A short proof of HRS-tilting

A SHOR T PR OOF OF HRS-TIL TI NG XIAO-WU CHEN Abstract. W e give a short pro of to the following tilting theo rem by Happel, Reiten and Smalø via an explicit construction: give n tw o abeli an categories A and B suc h that B is tilted f rom A , then A and B are derived equiv alent. 1. Introduction Let A b e a n ab elian ca tegory . Recall that a torsion p air on A means a pa ir ( T , F ) of full sub c ategories satis fying (T1). Hom A ( T , F ) = 0 for a ll T ∈ T and F ∈ F ; both sub categor ies T and F are closed under dire c t summands; (T2). for each ob ject X ∈ A , there is a short exact sequence 0 − → T − → X − → F − → 0 for s o me T ∈ T and F ∈ F . In a torsio n pair ( T , F ), it follows that the s ubcatego ry T is clo sed under extensions and factor o b jects; F is closed under extensio ns and s ub ob jects. The torsio n pair ( T , F ) is called a t ilting torsion p air , pr o vided that each o b ject in A embeds into an o b ject in T . Dually the torsion pair ( T , F ) is called a c otilting t orsio n p air , provided that ea c h ob ject in A is a factor ob ject of an ob ject in F ([3 , Chapter I, section 3]). Denote a complex in A by X • = ( X n , d n X ) n ∈ Z where d n X : X n − → X n +1 is the differential satisfying d n +1 X ◦ d n X = 0; its shift X • [1] is a co mplex given by ( X • [1]) n = X n +1 and d n X [1] = − d n +1 X . Denote by D ( A ) the (unbounded) derive d c ate gory of A , D + ( A ), D − ( A ) and D b ( A ) the full s ub categor y co ns isting of bo unded-below, bounded- a bov e and b ounded complexes, resp ectiv ely ([6, 4]). W e will a lw ays iden tify the ab elian category A a s the full sub category of D ( A ) consisting o f stalk complex e s concen tra ted a t degree zer o ([4, p.40, Prop osition 4.3 ]). Let ( T , F ) b e a torsion pair on A . F ollowing [3, Chapter I, section 2 ], set B to b e the full sub c ategory of D ( A ) consisting of co mplex es X • satisfying H 0 ( X • ) ∈ T , H − 1 ( X • ) ∈ F and H i ( X • ) = 0 for i 6 = 0 , 1. Note that T ⊆ B and F [1] ⊆ B . By [3, Chapter I, Prop osition 2.1] the ca teg ory B is the heart of certain t - structure on D ( A ) and thus by [1] it is an abelia n category (also see [2]); mor eo ver the pair ( F [1] , T ) is a tor sion pa ir on B . One might exp e ct that the resulting new ab elian catego ry B is derived equiv alent to A . But this is in ge ne r al false (by the ex a mple in [3, p.1 6]). How ever Happ el, Reiten and Sma lø show the following remark able res ult ([3, C ha pter I, Theorem 3 .3]). Theorem (Happe l-Reiten-Smalø) Le t ( T , F ) b e a t ilting torsion p air on A . Th en we have a natur al e quivalenc e of t riangulate d c ate gories D ( B ) ≃ D ( A ) which is c omp atible with the inclusion of B into D ( A ) . Similar r esults hold for D ∗ ( − ) with ∗ ∈ { + , − , b } . Key wor ds and phr ases. torsion pair, til ting, derived equiv alence. This pro ject w as supp orted b y Alexander von Hum b oldt Stiftung, and was also partially supp orted b y China P ostdo ctoral Science F oundation No. 2007042 0125 and No. 200801230. The autho r also gratefully ac knowledge s the supp ort of K. C. W ong Education F oundation, Hong Kong. E-mail: xwc hen@mail.ustc.edu.cn. 1 2 XIAO- WU CHEN In the ca se of Theorem the c a tegory B is said to b e tilte d from A . N o te that the original theorem only claims the equiv a lence betw een the b ounded der iv ed catego ries and require s the existence o f enough pro jective or injective ob jects. The quo ted version is improved by No ohi ([5, Theorem 7.6 ]). W e will give a short pro of of the theor em via an explicit co nstruction of the eq uiv alence functor. 2. The Proof o f Theorem Throughout ( T , F ) is a torsion pa ir on A and B is the resulting ab elian ca tegory . W e star t with a n easy observ ation. Lemma 2.1. Consider an c omplex T • = ( T n , d n T ) n ∈ Z with terms in T . Then it is exact in A if and only if it is exact in B . Pro of. Assume that T • is exact in A . Since T is close d under factor ob jects in A , the complex T • splits in to shor t exact sequences ξ n : 0 − → T ′ n i n − → T n p n − → T ′ n +1 − → 0 with T ′ n ∈ T and d n T = i n +1 ◦ p n . Since B is a hea rt of ce r tain t -str uc tur e on D ( A ), a sequence 0 − → B 0 f − → B 1 g − → B 2 − → 0 in B is short exact if and o nly if there is a triangle B 0 f − → B 1 g − → B 2 − → B 0 [1] in D ( A ) ([1] and [2, Chapter IV, § 4]). Note that shor t exact sequences in A induces triangles in D ( A ) ([4, p.6 2, Pro position 6.1]). Hence ξ n bec ome s hort exact sequences in B . Thus by splicing them together w e show that the complex T • is exa ct in B . The “if” part is prov ed similarly .  The following re s ult is needed. Lemma 2. 2. (the “only if” part of [3, Chapter I, Pro p osition 3.2 (i) ]) L et ( T , F ) b e a torsion p air on A and let B as b efor e. If the torsion p air ( T , F ) is tilting, then the r esulting torsion p air ( F [1] , T ) on B is c otilting. Remark 2.3. Note that the co n verse of the ab ov e lemma is also true as sta ted in [3, Chapter I, Pr o positio n 3.2 (i) ]. How ever it seems to the a uthor that a dual a rgumen t of this lemma is not working. Instead, thanks to Theor em a nd then by combining [3 , Chapter I, Pro posi- tion 3.4] and the “only if” par t of [3, Chapter I, Prop osition 3 .2 (ii) ] one deduces that the conv erse holds (here one needs the fact that the eq uiv alence in Theor em is co mpatible with the inclus ion B ֒ → D ( A )). Pro of o f Theorem: Denote by K ( A ) the homotopy categ o ry o f complex es in A , K ( T ) ( r esp. K ex ( A )) its full sub category consisting of complexes in T ( r esp. exact complex e s). The inclusion K ( T ) ֒ → K ( A ) induces the following exact functor F : K ( T ) /K ( T ) ∩ K ex ( A ) − → D ( A ) . Since the tors ion pair ( T , F ) is tilting and T is c losed under factor ob jects, we hav e for ea c h X ∈ A a shor t exact s equence 0 − → X − → T 0 − → T 1 − → 0 with T i ∈ T . Note further that T is closed under extensio ns, we infer that the conditions in [4, p.42, Lemma 4.6 2 )] are fulfilled, and thus for each co mplex X • in K ( A ) there is a quasi- is omorphism X • − → T • with T • ∈ K ( T ). This implies tha t the functor F is dense and by [6, p.2 83, 4-2 Th ´ eor` eme] it is fully-faithful, that is, the functor F is a n equiv alence of triangulated categorie s. B y Lemma 2.2 we may apply the dual arg umen t to obtain a natural equiv alenc e G : K ( T ) /K ( T ) ∩ K ex ( B ) − → D ( B ) . By Lemma 2.1 we hav e K ( T ) ∩ K ex ( A ) = K ( T ) ∩ K ex ( B ). Hence F G − 1 : D ( B ) − → D ( A ) is the r equired equiv alence, where G − 1 denotes a quasi-inv er se of G . T o see other equiv alences, let ∗ ∈ { + , − , b } a nd let K ∗ ( − ) denote the co rresp onding homotopy categor ies. Note that in the argument ab ov e, for a co mplex X • ∈ K ∗ ( A ) we may take a qua si-isomorphism X • − → T • with T • ∈ K ∗ ( T ) (for the case ∗ = + , just consult A SHOR T PROOF OF HRS-TIL TING 3 the pro of in [4, p.43, 1)]; for the case ∗ = − , bec a use T is closed under factor ob jects o ne may replace T • by its go od truncations; for the case ∗ = b , cons ult the pro of in [4, p.43, 1)] and note that since T is closed under factor ob jects, the a rgument ther ein is done within finitely many steps, consequently the obtained complex T • is b ounded). Thus we construct the equiv alences F ∗ and G ∗ as a bov e. This proves the c orresp onding equiv alences b etw een the der iv ed categories D ∗ ( − ). Finally we will show that the obtained equiv alence F G − 1 is co mpa tible with the inclusion B ֒ → D ( A ). This is subtle. Given a n ob ject B ∈ B , since the torsion pair ( F [1 ] , T ) is cotilting, we hav e a sho rt exact se q uence in B , η : 0 − → T − 1 d − → T 0 g − → B − → 0 with T i ∈ T , in other words, a triangle ξ : T − 1 d − → T 0 g − → B − → T − 1 [1] in D ( A ). Then b y construction F G − 1 ( B ) is is omorphic to the complex T • = · · · − → 0 − → T − 1 d − → T 0 − → 0 − → · · · . No te that the co mplex T • is the mapping cone of d and th us for m the triangle ξ we obtain T • is isomor phic to B ([4 , p.23, Prop ostion 1 .1 c)]), in particular T • ∈ B . Note the following natura l triang le T − 1 d − → T 0 − → T • − → T − 1 [1] and thus a sho rt exa ct s equence γ : 0 − → T − 1 d − → T 0 − → T • − → 0 in B . Compa ring the sho rt exact sequences η a nd γ we obtain a unique is o morphism θ B : B ≃ T • in B . W e cla im that θ is natural in B and then we obtain a na tur al iso morphism betw een the inclusio n functor B ֒ → D ( A ) a nd the c o mposite B ֒ → D ( B ) F G − 1 − → D ( A ) (here we identify T • with F G − 1 ( B )). In fact, given a morphism f : B − → B ′ in B , choose an exact sequence η ′ : 0 − → T ′− 1 d ′ − → T ′ 0 g ′ − → B ′ − → 0 w ith T ′ i ∈ T . F orm the complex T ′• and then obtain the shor t exact sequence γ ′ and the iso morphism θ B ′ as ab ov e. Iden tify G ( T • ) with B , G ( T ′• ) with B ′ . Since the functor G is fully-faithful, we hav e a chain map φ • : T • − → T ′• such that G ( φ • ) = f . This implies the following c o mm utative exa c t diagr am in B 0 T − 1 φ − 1 d T 0 φ 0 g f 0 0 T ′− 1 d ′ T ′ 0 g ′ B ′ 0 . F rom this it is direct to see that θ B ′ ◦ f = φ • ◦ θ B in B and thus in D ( A ). This finis hes the pro of.  References [1] A. Beilinson, J. Bernstein, P. Deligne, F aisceaux p erv ers, Ast´ eri sque 10 0 , So ci ´ et ´ e M ath ´ ematique de F rance, P ari s, 1982. [2] S.I. G elf and, Yu I. Ma n in, Methods of homological algebra, Spri nger, 1996. [3] D. Happel, I. Reiten, S.O. Smalø, Tilting in Ab elian Categories and Quasitilted Algebras, M emoirs Amer. Math. Soc., vol. 120 no. 575, Pr o vince, Rho de Island, 1996. [4] R. Har tshorne, Residue and Duality , Lecture Notes i n Math. 20 , Springer-V erlag, 1966. [5] B. Noohi, Explicit HRS-t ilting, Journal of Noncommut ative Geometry , to appear. [6] J.L. Verdier , Cat ´ egories d´ eriv´ ees, ´ etat 0, Springer Lecture Notes 569 (1977), 262-311. Xiao-W u Chen, Department of Mathematics, Universit y of Science and T ec hnology of China, Hefei 230026, P . R. China Curr ent addr ess : Institut f uer Mathematik, Universitaet Paderborn, 33095, Paderborn, Deutsch land