A) is an additive unctor Observe that Hom K(A) (, Y ) can also be deined as the quotient o Hom C(A) (, Y ) by the subroup Ht(, Y ) = { Y : is homotopic to 0} since Ht is an ideal in C(A), that is, Ht i or belons to Ht Let T : C(A) C(A) be the automorphism deined in the ollowin way: or any complex in C(A), T() n = n+1 and T(d) n = d n+1 ; or any morphism Y in C(A), T() n = n+1 The unctor T is additive and it is an automorphism Since T() is a morphism homotopic to zero i and only i is a morphism homotopic to zero, it induces an additive automorphism T : K(A) K(A) We denote [n] = T n () and d[n] = T n (d), or any n Z We will see that K(A) is a trianulated cateory with translation unctor T

8 Û Û Ù Ù Û Û Û Ù Û 8 LEMMA 31 The cohomoloy unctors H n : C(A) A induce well deined unctors H n : K(A) A Proo I, : Y are homotopic, then d n 1 Y h n + h n+1 d n = n n, so H n () = H n () DEFINITION 32 Let Y be a morphism in C(A) (i) The mappin cone o is the complex cone() such that cone() n = n+1 Y n, and the dierential d n : n+1 Y n n+2 Y n+1 is iven by ( ) d n d n+1 = 0 d n Y (ii) The mappin cylinder o is the complex cyl() such that cyl() n = n n+1 Y n, and the dierential d n : n n+1 Y n n+1 n+2 Y n+1 is iven by d n d n id 0 = 0 d n d n Y PROPOSITION 33 Any short exact sequence 0 Y Z 0 in C(A) its into a commutative diaram h 0 Y cone() α 0 ν cyl() π cone() 0 β γ 0 Y Z 0 deined in deree n by the ollowin diaram δ [1] 0 «id 0 0 ( id 0 ) Y n n+1 Y n n+1 «00 n n n+1 Y n n+1 Y n id 0 id id ( id 0) ( n 0 id ) ( 0 n ) n n Y n Z n n with exact rows, α, β homotopy inverse equivalences and γ a quasi-isomorphism

9 DERIVED CATEGORIES AND THEIR APPLICATIONS 9 Proo A direct computation shows that all the maps are morphisms o complexes It is clear that β α = id On the other hand, (id α β)(x, x, y) = (x, x, y) (0, 0, (x)+y) = (x, x, (x)) = (h d+d h)(x, x, y), where h(x, x, y) = (0, x, 0) Then α and β are homotopy equivalences, so they are quasi-isomorphisms Now, since β is a quasi-isomorphism, the 5-lemma applied to the lon exact sequences obtained rom the second and the third row, implies that γ is a quasi-isomorphism DEFINITION 34 (i) A short exact sequence 0 Y Z 0 in C(A) semi-splits i 0 n n Y n n Z n 0 splits, or all n (ii) A trianle in K(A) is a distinuished trianle i it is isomorphic, in K(A), to one o the orm Y h cone() δ [1] THEOREM 35 [Ve96, II132] The cateory K(A) is a trianulated cateory with translation unctor T Let H n : C(A) A be the unctor deined by H n () = Kerd n / Im d n 1 For any morphism homotopic to zero, H n () = 0 Then H n actors uniquely throuh K(A) The ollowin are immediate consequences o Proposition 33 PROPOSITION 36 Any short exact sequence in C(A) is quasi-isomorphic to a semi-split short exact sequence PROPOSITION 37 Any distinuished trianle in K(A) is quasi-isomorphic to one induced by a semi-split short exact sequence in C(A) PROPOSITION 38 The unctor H n : K(A) A is a cohomoloical unctor, or any n Z Proo Let Y Z [1] be a distinuished trianle in K(A) By deinition, it is isomorphic to Y h cone() δ [1] Since 0 Y h cone() δ [1] 0 is a short exact sequence in C(A), it induces an exact sequence H n (Y ) Hn (h) H n (cone()) Hn (δ) H n+1 () A simple computation shows that the connectin morphism H n+1 () H n+1 (Y ) associated to the exact sequence appearin in the previous proo is H n+1 (): recall that the connectin morphism is deined by chasin the ollowin diaram

10 Û Û Ù Û Û 10 h n+1 =( 0 0 Y n+1 n+2 Y n+1 id ) ( id 0) n+1 Y n n+1 0 «d n d n+1 = 0 d n Y so, i x n+1 is such that d n+1 (x) = 0, then x = δn (x, 0) Finally, d n (x, 0) = ( dn+1 (x), (x)) = (0, (x)) = hn+1 ((x)) REMARK 39 Let 0 Y Z 0 be a semi-split short exact sequence in C(A), that is, Y n n Z n Then Y is isomorphic to the complex n 1 Z n 1 d n 1 h n 1! 0 d n 1 Z n Z n where h : Z [1] is a morphism o complexes and Y Z h [1] d n h n 0 d n Z «n+1 Z n+1 is a distinuished trianle in K(A) Moreover, the connectin morphism δ in the lon exact sequence is just H n (h) H n () Hn () H n (Y ) Hn () H n (Z) δn H n+1 () 4 Derived cateories As we said in the introduction, the derived cateory D(A) o A is obtained rom C(A) by ormally invertin all quasi-isomorphisms So D(A) is the localization o C(A) with respect to the class o quasi-isomorphisms in C(A): there exists a unctor Q : C(A) D(A) sendin quasi-isomorphisms to isomorphisms which is universal or such property, that is, or any unctor F : C(A) B sendin quasiisomorphisms to isomorphisms, there exists a unctor G : D(A) B makin the ollowin diaram commutative C(A) F G Q B D(A) The cateory D(A) can be obtained just by addin ormally inverses or all quasiisomorphisms But in this case, morphisms in D(A) are just ormal expressions o the orm 1 s1 1 2 s 1 2 n s 1 n, where all i are morphism in C(A) and s i are quasi-isomorphisms en C(A) To work with these kind o expressions, we would like to ind a common denominator It

11 DERIVED CATEGORIES AND THEIR APPLICATIONS 11 was Verdier s observation that we can obtain a better description o the morphisms by a calculus o ractions i we construct D(A) in several steps: C(A) K(A) D(A) that is, we consider complexes and their morphisms, then complexes and homotopy classes o morphisms o complexes, and inally, we invert all quasi-isomorphisms in K(A) Then D(A) = K(A)[S 1 ] is the localization o K(A) with respect to the class S o all quasi-isomorphisms DEFINITION 41 A class S o maps in a cateory C is said to be a multiplicative system i it satisies the ollowin conditions: S1 For any object in C, id S I s, t are composable maps in S, then s t S S2 Given s Y, Z and W Y with s S, there exist morphisms in C completin the ollowin diarams in a commutative way s Y Y u W Z t s Y with t, u S, that is, t 1 = s 1 and s 1 = u 1 S3 Given, : Y, there exists W s with s = s i and only i there exists Y t Z with t = t, s, t S The multiplicative system S is said to be saturated i it satisies: S4 A morphism s Y belons to S i and only i there exist morphisms Y Z and W such that s S and s S I C is a trianulated cateory with translation unctor T, the multiplicative system S is said to be compatible with the trianulation i it satisies: S5 A morphism s belons to S i and only i T(s) belons to S S6 Given a morphism o trianles (α, β, γ) : (, Y, Z,,, h) (, Y, Z,,, h ), i α and β belon to S, then γ belons to S I S is a multiplicative system, the morphisms o the localization o C with respect to S can be described by a calculus o ractions I S is saturated and Q : C C[S 1 ] is the localization unctor, then s belons to S i and only i Q(s) is an isomorphism Finally, i S is compatible with the trianulation, C[S 1 ] is a trianulated cateory, with distinuished trianles isomorphic to imaes o distinuished trianles in C We reer to [Ve96, II2] or more details PROPOSITION 42 Let S be the class o quasi-isomorphisms in K(A) Then S is a saturated multiplicative system in K(A), compatible with the trianulation Proo See or instance [Ha66, I4]

12 12 REMARK 43 (1) The class S o quasi-isomorphism in C(A) is not a multiplicative system (2) The localization o C(A) with respect to S is isomorphic to the localization o K(A) with respect to S Now we can describe D(A) The objects are those o K(A) The morphisms Y in D(A) are equivalence classes o pairs s 1 = ( U s Y ) with s S The equivalence relation is deined as ollows: i there exists a commutative diaram ( U s Y ) ( V t Y ) U α h W u Y β V s t with u S, that is, s 1 t 1 i and only i (α s) 1 (α ) = u 1 h = (β t) 1 (β ) The composition o morphisms in D(A) can be visualized by the diaram W h u that is, U s Y V (t 1 ) (s 1 ) = (u t) 1 (h ), where the existence o u and h comes rom S2 Finally, the unctor Q : K(A) D(A) is deined as the identity in objects and sends a morphism Y to the equivalence class o the pair id 1 Y Y idy Y ) t Z = ( Since S is saturated, a morphism Y in K(A) is a quasi-isomorphism i and only i Q() is an isomorphism REMARK 44 (i) A complex in K(A) is quasi-isomorphic to zero i and only i Q() = 0 in D(A)

13 DERIVED CATEGORIES AND THEIR APPLICATIONS 13 (ii) Let Y be a morphism in C(A) Then Q() = 0 in D(A) i and only i there exists a quasi-isomorphism Y s Z such that s is homotopic to zero in C(A) Moreover, i is a monomorphism (epimorphism) in K(A), then Q() is so (iii) The cohomoloical unctor H n : K(A) A sends quasi-isomorphisms to isomorphisms, so it actors throuh D(A), inducin a cohomoloical unctor H n : D(A) A, or any n Z The automorphism T : K(A) K(A) sends quasi-isomorphisms to quasiisomorphisms, so it induces an automorphism T : D(A) D(A) THEOREM 45 The cateory D(A) is a trianulated cateory with translation unctor T and Q : K(A) D(A) is an additive unctor o trianulated cateories Proo The class o quasi-isomorphisms in K(A) is a multiplicative system compatible with the trianulation, so the localization is trianulated and the distinuished trianles are those isomorphic to imaes by Q o distinuished trianles in K(A) Clearly Q commutes with the translation unctor T and sends distinuished trianles to distinuished trianles Since A C(A) is ully aithul, we identiy objects and morphisms in A with objects and morphisms in C(A) concentrated in deree zero PROPOSITION 46 The composition A C(A) K(A) D(A) is a ully aithul unctor Proo Denote q : A D(A) the composition unctor For any object in A, q() is the complex concentrated in deree zero, and or any morphism Y, q() is the equivalence class o the pair ( Y idy Y ) Observe that the composition o q with the cohomoloical unctor H 0 : D(A) A is the identity o A The unctor q is aithul: i q() = 0, there exists a quasi-isomorphism s such that s is homotopic to zero; then H 0 (s) H 0 () = 0, but H 0 (s) is an isomorphism, and hence = H 0 () = 0 The unctor q is ull: let (q() Z s q(y )) be a representative o the equivalence class o a morphism in D(A) rom q() to q(y ) Since s is a quasi-isomorphism, the complex Z has cohomoloy Y in deree zero, and zero otherwise Then, the morphism o complexes Z Z 1 d 1 Z 0 d 0 Z 1 d 1 Z 2 u u 0 U 0 Z 0 / Imd 1 h 0 Z 1 d 1 Z 2 is a quasi-isomorphism and Y (u s)0 Z 0 / Imd 1 is the kernel o Z 0 1 h0 / Im d Z 1 Now, h 0 u 0 = d 0 = 0, so there exists a unique morphism Y in A such

14 14 that u 0 s 0 = u 0 Finally the commutative diaram says that s 1 and id 1 Y Z u s u U Y u s u s Y id Y are equivalent, so q is ull From now on, we will identiy A with the ull subcateory q(a) We already know that Hom D(A) (, Y ) Hom A (, Y ) or any pair o objects, Y in A Now we want to describe Hom D(A) ([m], Y [n]) or any m, n Z It is clear that Hom D(A) ([m], Y [n]) Hom D(A) (, Y [n m]), so we only have to study Hom D(A) (, Y [n]) or n 0 Followin Yoneda, let Ext n A(, Y ) be the set o isomorphism classes o exact sequences ψ : 0 Y Z n Z 1 0 in A It is clear that we can deine a map Ext n A (, Y ) Hom D(A)(, Y [n]) by sendin ψ to the equivalence class o the morphism ( Z s Y [n]) iven by Z 0 Z n Z 1 0 s Y [n] 0 Y with s a quasi-isomorphism On the other hand, let ( Z s Y [n]) be a representative o a map rom to Y [n] Since s is a quasi-isomorphism, the complex Z has cohomoloy zero except in deree n Consider the ollowin quasi-isomorphism Z Z n 1 d n 1 Z n Z n+1 u U 0 Z n / Imd n 1 Z n+1 Observe that i n < 0, then u = 0, so s 1 (u s) 1 (u ) 0 Hence Hom D(A) (, Y [n]) = 0 or any n < 0 I n > 0, consider the quasi-isomorphism W v U iven by

15 DERIVED CATEGORIES AND THEIR APPLICATIONS 15 0 Z n / Im d n 1 Z n+1 Z 1 Im d Z n / Im d n 1 Z n+1 Z 1 Z 0 Z 1 and observe that there exists a commutative diaram where t, are iven by Z u s u U Y [n] u s v t W W 0 Z n /Imd n 1 Z 1 Imd 1 0 t Y [n] 0 Y Finally the morphism s 1 t 1 rom to Y [n] can be associated with the exact sequence appearin in the irst row o the diaram 0 Y Z n /Im d n 1 Z n+1 Z 2 E 0 0 Y Z n /Im d n 1 Z n+1 Z 2 Z 1 Im d 1 0 This shows that there is a close connection between Hom D(A) (, Y [n]) and Ext n A (, Y ) In act, the ollowin theorem holds THEOREM 47 Let, Y be objects in A Then (i) Hom D(A) ([m], Y [n]) = Hom D(A) (, Y [n m]) or all n, m Z; (ii) Hom D(A) (, Y ) = Hom A (, Y ); (iii) Hom D(A) (, Y [n]) = Ext n A(, Y ) or all n > 0; (iv) Hom D(A) (, Y [n]) = 0 or all n < 0 An abelian cateory A is said to be semisimple i any short exact sequence in A splits

16 16 THEOREM 48 The derived cateory D(A) is abelian i and only i A is semisimple Proo I A is semisimple then D(A) is equivalent to the abelian semisimple cateory Z A, as we shall see in the second example o the ollowin section Assume that D(A) is abelian We know rom Proposition 28 that any monomorphism splits, and that any epimorphism splits Let Y be a morphism in D(A) Then is equal to the composition π Im ι Y Since π is an epimorphism and ι is a monomorphism, they split Hence there exist Im s and Y t Im such that i v = s t then = v Assume that A is not semisimple and let U V be a non-split monomorphism Then there exists a morphism q(v ) v q(u) in D(A) such that q() = q() v q() Now, q is ully aithull, so there exists a morphism V u U in A such that = u But is a monomorphism, so id U = u, a contradiction 5 Examples 51 Hereditary cateories Let A be an hereditary cateory, that is, Ext 2 A(, ) = 0 For instance, the cateory o abelian roups is hereditary In this case we can easily describe objects, morphisms and trianles in D(A) We start with the description o the objects o D(A) Let = ( n, d n ) be a complex in C(A) The vanishin o Ext 2 A(, ) implies that Ext 1 A(Z, ) is riht n 1 dn 1 exact or any Z in A Let n and consider the short exact sequences Since 0 Kerd n 1 n 1 Im d n Im d n 1 Kerd n H n () 0 Ext 1 A (Hn (), n 1 ) Ext 1 A (Hn (), Im d n 1 ) 0 is exact, there exists a morphism between short exact sequences 0 n 1 h n E n n H n () 0 s n 0 Im d n 1 Kerd n H n () 0 which induces the ollowin quasi-isomorphisms 0 H n 1 () H n () ` n 0 0 h n 0 0 «0 H n+1 () E n 1 n 1 E n n E n+1 n+1 ` s n id n 1 d n 1 n d n n+1 where s n is the composition E n sn Kerd n n So ( n, d n ) is quasi-isomorphic to (H n (), 0) = n Hn ()[ n]

17 DERIVED CATEGORIES AND THEIR APPLICATIONS 17 Let, Y be objects in A Then Hom D(A) ([n], Y [m]) Hom D(A) (, Y [m n]) Ext m n A (, Y ) which is zero except or m n {0, 1} Finally, Hom D(A) ([n], Y [n]) Hom A (, Y ) and Hom D(A) ([n], Y [n + 1]) Ext 1 A (, Y ) Concernin trianles, let [n] Y [n] be a morphism in D(A) The previous computation applied to the complex cone() : 0 Y 0 says that it is quasi-isomorphic to the complex 0 Ker 0 Coker 0 Hence any distinuished trianle is isomorphic to one o the orm [n] Y [n] Ker[n + 1] Coker[n] [n + 1] In particular, i 0 Y Z 0 is a short exact sequence in A, then Y Z δ [1] is a distinuished trianle in D(A), and δ Hom D(A) (Z, [1]) is the morphism associated to the iven short exact sequence 52 Semisimple cateories Let A be a semisimple cateory In this case, A is hereditary, so the conclusions in the previous example hold But now Ext 1 A (, ) = 0, so Hom D(A) ([n], Y [n]) Hom A (, Y ) and Hom D(A) ([n], Y [m]) = 0 or all m n Hence D(A) is equivalent to Z A 53 A = mod k(a 2 ) Let A be the ull subcateory o initely enerated let modules over the path alebra associated to the quiver A 2 : 1 2 In this case we only have three non-isomorphic indecomposable modules: the simple projective module S 2, the projective module o lenth two P 1, and the simple module S 1 Moreover, i, Y {S 1, S 2, P 1 }, then Hom A (, Y ) = k i (, Y ) = (S 2, P 1 ) or (P 1, S 1 ) and it is zero otherwise, and Ext 1 A(, Y ) = k i (, Y ) = (S 1, S 2 ) and it is zero otherwise Then D(A) has the ollowin picture S 1[ 1] S 2[0] P 1[0] S 1[0] S 2[1] P 1[1] S 1[1] where the composition o any two consecutive arrows is zero Observe that this cateory has neither monomorphisms nor epimorphisms All distinuished trianles can be visualized in the picture as the diarams o three consecutive arrows

18 18 6 Morita theory in D(A) Given an alebra A over a commutative rin k, we shall denote D(A) or the derived cateory D(Mod A) Given two k-alebras A and B the natural question is: when are D(A) and D(B) equivalent cateories? (trianle equivalent?) O course, i A and B are Morita equivalent (that is, Mod-A and Mod-B are k-linearly equivalent) then D(A) and D(B) are equivalent Are there other equivalences? The answer is yes In act, we shall present some examples later Rickard developed [Rick89], [Rick91] a Morita theory or derived cateories based on the notion o tiltin complex As we shall see this is a eneralization o the notion o tiltin module A summary o the history o the subject is developed or example in [KZ98] Keller s approach [Ke94] is a little dierent and we will ollow it This is also the point o view o [DG02] DEFINITION 61 (1) A unctor F between trianulated cateories S and T is said exact i it commutes whith shits and preserves distinuished trianles, that is, F is equipped with a natural isomorphism i : F([1]) F()[1], Obj(S) such that or every distinuished trianle in S, Y Z h [1] F() F() F(Y ) F() i F(h) F(Z) F()[1] is a distinuished trianle in T (2) An equivalence between two trianulated cateories is an equivalence o cateories which is exact and whose inverse unctor is also exact REMARK 62 I F : S T is exact then it is automatically additive We know rom classical Morita theory that two k-alebras A and B are Morita equivalent i and only i there exists a bimodule A P B such that it is initely enerated projective, balanced and enerator A-module and B End A (P) as a B- bimodule We have that P is a enerator or Mod-A, that is or every M Mod-A, there exists a set I and an epimorphism P (I) M For example, A is a enerator A-module but this proenerator ives the trivial equivalence The act that P is projective implies that P is a direct summand o A (n), or some n N, in particular Hom A (P, ) commutes with direct sums The notion o tiltin complex appears naturally i we look at the properties veriied by A[0] in D(A) The ree rank one A-module A considered as a cochain complex concentrated in deree 0 veriies that, n 0, Hom D(A) (A[n], A) = Hom D(A) (A, A[ n]) = Ext n A (A, A) = 0 DEFINITION 63 A complex in D(A) is a enerator o this cateory i and only i the smallest ull trianulated subcateory o D(A) containin and closed by ininite direct sums is D(A) EAMPLE 64 A[0] is a enerator o D(A)

19 DERIVED CATEGORIES AND THEIR APPLICATIONS 19 DEFINITION 65 A tiltin complex T or a k-alebra A is a bounded cochain complex o projective A-modules which enerates the derived cateory D(A) and such that the raded rin o endomorphisms Hom D(A) (T, T) is concentrated in deree 0 As a special case o tiltin complexes we have the tiltin modules over a inite dimensional k-alebra (k is a ield) (when thinkin them as their projective resolutions) DEFINITION 66 Let A be a inite dimensional k-alebra and T a initely enerated A-module We say that T is a tiltin module i (1) pdim A (T) 1 (2) Ext 1 A(T, T) = 0 (that is, there are no sel-extensions o T) (3) There exists a short exact sequence o A-modules 0 A T 1 T 2 0 such that T i are direct summands o inite direct sums o T (that is, T i add(t)) or i = 1, 2 REMARK 67 I T is a projective A-module the irst and second conditions are automatic In that case the exact sequence in 3 splits, so A is a direct summand o T n, or some n The third condition is veriied i and only i T is a enerator in Mod-A As a consequence A M End A (T) When A is sel-injective, every A-module o inite projective dimension is projective Then or these alebras, tiltin complexes are the same as Morita equivalences The ollowin theorem is due to Rickard [Rick89] THEOREM 68 Given two k-alebras A and B such that A or B is k-lat, then the ollowin are equivalent: (1) The unbounded derived cateories o A and B are equivalent as trianulated cateories (2) There is a tiltin complex T in D(A) whose endomorphism rin Hom D(A) (T, T) is isomorphic to B (3) There exists a cochain complex o B-A-bimodules M such that the derived tensor product L B M : D(B) D(A) is an equivalence o cateories We shall ive a proo o this theorem ollowin Keller For this we must recall some preliminaries irst 7 Derived cateory o a dierential raded alebra 71 DG alebras Let A = p Z A p be a dierential raded alebra (DG alebra or short), that is, A is Z-raded alebra with a morphism d : A A o deree one such that d(ab) = d(a)b + ( 1) p ad(b), i a A p, b A The map d is called dierential

20 20 EAMPLE 71 (1) Let A = A 0 be a k-alebra, and d = 0 (2) Given a k-alebra B and a complex (, d) o B-modules, let us take A = Hom B (, ), that is, A = p Z A p with A p = i Z Hom B( i, i+p ) and dierential deined by (δ()) i = d i ( 1) p i+1 d, or = ( i ) i Z A p 72 DG modules Let A be a DG alebra DEFINITION 72 A dierential raded A-module (DG A-module or short) is a Z-raded riht A-module M = p Z M p toether with a k-linear dierential d : M M (o deree one) such that d(ma) = d(m)a + ( 1) p md(a), m M p, a A A morphism o DG modules M and N is an A-linear map : M N o deree zero wich commutes with the dierentials EAMPLE 73 (1) The DG modules or the DG alebra A o the irst example o the previous subsection are the same as the cochain complexes o A-modules (2) Let A be the DG alebra o the second example o the previous subsection Each complex (N, d) o B-modules ives rise to a DG A-module Hom B (, N) with A-action ( j ) j Z ( i ) i Z = ( i+p i ) i Z, ( j ) Hom A (, N), ( i ) Hom A (, ) p Also, (, d) is a DG A-module by means o the action: Hom A (, ), ( i )(x j ) = ( i (x i )) 73 The homotopy cateory H(A) We recall that the homotopy cateory H(A) is deined as ollows Its class o objects is iven by Obj(H(A)) = Obj(DGMod-A) and the set o morphisms Hom H(A) (M, N) is deined as Hom A (M, N)/ where is the equivalence relation iven by identiyin homotopic maps There is a shit operator deined by S : H(A) H(A) (S(M)) p = M p+1 and d S(M) = d M, M Obj(H(A)) We recall that endowed with S and all the trianles isomorphic to the standard trianles, H(A) becomes a trianulated cateory EAMPLE 74 For the irst example o the previous subsection we have that H(A) is the standard homotopy cateory o cochain complexes o A-modules

21 DERIVED CATEGORIES AND THEIR APPLICATIONS The derived cateory We also recall that D(A) = H(A)[Σ 1 ], where Σ denotes the class o all the homotopy classes o quasi-isomorphisms EAMPLE 75 For the irst example o the previous subsection we have that D(A) may be identiied with the standard derived cateory o cochain complexes o A-modules REMARK 76 We notice that D(A) has ininite direct sums (ordinary sums o DG A-modules) Consider now the ree DG A-module A and let M be a DG A-module Then Hom H(A) (A, M) H 0 (M) (1) is bijective In particular, each quasi-isomorphism s : M M induces a bijection Hom H(A) (A A, M) Hom H(A) (A A, M ) (1) Since H 0 (M) Hom D(A) (A, M) then Hom H(A) (A, M) Hom D(A) (A, M) DEFINITION 77 A DG A-module M is said to be closed i Hom H(A) (M, N) Hom D(A) (M, N) or all N We shall denote by H p (A) the ull subcateory o H(A) ormed by closed objects EAMPLE 78 (1) Free DG A-modules o inte type are closed (2) Complexes o projective A-modules are closed (3) Suppose that M and N are closed and let : M N be a morphism o DG A-modules Consider the mappin cone cone() = S(M) N Then cone() is also closed REMARK 79 As a consequence H p (A) is a trianulated subcateory o H(A) Why are we interested in considerin the subcateory H p (A)? PROPOSITION 710 (1) For all M H(A) there is a quasi-isomorphism p M M with p M closed (2) The map M p M may be completed to a trianulated unctor commutin with ininite direct sums and such that it ives a trianulated equivalence D(A) H p (A) (3) H p (A) is the smallest ull subcateory o H(A) containin A and closed under ininite direct sums We will say that p M is the projective resolution o M EAMPLE 711 For M and A concentrated in deree zero, choose p M to be the homotopy class o any projective resolution o M Then p M is closed since all epimorphisms split EAMPLE 712

22 22 (1) I M is a riht bounded complex, p M is a projective resolution o the complex M (see [Ha66]) (2) For A = A 0, d = 0 and arbitrary M the description o p M has been obtained by Spaltenstein ([Sp88]) (3) For A and M arbitrary, see [Ke94] REMARK 713 The two last items o the previous proposition imply that D(A) coincides with the smallest ull subcateory containin A and closed by ininite direct sums 75 Let derived tensor unctors Let A and B be two DG alebras and let B A be a DG B-A-bimodule, that is, = p Z p with a k-linear map d : o deree one such that d(bxa) = d(b)xa + ( 1) p bd(xa) = d(b)xa + ( 1) p bd(x)a + ( 1) p+q bxd(a), or b B p, x q and a A Deine the DG alebra B op k A by (B op k A) n = p+q=n B p k A q and the map d is iven by d n : (B op k A) n (B op k A) n+1 d(b a) = d B (b) a + ( 1) p b d A (a), or b B p and a A q The product is iven by (b a)(b a ) = ( 1) qp b b aa, or b B p, b B p, a A q and a A q We notice then that is a riht DG B op k A-module since x(b a) = ( 1) rp bxa, b B p, x r, a A q Let N be a riht DG B-module We deine N k as the DG A-module with action o A in as beore and with the DG structure iven by (N k ) m = p+q=m N p k q, d(n x) = d N (n) x + ( 1) p n d (x), or n N p and x q The k-submodule enerated by all dierences nb x n bx is stable under d and under multiplication by elements o A So N B, the quotient module o N k by this submodule, is a well deined DG A-module Moreover, this construction is unctorial in N and The unctor ( ) B : DG-Mod-B DG-Mod-A yields a trianulated unctor rom H(B) to H(A), denoted by the same symbol We deine the let derived tensor product ( ) L B : D(B) D(A) by N L B := pn B Notice that ( ) L B commutes with direct sums since p( ) and ( ) B do LEMMA 714 The unctor F = ( ) L B is an equivalence i and only i the ollowin conditions hold:

23 DERIVED CATEGORIES AND THEIR APPLICATIONS 23 (a) The unctor F induces bijections Hom D(B) (B, B[n]) Hom D(A) ( A, A [n]), n Z (b) The unctor Hom D(A) ( A, ) commutes with (ininite) direct sums (c) The smallest ull trianulated subcateory o D(A) containin A and closed under (ininite) direct sums coincides with D(A) Proo The conditions are necessary since they hold in D(B) or B and they must be preserved by equivalences In order to prove that they are suicient let us consider the ull subcateory C o objects V in D(B) or which the maps Hom D(B) (B[n], V ) Hom D(A) ( A [n], F(V )) are bijective or all n Z This cateory is clearly closed under the shit and its inverse Usin the 5-lemma we check that it is a trianulated subcateory Usin (b) we et that it is closed under (ininite) direct sums Also, usin (a) we see that C contains B B Thus we must have that C = D(B) So, the ull subcateory C o objects U in D(B) such that Hom D(B) (U, V ) Hom D(A) (F(U), F(V )) is bijective or all V in D(B) contains B Aain it is closed under the shit and its inverse, and also closed under (ininite) direct sums It is a trianulated cateory by the 5-lemma Thus C = D(B) and as a consequence F is ully aithul Condition (c) shows that F is surjective EAMPLE 715 Suppose that φ : A B is a quasi-isomorphism, that is, a morphism o DG alebras inducin an isomorphism H (A) H (B) Then ( ) L B B A : D(B) D(A) is an equivalence In act, B A is isomorphic to A A in D(A), so the conditions (b) and (c) o the lemma beore hold In order to prove (a) consider the commutative diaram Hom D(B) (B, B[n]) Hom D(A) (B A, B[n] A ) H n (B) Hom H(A) (A, B[n] A ) Similarly, ( ) L A B B : D(A) D(B) is an equivalence We may perorm compositions o let derived unctors o this kind LEMMA 716 I A, B and C are DG k-alebras, A is a lat k-module, C Y B is a DG C-B-bimodule, we have or some DG C-A-bimodule Z (( ) L C Y ) L B ( ) L C Z,

24 24 Proo Take P = p ( B A ), considerin B A as a DG riht B op k A-module The morphism o unctors ( ) L B P ( ) L B is clearly invertible since the composition P F(B) = B L B P B B is an isomorphism and, usin a result o [Ke94], a morphism o unctor between trianulated cateories is invertible i and only i it is an isomorphism when one applies the unctors to a enerator o the cateory Also the morphism N L B P = ( pn) B P N B P is invertible in D(A) or each N D(B), usin the same result and the acts that P is closed over B op k A and that the unctor ( ) B (B op k A) ( ) k A preserves quasi-isomorphisms by the k-latness o A Thus (L L C Y ) L B (L L C Y ) B P p L C Y B P = L L C (Y B P), and we take Z = Y B P 8 Applications to tiltin theory Let k be a commutative rin, B a k-alebra, A a lat k-alebra Recall Rickard s theorem: THEOREM 81 Given two k-alebras A and B such that A or B is k-lat, then the ollowin are equivalent (1) The unbounded derived cateories o A and B are equivalent as trianulated cateories (2) There exists a cochain complex o B-A-bimodules such that the derived tensor product L B : D(B) D(A) is an equivalence o cateories Proo (1) (2) Let F : D(B) D(A) be the iven trianulated equivalence Take T = p(f(b B )) and B = Hom A (T, T) There are canonical isomorphisms Since T is closed in H(A), we also have H n ( B) Hom H(A) (T, T[n]), n Z Hom H(A) (T, T[n]) Hom D(A) (T, T[n]) Hom D(B) (B, B[n]), n Z Thus H n ( B) i n 0 and H 0 ( B) may be identiied with H 0 (B) = B I we view A as a DG alebra concentrated in deree zero we may view T as a DG B-Abimodule We claim that ( ) L B T : D( B) D(A) is an equivalence Since F is

25 DERIVED CATEGORIES AND THEIR APPLICATIONS 25 an equivalence, conditions (b) and (c) o lemma 714 clearly hold For (a) use the commutative diaram Hom D( B) ( B, B[n]) H n ( B) Hom D(A) (T, T[n]) Hom H(A) (T, T[n]) To establish a connection between B and B, let us introduce the DG subalebra C B with C n = B n or n 1, C 0 = Z 0 B and C n = 0 or n 1 Note that there are canonical morphisms Z 0 /dz 1 = H 0 ( B) = B C B which are both quasi-isomorphisms So, by example 715, we have a chain o equivalences D(B) ( ) L B B D(C) ( ) L B C T ( ) L B D( B) D(A) Applyin the previous lemma twice we et the required complex o B-A-bimodules 9 Appendix In this section we show an example o tiltin equivalence which is not a Morita equivalence The example is due to Schwede ([Sch04]) Consider a ield k and take the k-alebra A deined by: A = {upper trianular matrices in M 3 (k)} Up to isomorphism, there are three indecomposable A-projective modules: P 1 = {(y 1, y 2, y 3 ) y i k}, P 2 = {(0, y 2, y 3 ) y i k} and P 3 = {(0, 0, y 3 ) y i k} The P i are the projective covers o the simple modules S 1 = P 1 /P 2, S 2 = P 2 /P 3 and S 3 = P 3 In particular, S 3 is projective and pdim A (S i ) = 1 or i = 1, 2 Let us take T = P 1 P 2 S 2, which is clearly not projective The ollowin projective resolution o S 2 : 0 P 1 P 2 P 3 P 1 P 2 P 2 S 2 0 may be used to compute Ext 1 A (T, T) = Ext1 A (S2, T) = 0 The module P 1 P 2 P 3 is A-ree o rank one, then T is a tiltin A-module The k-alebra End A (T) may be identiied, by a direct computation, to the subalebra o M 3 (k) consistin o upper trianular matrices (x ij ) 1 i,j 3 such that x 23 = 0 Now A and End A (T) are NOT Morita equivalent since their lattices o projective modules dier Acknowledement: we want to thank Estanislao Herscovich or his help in the preparation o this notes

26 26 Reerences [Ve77] Deline, P Cohomoloie étale Séminaire de Géométrie Alébrique du Bois-Marie SGA Avec la collaboration de J F Boutot, A Grothendieck, L Illusie et J L Verdier Lecture Notes in Mathematics, Vol 569 Spriner-Verla, Berlin-New York, 1977 [DG02] Dwyer, W; Greenless, P Complete modules and torsion modules Amer J Math 124, (2002), pp [GM03] Geland, S I; Manin, Y I Methods o homoloical alebra Second edition Spriner Monoraphs in Mathematics Spriner-Verla, Berlin, 2003 [Ha66] Hartshorne, R Residues and duality Lecture notes o a seminar on the work o A Grothendieck, iven at Harvard 1963/64 With an appendix by P Deline Lecture Notes in Mathematics, No 20 Spriner-Verla, Berlin-New York, 1966, vii+423 pp [Ke94] Keller, B Derived DG-cateories Ann Sci Éc Norm Sup (4) 27, (1994), pp [K96] Keller, B Derived cateories and their uses Handbook o alebra, Vol 1, , North- Holland, Amsterdam, 1996 [KZ98] Köni, S; Zimmermann, A Derived equivalences or roup rins Lecture Notes in Math 1685, Spriner-Verla, 1998 [Kr05] Krause, H Derived cateories, resolutions, and Brown representability ariv mathkt/ [P62] Puppe, D On the ormal structure o stable homotopy theory Colloq alebr Topoloy, Aarhus 1962, (1962) [Rick89] Rickard, J Morita theory or derived cateories J London Math Soc (2) 39, (1989), [Rick91] Rickard, J Derived equivalences as derived unctors J London Math Soc (2) 43, (1991), [Sp88] Spaltenstein, N Resolutions o unbounded complexes Compositio Math 65 (1988), no 2, [Sch04] Schwede, S Morita theory in abelian, derived and stable model cateories Structured rin spectra, 33 86, London Math Soc Lecture Note Ser, 315, Cambride Univ Press, Cambride, 2004 [Ve96] Verdier, JL Des catéories dérivées des catéories abéliennes Astérisque 239 (1996) María Julia Redondo Instituto de Matemática, Universidad Nacional del Sur, Av Alem 1253, (8000) Bahía Blanca, Arentina mredondo@cribaeduar Andrea Solotar Departamento de Matemática, FCEyN, Universidad Nacional de Buenos Aires, Ciudad Universitaria, Pabellón I, (1428) Ciudad Autónoma de Buenos Aires, Arentina asolotar@dmubaar Recibido: 26 de octubre de 2006 Aceptado: 10 de abril de 2007