COMPLETING PERFECT COMPLEXES

Transcription

1 COMPLETING PERFECT COMPLEXES HENNING KRAUSE, WITH APPENDICES BY TOBIAS BARTHEL AND BERNHARD KELLER Dedcated to the memory of Ragnar-Olaf Buchwetz. Abstract. Ths note proposes a new method to complete a trangulated category, whch s based on the noton of a Cauchy sequence. We apply ths to categores of perfect complexes. It s shown that the bounded derved category of fntely presented modules over a rght coherent rng s the completon of the category of perfect complexes. The result extends to non-affne noetheran schemes and gves rse to a drect constructon of the sngularty category. The parallel theory of completon for abelan categores s compatble wth the completon of derved categores. There are three appendces. The frst one by Tobas Barthel dscusses the completon of perfect complexes for rng spectra. The second one by Tobas Barthel and Hennng Krause refnes for a separated noetheran scheme the descrpton of the bounded derved category of coherent sheaves as a completon. The fnal appendx by Bernhard Keller ntroduces the concept of a morphc enhancement for trangulated categores and provdes a proper foundaton for completng a trangulated category. 1. Introducton Ths note proposes a new method to complete a trangulated category, and we apply ths to categores of perfect complexes [17]. For any category C, we ntroduce ts sequental completon Ĉ, whch s a categorcal analogue of the constructon of the real numbers from the ratonals va equvalence classes of Cauchy sequences, followng Cantor and Méray [5, 27]. When a rng Λ s rght coherent, then the category mod Λ of fntely presented modules s abelan and one can consder ts bounded derved category D b (mod Λ), whch contans the category of perfect complexes D per (Λ) as a full trangulated subcategory. The followng theorem descrbes D b (mod Λ) as a completon of D per (Λ). Theorem 1.1. For a rght coherent rng Λ there s a canoncal trangle equvalence D per (Λ) b D b (mod Λ) whch sends a Cauchy sequence n D per (Λ) to ts colmt. The descrpton of D b (mod Λ) as a completon extends to non-affne schemes. Thus for a noetheran scheme X there s a canoncal trangle equvalence D per (X) b D b (coh X). Date: June 15, Mathematcs Subject Classfcaton. 18E30 (prmary); 14F05, 16E35, 55P42 (secondary). Key words and phrases. Completon, Cauchy sequence, derved category, trangulated category, morphc enhancement, perfect complex, coherent rng, noetheran scheme, rng spectrum. Tobas Barthel was supported by the the Dansh Natonal Research Foundaton (DNRF92) and the European Unon s Horzon 2020 research and nnovaton programme under the Mare Sklodowska-Cure grant agreement No

2 2 HENNING KRAUSE In partcular, ths provdes a drect constructon of the sngularty category (n the sense of Buchwetz and Orlov [3, 36]) as the Verder quotent D per (X) b D per (X). The completon Ĉ of a category C comes wth an embeddng C Ĉ so that the objects n Ĉ are precsely the colmts of Cauchy sequences n C, and Ĉ dentfes wth a full subcategory of the Ind-completon of C n the sense of Grothendeck and Verder [14]. When C s trangulated, there s a natural fnteness condton such that Ĉ nherts a trangulated structure wth exact trangles gven as colmts of Cauchy sequences of exact trangles n C. Ths nvolves the noton of a phantom morphsm and Mlnor s exact sequence [28]. In order to explan ths, let us assume for smplcty that C dentfes wth the full subcategory of compact objects of a compactly generated trangulated category T. Fx a class X of sequences X 0 X 1 X 2 n C that s stable under suspensons. We consder ther homotopy colmts and have f and only f Ph(hocolm X, hocolm Y j ) = 0 for all X, Y X j lm 1 colm Hom(X, Y j ) = 0 for all X, Y X, j where Ph(U, V ) denotes the set of phantom morphsms U V. It s ths fnteness condton whch s satsfed for categores of perfect complexes, and t enables us to establsh a trangulated structure for the completon of C wth respect to X. The dea of completng a trangulated category C s not new; the method s always to dentfy the completon D wth a category of certan cohomologcal functors C op Ab. Note that the category of all cohomologcal functors s equvalent to the Ind-completon of C. In most cases, C dentfes wth the category of compact objects of a compactly generated trangulated category T, and D s another trangulated subcategory of T. Let us menton the paper of Neeman [29], whch has phantomless n ts ttle and uses a modfcaton of the axoms of a trangulated category. In [37], Rouquer dentfes varous natural choces of cohomologcal functors C op Ab. The recent work of Neeman [34, 35] employs the noton of approxmablty ; t s crucal for understandng the case of non-affne schemes and recommended as an alternatve approach va Cauchy sequences. We also nclude a dscusson of completons for abelan categores. Agan, some fnteness condton s needed so that the completon s abelan. For nstance, we show for a noetheran algebra Λ over a complete local rng that the completon of the category fl Λ of fnte length modules dentfes wth the category of artnan Λ-modules. Usng Matls dualty, ths yelds for Γ = Λ op trangle equvalences D b (mod Λ) op D b ( fl Γ) D b (fl Γ) b. Ths paper has three appendces. The frst one by Tobas Barthel dscusses completons for stable homotopy categores. In partcular, we see that Theorem 1.1 generalses to rng spectra. The second one by Tobas Barthel and Hennng Krause refnes for a separated noetheran scheme the descrpton of the bounded derved category of coherent sheaves as a completon. It s shown that the objects are precsely the colmts of Cauchy sequences of perfect complexes that satsfy an ntrnsc boundedness condton.

3 COMPLETING PERFECT COMPLEXES 3 In the fnal appendx, Bernhard Keller ntroduces the noton of a morphc enhancement of a trangulated category, followng [19]. Ths allows us to capture the noton of standard trangle and of coherent morphsm between standard trangles, generalsng analogous approaches va stable model categores or stable dervators. Morphc enhancements provde the approprate settng for turnng a completon nto a trangulated category. In fact, we see that n Theorem 1.1 the completon of the morphc enhancement of D per (Λ) dentfes wth the morphc enhancement of D b (mod Λ). Acknowledgement. Ths work benefted from dscussons at an Oberwolfach workshop n March Followng the sprt of ths workshop, t s ntended as a contrbuton to prsmatc algebra, so of potental common nterest to stable homotopy theory, representaton theory, and algebrac geometry. I wsh to thank Amnon Neeman for varous helpful comments on ths work, n partcular for drawng my attenton to the related noton of approxmablty, for provdng the proof of Lemma B.5, and for pontng out problems n some prevous versons of ths manuscrpt. Also, the nterest and comments of Greg Stevenson are very much apprecated. I am grateful to Tobas Barthel and Bernhard Keller for many stmulatng comments and for agreeng to nclude ther deas n form of an appendx. 2. The sequental completon of a category Let N = {0, 1, 2,...} denote the set of natural numbers, vewed as a category wth a sngle morphsm j f j. Now fx a category C and consder the category Fun(N, C) of functors N C. An object X s nothng but a sequence of morphsms X 0 X 1 X 2 n C, and the morphsms between functors are by defnton the natural transformatons. We call X a Cauchy sequence f for all C C the nduced map Hom(C, X ) Hom(C, X +1 ) s nvertble for 0. Ths means: C C n C N j n C Hom(C, X ) Hom(C, X j ). Let Cauch(N, C) denote the full subcategory consstng of all Cauchy sequences. A morphsm X Y s eventually nvertble f for all C C the nduced map Hom(C, X ) Hom(C, Y ) s nvertble for 0. Ths means: C C n C N n C Hom(C, X ) Hom(C, Y ). Let S denote the class of eventually nvertble morphsms n Cauch(N, C). Defnton 2.1. The sequental completon of C s the category Ĉ := Cauch(N, C)[S 1 ] that s obtaned from the Cauchy sequences by formally nvertng all eventually nvertble morphsms, together wth the canoncal functor C Ĉ that sends an object X n C to the constant sequence X d X d. A sequence X : N C nduces a functor and ths yelds a functor X : C op Set, Ĉ Fun(C op, Set), C colm Hom(C, X ), X X, because the assgnment X X maps eventually nvertble morphsms to somorphsms. We wll show that ths functor s fully fathful.

4 4 HENNING KRAUSE Let D be a category and S a class of morphsms n D. There s an explct descrpton of the localsaton D[S 1 ] provded that the class S admts a calculus of left fractons n the sense of [12], that s, the followng condtons are satsfed: (LF1) The dentty morphsm of each object s n S. The composton of two morphsms n S s agan n S. (LF2) Each par of morphsms X σ X Y wth σ S can be completed to a commutatve dagram σ X Y τ X Y such that τ S. (LF3) Let α, β : X Y be morphsms. If there s σ : X X n S such that ασ = βσ, then there s τ : Y Y n S such that τα = τβ. If S admts a calculus of left fractons, then the morphsms n D[S 1 ] are of the form σ 1 α gven by a par of morphsms X α Y σ Y n D wth σ S, where we dentfy a morphsm n D wth ts mage under the canoncal functor D D[S 1 ]. For pars (α 1, σ 1 ) and (α 2, σ 2 ) we have σ1 1 α 1 = σ2 1 α 2 n D[S 1 ] f and only f there exsts a commutatve dagram Y 1 α 1 σ 1 α σ X Y Y α 2 σ 2 Y 2 wth σ n S; see [12]. Lemma 2.2. The eventually nvertble morphsms n Cauch(N, C) admt a calculus of left fractons. We need some preparatons for the proof of ths lemma. Gven functors f : N N and X : N C, let X f denote the composte X f. Call f cofnal f n f(n) for all n N. In ths case there s a canoncal morphsm f X : X X f. A straghtforward computaton of fltered colmts n Set yelds the followng. Lemma 2.3. Let X, Y : N C be functors. (1) Gven a morphsm φ: X Ŷ, there exsts a morphsm α: X Y and a cofnal f : N N such that f Y φ = α. (2) Gven morphsms α, β : X Y such that α = β, there exsts a cofnal f : N N such f Y α = f Y β. Let S denote the class of eventually nvertble morphsms. If X : N C s Cauchy and f : N N s cofnal, then X f s Cauchy and the canoncal morphsm f X : X X f s n S. Moreover, for any σ : X Y n S there exsts a cofnal f : N N and a morphsm σ : Y X f such that σ σ = f X. Ths follows by applyng Lemma 2.3 to σ 1. Proof of Lemma 2.2. The condton (LF1) s clear. To check (LF2) fx a par of morphsms X σ X α Y wth σ S. Choose σ : X X f such that σ σ = f X.

5 COMPLETING PERFECT COMPLEXES 5 Then we obtan the followng commutatve square wth f Y S. σ X σ α α f Y X X f Y f To check (LF3) fx a par of morphsms α, β : X Y. Let σ : X X n S such that ασ = βσ. Ths mples α = β, and t follows from Lemma 2.3 that there exsts a cofnal f : N N such that f Y α = f Y β. Note that f Y S. Proposton 2.4. The canoncal functor Ĉ Fun(Cop, Set) s fully fathful; t dentfes Ĉ wth the colmts of Cauchy sequences of representable functors. Also, the canoncal functor C Ĉ s fully fathful. Proof. We use the fact that the class S of eventually nvertble morphsms n Cauch(N, C) admts a calculus of left fractons. Then every morphsm n Ĉ s of the form σ 1 α gven by a par of morphsms X α Y σ Y n Cauch(N, C) wth σ S. Fx Cauchy sequences X, Y : N C. We need to show that the canoncal map Hom(X, Y ) Hom( X, Ŷ ), σ 1 α σ 1 α, s a bjecton. The map s surjectve, because Lemma 2.3 yelds for any morphsm φ: X Ŷ a cofnal f : N N and α: X Y f such that φ = f Y 1 α. Now fx a par of morphsms σ1 1 α 1 and σ2 1 α 2 n Ĉ such that σ 1 α 1 1 = σ 2 1 α 2. Use (LF2) to complete σ 1 and σ 2 to a commutatve dagram Y Y Y σ 2 σ 1 τ 2 τ 1 wth τ 1, τ 2 S. Then we have τ 1 α 1 = τ 2 α 2 and there exsts a cofnal f : N N such that f Z τ 1 α 1 = f Z τ 2 α 2. Ths mples σ1 1 α 1 = σ2 1 α 2 n Ĉ. Colmts n Fun(C op, Set) are computed pontwse. Thus for X n Fun(N, C) the functor X s the colmt of the sequence Hom(, X 0 ) Hom(, X 1 ) Hom(, X 2 ) n Fun(C op, Set). It follows that Ĉ dentfes wth the colmts of Cauchy sequences of representable functors. Fnally, the canoncal functor C Ĉ s fully fathful, snce the composton wth Ĉ Fun(C op, Set) s fully fathful by Yoneda s lemma. Corollary 2.5. For X, Y Ĉ we have a natural bjecton Hom(X, Y ) lm Z f Y colm j Hom(X, Y j ). Proof. Combnng Proposton 2.4 and Yoneda s lemma, we have Hom(X, Y ) = Hom(colm Hom(, X ), colm Hom(, Y j )) j = lm Hom(Hom(, X ), colm Hom(, Y j )) j = lm colm Hom(X, Y j ). j Call C sequentally complete f every Cauchy sequence n C has a colmt n C. Clearly, C s sequentally complete f and only f the canoncal functor C Ĉ s an equvalence. We do not know whether Ĉ s always sequentally complete.

6 6 HENNING KRAUSE Remark 2.6. From Proposton 2.4 t follows that the sequental completon of C dentfes wth a full subcategory of the Ind-completon Ind(C) n the sense of [14, 8]. Remark 2.7. Let F : C D be a functor. (1) Suppose that F admts a left adjont. Then F preserves Cauchy sequences and nduces therefore a functor F : Ĉ D such that (2.1) F (X) = colm F (X ) for X Ĉ. (2) Suppose that D admts fltered colmts. Then F extends va (2.1) to a functor Ĉ D. (3) If (F, G) s an adjont par of functors that preserve Cauchy sequences, then ( F, Ĝ) s an adjont par snce Hom( F (X), Y ) = Hom(colm F (X ), colm Y j ) j = lm colm Hom(F (X ), Y j ) j = lm colm Hom(X, G(Y j )) j = Hom(X, Ĝ(Y )). The noton of a Cauchy sequence goes back to work of Bolzano and Cauchy (provdng a crteron for convergence), whle the constructon of the real numbers from the ratonals va equvalence classes of Cauchy sequences s due to Cantor and Méray [5, 27]. The sequental completon generalses ths constructon. Example 2.8. Vew the ratonal numbers (Q, ) wth the usual orderng as a category. Its sequental completon dentfes wth (R { }, ). Proof. A Cauchy sequence x Cauch(N, Q) s by defnton a sequence x 0 x 1 x 2 of ratonal numbers that s ether bounded, so converges to x R, or t s unbounded and we set x =. Gven a morphsm x y n Cauch(N, Q) that s eventually nvertble, we have x = ȳ. Conversely, f x = ȳ, then we defne u Cauch(N, Q) by u = mn(x, y ) and have morphsms x u y that are both eventually nvertble. Thus the assgnment x x yelds an equvalence (.e. an somorphsm of posets) Q R { }. Let I = {0 < 1} denote the poset consstng of two elements. For any category C, the category of morphsms n C dentfes wth C I = Fun(I, C). Example 2.9. Let C be a category that admts an ntal object. Then there s a canoncal equvalence ĈI ĈI. Proof. The equvalence F : Fun(I, Fun(N, C)) Fun(I N, C) Fun(N, Fun(I, C)) restrcts to an equvalence F 0 : Fun(I, Cauch(N, C)) Cauch(N, Fun(I, C)). In order to see ths, let φ: X Y be a morphsm n Fun(N, C) and α: C D a morphsm n C. Suppose X and Y are n Cauch(N, C). Thus there s n N such that Hom(C, X ) Hom(C, X +1 ) and Hom(D, Y ) Hom(D, Y +1 ) for n. Then Hom(α, φ ) Hom(α, φ +1 ) for n. Thus F (φ) s n Cauch(N, Fun(I, C)). On the other hand, f F (φ) s n Cauch(N, Fun(I, C)), then we choose α = d C and α: I D (I the ntal object n C) to see that X and Y are n Cauch(N, C). It s easly checked that F 0 nduces a functor ĈI ĈI, and we clam that t s an

7 COMPLETING PERFECT COMPLEXES 7 equvalence. Recall from Remark 2.6 that there s a canoncal embeddng Ĉ Ind(C). So t remans to use the fact that Ind(C) I Ind(C I ), whch follows from Propostons and n [14]. Let us generalse the defnton of the completon Ĉ, because later one we need to modfy the underlyng choce of Cauchy sequences. We fx a class X of objects n Fun(N, C). The completon of C wth respect to X s the category ĈX wth class of objects X and Hom(X, Y ) = lm colm j Hom(X, Y j ) for X, Y X. It follows from the defnton that the assgnment X colm Hom(, X ) nduces a fully fathful functor ĈX Fun(C op, Set). Clearly, Ĉ dentfes wth ĈX when X equals the class of Cauchy sequences, by Corollary 2.5. If C s an exact category and X denotes the class of sequences X such that each X X +1 s an admssble monomorphsm, then C := ĈX admts a canoncal exact structure and s called countable envelope of C [18, Appendx B]. 3. The sequental completon of an abelan category Let C be an addtve category. Lemma 3.1. The sequental completon Ĉ s an addtve category and the canoncal functor C Ĉ s addtve. Proof. The asserton follows from the fact that Cauch(N, C) s addtve and that the eventually nvertble morphsms admt a calculus of left fractons [12, I.3.3]. It follows that the assgnment X X yelds a fully fathful addtve functor Ĉ Add(C op, Ab) nto the category of addtve functors C op Ab. Lemma 3.2. If C admts kernels, then Ĉ admts kernels and C Ĉ s left exact. Proof. A morphsm X Y n Ĉ s up to somorphsm gven by a morphsm φ: X Y n Cauch(N, C). Then K := (Ker φ ) N s a Cauchy sequence, and ths yelds the kernel n Ĉ, because the sequence 0 K X Ŷ s exact n Add(Cop, Ab). Let A be an abelan category. We wrte fl A for the full subcategory of objects havng fnte composton length, and art A denotes the full subcategory of artnan objects. Example 3.3. Let A be a Grothendeck abelan category. Suppose that there are only fntely many somorphsm classes of smple objects and that the njectve envelope of each smple object s artnan. Then the sequental completon of fl A dentfes wth art A. Proof. Set C = fl A. The assumpton on A mples that an object X s artnan f and only f X s locally fnte (.e. a fltered colmt of fnte length objects) and the socle soc X has fnte length. In that case the socle seres of X yelds a Cauchy sequence soc 0 X soc 1 X n C wth colm (soc X) = X. Now let X Ĉ. The assgnment X X := colm X yelds a fully fathful functor Ĉ A. Clearly, X s locally fnte, and soc X = colm (soc X ) mples that soc X has fnte length, snce there are only fntely many smple objects n A. Thus X s artnan. We have already seen that every artnan object s of the form X for some Cauchy sequence X n C.

8 8 HENNING KRAUSE The precedng example suggests a general crteron such that the sequental completon of an abelan category s abelan. Let C be a length category. Thus C s an abelan category and every object has fnte length. We call C Ext-fnte f for every par of smple objects S and T the End(S)-module Ext 1 (S, T ) has fnte length. Proposton 3.4. Let C be an Ext-fnte length category havng only fntely many somorphsm classes of smple objects. Then Ĉ s an abelan category wth njectve envelopes satsfyng the (AB5) condton. Moreover, the canoncal functor C Ĉ nduces an equvalence C fl Ĉ. Proof. We embed C nto a Grothendeck abelan category va the functor C A := Lex(C op, Ab), X Hom(, X), where Lex(C op, Ab) denotes the category of left exact functors C op Ab; see [11]. Then Ĉ dentfes wth a subcategory of A va Proposton 2.4, and we clam that Ĉ = A 0 := {X A soc X C}. The assumpton on C mples for every X A 0 that X n := soc n X has fnte length for all n > 0. Ths s shown by nducton on n and we may assume that X s njectve. Then we have for any smple S A Hom(S, X n+1 /X n ) Ext 1 (S, X n ). Thus X n C mples X n+1 C. Next observe that A 0 s closed under takng quotents, and therefore a Serre subcategory of A, so also closed under subobjects and extensons. If X E(X) s an njectve envelope n A, then soc X soc E(X). Thus A 0 s an abelan category wth njectve envelopes, and the condton (AB5) for A mples the same for A 0. It remans to prove the above clam. If X = colm X s the colmt of a Cauchy sequence n C, then soc X has fnte length, snce there are only fntely many smple objects n C. On the other hand, f X A 0, then the socle seres of X yelds a Cauchy sequence n C wth colm (soc X) = X. Thus X Ĉ. Example 3.5. Let Λ be a rng and C Mod Λ a full subcategory of ts module category that contans Λ. Then C s sequentally complete. Proof. Let X Cauch(N, C). We have Hom(Λ, X ) = X, and therefore colm X belongs to C. 4. The sequental completon of a trangulated category Let T be a trangulated category and suppose that countable coproducts exst n T. Let φ 0 φ 1 φ 2 X 0 X 1 X 2 be a sequence of morphsms n T. A homotopy colmt of ths sequence s by defnton an object X that occurs n an exact trangle Σ 1 X 0 X d φ 0 X X. We wrte hocolm X for X and observe that a homotopy colmt s unque up to a (non-unque) somorphsm [2]. Recall that an object C n T s compact f Hom(C, ) preserves all coproducts. A morphsm X Y s phantom f any composton C X Y wth C compact s zero [7, 8]. The phantom morphsms form an deal and we wrte Ph(X, Y ) for the subgroup of all phantoms n Hom(X, Y ). Let us denote by T/ Ph the addtve category whch s obtaned from T by annhlatng all phantom morphsms.

9 COMPLETING PERFECT COMPLEXES 9 Lemma 4.1. Let C T be compact. Any sequence X 0 X 1 X 2 n T nduces an somorphsm colm Hom(C, X ) Proof. See [20, 5.1] or [30, Lemma 1.5]. Hom(C, hocolm X ). φ 2 φ 1 Recall that for any sequence A 2 A1 A0 of maps between abelan groups the nverse lmt and ts frst derved functor are gven by the exact sequence 0 lm A 0 A d φ 0 A lm 1 A 0. The followng result goes back to work of Mlnor [28] and was later extended by several authors, for nstance n [7, 8]. Lemma 4.2. Let X = hocolm X be a homotopy colmt n T such that each X s a coproduct of compact objects. Then we have for any Y n T a natural exact sequence and an somorphsm 0 Ph(X, Y ) Hom(X, Y ) lm Hom(X, Y ) 0 Ph(X, ΣY ) = lm 1 Hom(X, Y ). Proof. Apply Hom(, Y ) to the exact trangle defnng hocolm X and use that a morphsm X Y s phantom f and only f t factors through the canoncal morphsm X 0 ΣX. Let C T be a full addtve subcategory consstng of compact objects and consder the restrcted Yoneda functor T Add(C op, Ab), X h X := Hom(, X) C. Note that for any sequence X 0 X 1 X 2 n C we have by Lemma 4.1 (4.1) X = colm Hom(, X ) = h hocolm X. Lemma 4.3. Let X = hocolm X be a homotopy colmt n T such that each X s a coproduct of objects n C. Then we have for any Y n T a natural somorphsm Hom(X, Y ) Ph(X, Y ) Proof. Usng the precedng lemmas, we have Hom(X, Y ) Ph(X, Y ) Hom(h X, h Y ). = lm Hom(X, Y ) = lm Hom(h X, h Y ) = Hom(colm h X, h Y ) = Hom(h X, h Y ). Proposton 4.4. Let C T be a full addtve subcategory consstng of compact objects. Takng a sequence X 0 X 1 X 2 n C to ts homotopy colmt nduces a fully fathful functor Ĉ T/ Ph. Proof. We have the functor Ĉ Add(C op, Ab), X X, whch s fully fathful by Proposton 2.4. Now combne (4.1) and Lemma 4.3.

10 10 HENNING KRAUSE Defnton 4.5. Let C be a trangulated category and X a class of sequences (X ) N n C that s stable under suspensons,.e. (Σ n X ) N s n X for all n Z. We say that X s phantomless f for any par of sequences X, Y n X we have (4.2) lm 1 colm Hom(X, Y j ) = 0. j The followng lemma justfes the term phantomless. Lemma 4.6. Let C T be a full trangulated subcategory consstng of compact objects and X a class of sequences (X ) N n C that s stable under suspensons. Consder the full subcategory Then the followng are equvalent: D := {hocolm X T X X} T. (1) The class X s phantomless. (2) We have Ph(U, V ) = 0 for all U, V D. (3) The functor D Add(C op, Ab) takng X to Hom(, X) C s fully fathful. (4) The assgnment X hocolm X yelds an equvalence ĈX D. In ths case the homotopy colmt of a Cauchy sequence n C s actually a colmt n D, provded that X contans all constant sequences consstng of denttes only. Proof. (1) (2): Combne Lemmas 4.1 and 4.2. (2) (3): Apply Lemma 4.3. (2) (4): Apply Proposton 4.4. The fnal asserton follows from the dentty (4.1) for any sequence X n C, snce D dentfes wth a full subcategory of Add(C op, Ab). Recall that a trangulated category s algebrac f t s trangle equvalent to the stable category St(A) of a Frobenus category A. A morphsm between exact trangles X Y Z ΣX X Y Z ΣX n St(A) wll be called coherent f t can be lfted to a morphsm 0 X Ỹ Z 0 0 X Ỹ Z 0 between exact sequences n A (so that the canoncal functor A St(A) maps the second to the frst dagram). Note that any commutatve dagram X Y X Y n St(A) can be completed to a coherent morphsm of exact trangles as above. The followng theorem establshes a trangulated structure for the sequental completon of a trangulated category C. Let us stress that we use a relatve verson of ths result for our applcatons, as explaned n Remark 4.10 below; t depends on the choce of a class X of sequences n C whch s phantomless.

11 COMPLETING PERFECT COMPLEXES 11 Theorem 4.7. Let C be an algebrac trangulated category, vewed as a full subcategory of ts sequental completon Ĉ. Suppose that the class of Cauchy sequences s phantomless. Then Ĉ admts a unque trangulated structure such that the exact trangles are precsely the ones somorphc to colmts of Cauchy sequences that are gven by coherent morphsms of exact trangles n C. Theorem C.8 provdes a substantal generalsaton, from algebrac trangulated categores to trangulated categores wth a morphc enhancement. Proof. The proof s gven n several steps. (1) The assumpton on C to be algebrac mples that C dentfes wth the stable category St(A) of a Frobenus category A. Let A denote the countable envelope of A whch s a Frobenus category contanng A as a full exact subcategory; see [18, Appendx B]. Then C dentfes wth a full trangulated subcategory of compact objects of T := St(A ). For any sequence of coherent morphsms η 0 η 1 η 2 of exact trangles η : X Y Z ΣX n C there s n T an nduced exact trangle (4.3) hocolm X hocolm Y hocolm Z Σ(hocolm X ). Let us sketch the argument. We can lft the sequence (η ) N to a sequence of exact sequences η : 0 X Ỹ Z 0 n A and obtan a commutatve dagram wth exact rows X 0 Ỹ 0 X 0 Ỹ 0 Z 0 0 Z 0 n A. The vertcal morphsm are nduced by the morphsms η η +1, and takng mappng cones of the vertcal morphsms yelds the desred exact trangle (4.3). (2) The assumpton on Cauchy sequences n C to be phantomless mples that the functor Ĉ T takng a sequence to ts homotopy colmt nduces an equvalence Ĉ D := {hocolm X T X Cauch(N, C)}. Ths follows from Lemma 4.6. In partcular, the homotopy colmt of a Cauchy sequence n C s actually a colmt n D. (3) We clam that D s a trangulated subcategory of T and that the exact trangles n D are up to somorphsm the colmts of Cauchy sequences gven by coherent morphsms of exact trangles n C. For a Cauchy sequence gven by coherent morphsms of exact trangles X Y Z ΣX n C, we form n D ts colmt and obtan an exact trangle (4.3); t does not depend on any choces. Conversely, fx an exact trangle η : X Ȳ Z Σ X n D that s gven by X, Y Cauch(N, C) wth X = colm X and Ȳ = colm Y. The morphsm X Ȳ s up to somorphsm gven by a morphsm φ: X Y n Cauch(N, C), so of the form colm φ. Now complete the φ : X Y to a sequence of coherent morphsms between exact trangles X Y Z ΣX n C. It s easly checked that (Z ) N s a Cauchy sequence; set Z := colm Z. Ths yelds an exact trangle η : X Ȳ Z Σ X n T, keepng n mnd the above remark about homotopy colmts of exact trangles. It follows that D s closed under the formaton of cones and therefore a trangulated subcategory of T. Clearly, η and η are somorphc

12 12 HENNING KRAUSE trangles. Thus any exact trangle n D s up to somorphsm a colmt of exact trangles n C. Corollary 4.8. Let T be an algebrac trangulated category wth countable coproducts and C T a full trangulated subcategory consstng of compact objects. Suppose the class of Cauchy sequences n C s phantomless. Then the full subcategory {hocolm X T X Cauch(N, C)} T s a trangulated subcategory whch s trangle equvalent to Ĉ. The exact trangles are precsely the ones somorphc to colmts of Cauchy sequences gven by coherent morphsms of exact trangles n C. For a generalsaton of Corollary 4.8 from algebrac trangulated categores to trangulated categores wth a morphc enhancement, see Secton C.7. Remark 4.9. To be phantomless s a condton whch can be checked for a specfc sequence Y = (Y ) N n C. Gven C C, call a subgroup U Ŷ (C) = colm Hom(C, Y j ) j of fnte defnton f t arses as the mage of a map Ŷ (D) Ŷ (C) for some morphsm C D n C; see [15]. The descendng chan condton (dcc) for subgroups of fnte defnton mples that (4.2) holds for all sequences X n C, snce t mples the Mttag-Leffler condton for Ŷ (X 2) Ŷ (X 1) Ŷ (X 0). The dcc for subgroups of fnte defnton s equvalent to Y beng Σ-pure-njectve when vewed as an object n Ind(C), by [9, 3.5]. On the other hand, when T s a compactly generated trangulated category, then Z T s pure-njectve f and only f Ph(, Z) = 0, by [22, Theorem 1.8]. Let C be a trangulated category and fx a cohomologcal functor H : C A nto an abelan category. Set H n := H Σ n for n Z. We call a sequence (X ) N n C bounded f colm H n (X ) = 0 for n 0 and wrte Ĉ b := {X Ĉ X bounded} for the full subcategory of bounded objects. Remark Suppose for all C C that H n (C) = 0 for n 0. Then we may restrct ourselves to bounded Cauchy sequences, and f ths class s phantomless, then the concluson of Theorem 4.7 holds for Ĉb. More generally, fx a class X Fun(N, C) that s closed under suspensons and cones. When X s phantomless, then the concluson of Corollary 4.8 holds for Ĉ X {hocolm For more detals, cf. Secton C.5. X T X X}. Example Let Λ be a quas-frobenus rng of fnte representaton type. Then the class of all sequences n the stable category St(mod Λ) s phantomless. In fact, ths holds for any locally fnte trangulated category [24, 40] and can be deduced from Remark 4.9. The followng example s a contnuaton of Example 3.3. For an abelan category A let D b (A) denote ts bounded derved category.

13 COMPLETING PERFECT COMPLEXES 13 Example Let k be a commutatve rng and A a k-lnear Grothendeck abelan category such that Hom(X, Y ) s a fnte length k-module for all X, Y fl A. Suppose that there are only fntely many somorphsm classes of smple objects and that the njectve envelope of each smple object s artnan. Then the class of Cauchy sequences n D b (fl A) s phantomless and we have trangle equvalences D b ( fl A) D b (art A) D b (fl A) b. Proof. The frst equvalence s clear from Example 3.3; so we focus on the second one. We may assume that all objects n A are locally fnte. For all X, Y D b (fl A) the k-module Hom(X, Y ) has fnte length, snce Ext n (S, T ) has fnte length for all smple S, T and n 0 by our assumptons on A. It follows that the class of Cauchy sequences n D b (fl A) s phantomless by the Mttag-Leffler condton. We wsh to apply Corollary 4.8 and choose for T the category K(Inj A) of complexes up to homotopy, where Inj A denotes the full subcategory of njectve objects n A. Set nj A = Inj A art A. Then D b (fl A) dentfes wth T c va F : D b (art A) K +,b (nj A) K(Inj A), by [23, Proposton 2.3]. Set C := D b (fl A). Then Corollary 4.8 yelds a trangle equvalence Ĉ D := {hocolm X T X Cauch(N, C)}. We clam that F nduces an equvalence D b (art A) D b := {X D colm H n (X ) = 0 for n 0}. Any object M art A s the colmt of the Cauchy sequence (soc M) N n fl A by Example 3.3, and ths yelds a Cauchy sequence n D b (fl A). Thus F maps art A nto D b, and therefore also D b (art A), snce F s an exact functor and D b (art A) s generated by art A as a trangulated category. Conversely, let X = colm X be an object n D b. We may assume that the complex X s homotopcally mnmal, as n [23, Appendx B]. The Cauchy condton mples for each smple S A and n Z that Hom(S, Σ n X) has fnte length over k, so the degree n component of X s artnan. Thus X belongs to K +,b (nj A), and ths yelds the clam. 5. Homologcally perfect objects Let T be a compactly generated trangulated category and denote by T c the full subcategory of compact objects. We fx a cohomologcal functor H : T A nto an abelan category. For n Z set H n := H Σ n. Defnton 5.1. We say that an object X n T s homologcally perfect (wth respect to H) f X can be wrtten as homotopy colmt of a sequence X 0 X 1 X 2 n T c such that the followng holds: (HP1) The sequence (X ) N s a Cauchy sequence n T c, that s, for every C T c Hom(C, X ) Hom(C, X +1 ) for 0. (HP2) For every n Z we have H n (X ) H n (X +1 ) for 0. (HP3) For almost all n Z we have H n (X ) = 0 for 0. In our applcatons the homologcally perfect objects admt an ntrnsc descrpton whch does not depend on the choce of H. The followng lemma makes ths precse when the cohomologcal functor H s gven by a compact generator. For noetheran schemes that are non-affne, the proof s more nvolved and we refer to Theorem B.1.

14 14 HENNING KRAUSE Lemma 5.2. Let G be a compact object n T that generates T c as a trangulated category. Then for X T the followng are equvalent: (1) The object X s homologcally perfect wth respect to H = Hom(G, ). (2) The object X can be wrtten as homotopy colmt of a Cauchy sequence n T c, and for every C T c we have Hom(C, Σ n X) = 0 for n 0. Proof. The assumpton on H mples that the condtons (HP1) and (HP2) are equvalent. Also, (HP3) s equvalent to the condton that for every C T c we have Hom(C, Σ n X) = 0 for n 0, snce H n (X) = colm H n (X ) by Lemma 4.1. Now fx a rng Λ. We wrte D(Λ) for the derved category of the abelan category of all Λ-modules. Let D per (Λ) denote the full subcategory of perfect complexes, that s, objects somorphc to bounded complexes of fntely generated projectve modules. The trangulated category D(Λ) s compactly generated and the compact objects are precsely the perfect complexes. Let mod Λ denote the category of fntely presented Λ-modules and proj Λ denotes the full subcategory of fntely generated projectve modules. When Λ s a rght coherent rng, then mod Λ s abelan and we consder ts derved category D b (mod Λ) usng the followng dentfcatons K b (proj Λ) K,b (proj Λ) D per (Λ) D b (mod Λ) where the top row conssts of categores of complexes of modules n proj Λ up to homotopy. Note that D per (Λ) = D b (mod Λ) when Λ has fnte global dmenson, and the converse holds when Λ s rght noetheran. We provde an ntrnsc descrpton of the objects from D b (mod Λ), whch uses for any complex X the sequence of truncatons gven by σ n 1 X σ n X σ n+1 X σ n X 0 0 X n X n+1 d d X X n 2 X n 1 X n X n+1. Lemma 5.3. Let Λ be a rght coherent rng. Then X n D(Λ) s homologcally perfect f and only f X belongs to D b (mod Λ). Proof. Let X be a complex n K,b (proj Λ) = D b (mod Λ) and wrte X as homotopy colmt of ts truncatons X = σ X whch le n K b (proj Λ). It s clear that X s homologcally perfect. In fact, D per (Λ) s generated by Λ; so t suffces to check the functor H n = Hom(Λ, Σ n ) for every n Z. We have H n (X ) H n (X +1 ) for 0 and H n X = 0 for n 0. On the other hand, f X s homologcally perfect, then H n X s fntely presented for all n, so X les n D b (mod Λ). 6. The bounded derved category Let Λ be a rng. We consder the category mod Λ of fntely presented Λ-modules and ts bounded derved category D b (mod Λ). Our am s to dentfy D b (mod Λ) wth a completon of D per (Λ); compare ths wth Rouquer s [37, Corollary 6.4].

15 COMPLETING PERFECT COMPLEXES 15 Lemma 6.1. Let Λ be a rng and set P = proj Λ. Then the functor s fully fathful. K,b (P) Add(K b (P) op, Ab), X h X := Hom(, X) K b (P), Proof. We vew K,b (P) as a subcategory of D(Λ). Let X, Y be objects n K,b (P) and wrte X as homotopy colmt of ts truncatons X = σ X whch le n K b (P). Let C denote the cone of X X +1. Ths complex s concentrated n degree +1; so Hom(C, Y ) = 0 for 0. Thus X X +1 nduces a bjecton Ths mples Hom(X +1, Y ) Hom(X, Y ) for 0. Hom(X, Y ) lm Hom(X, Y ) and therefore Ph(X, Y ) = 0 by Lemma 4.2. From Lemma 4.3 we conclude that Hom(X, Y ) Hom(h X, h Y ). Let C be a trangulated category and fx a cohomologcal functor H : C A. Recall that an object X n Ĉ s bounded f colm H n (X ) = 0 for n 0, and Ĉ b denotes the full subcategory of bounded objects n Ĉ. From Theorem 4.7 and Remark 4.10 we know that Ĉb admts a canoncal trangulated structure when C s algebrac and bounded Cauchy sequences are phantomless. Theorem 6.2. For a rght coherent rng Λ there s a canoncal trangle equvalence D per (Λ) b D b (mod Λ) whch sends a Cauchy sequence n D per (Λ) to ts colmt. Proof. We consder the functor D b (mod Λ) Add(D per (Λ) op, Ab), X Hom(, X) D per (Λ), whch s fully fathful by Lemma 6.1. On the other hand, we have the functor D per (Λ) b Add(D per (Λ) op, Ab), X X, whch s fully fathful by Proposton 2.4. Both functors have the same essental mage by Lemma 5.3, because we can dentfy ths wth a full subcategory of D(Λ) by Lemma 4.6. Ths yelds a trangle equvalence, snce the trangulated structures of both categores dentfy wth the one from D(Λ); see Corollary 4.8 plus Remark Remark 6.3. The trangulated category D per (Λ) admts a morphc enhancement whch s gven by D per (Λ 1 ), wth Λ 1 the rng of upper trangular 2 2 matrces over Λ. Ths enhancement can be completed and yelds a morphc enhancement of Dper (Λ) b that dentfes wth the morphc enhancement of D b (mod Λ); see Secton C.8. Ths observaton enrches the trangle equvalence of Theorem 6.2. For a noetheran algebra over a complete local rng, there s another descrpton of D b (mod Λ) whch s obtaned by completng the category of fnte length modules over Λ op. Proposton 6.4. Let Λ be a noetheran algebra over a complete local rng and set Γ = Λ op. Then there are trangle equvalences D b (mod Λ) op D b ( fl Γ) D b (fl Γ) b. Proof. Matls dualty gves an equvalence (mod Λ) op art Γ, so D b (mod Λ) op D b (art Γ), and we have art Γ fl Γ by Example 3.3. Ths yelds the frst functor, and the second s from Example 4.12.

16 16 HENNING KRAUSE 7. Pseudo-coherent objects Let T be a trangulated category and H : T A a cohomologcal functor nto an abelan category. For n Z set H n := H Σ n. Also let and T >n := {X T H X = 0 for all n} T n := {X T H X = 0 for all > n}. We suppose for all X, Y T and n Z the followng: (TS1) There s an exact trangle τ n X X τ >n X Σ(τ n X) wth τ n X T n and τ >n X T >n. (TS2) Hom(X, Y ) = 0 for X T n and Y T >n. Thus the category T s equped wth a t-structure [1]. Note that for any morphsm X Y n T we have τ >n X τ >n Y H X H Y for all > n. Now suppose that T s compactly generated and wrte T c for the full subcategory of compact objects. Defnton 7.1. An object X T s called pseudo-coherent (wth respect to the chosen t-structure) f X can be wrtten as homotopy colmt of a sequence X 0 X 1 X 2 n T c such that τ > X τ > X for all 0. We say that X has bounded cohomology f H n X = 0 for n 0. Lemma 7.2. The functor T Add((T c ) op, Ab), X h X := Hom(, X) T c, s fully fathful when restrcted to pseudo-coherent objects wth bounded cohomology. Proof. Let X, Y be objects n T. Suppose that X = hocolm X s pseudo-coherent and H n Y = 0 for n 0. Let C denote the cone of X X +1. The nduced morphsm τ > X τ > X +1 s an somorphsm snce τ > τ > (+1) = τ >. Thus C T and therefore Hom(X +1, Y ) Hom(X, Y ) for 0. It follows from Lemma 4.2 that Ph(X, Y ) = 0, so Hom(X, Y ) Hom(h X, h Y ) by Lemma 4.3. Example 7.3. Let Λ be a rng and T = D(Λ) the derved category of the category of all Λ-modules wth the standard t-structure. Then the canoncal functor K (proj Λ) D(Λ) dentfes K (proj Λ) wth the full subcategory of pseudocoherent objects n D(Λ). Proof. For X K (proj Λ) and 0 set X := σ X. Then we have X = hocolm X = X and τ > X τ > X for all 0. Thus X s pseudo-coherent. The other mplcaton s left to the reader. The example shows that for a rght coherent rng Λ and any object X n T = D(Λ) the followng are equvalent: (PC) X s pseudo-coherent and has bounded cohomology. (HP) X s homologcally perfect.

17 COMPLETING PERFECT COMPLEXES 17 Ths seems to be a common phenomenon (cf. Propostons 8.1 and A.1) though we do not have a general proof. Let C be a trangulated category and fx a cohomologcal functor H : C A. Call a sequence (X ) N n C strongly bounded f colm H n (X ) = 0 for n 0, and f for every n Z we have H n (X ) H n (X +1 ) for 0. By abuse of notaton, we wrte Ĉb for the full subcategory of strongly bounded objects n Ĉ.1 Lemma 7.4. Suppose that (PC) (HP) for all X T, and set C := T c. Then the functor F : Ĉb T, X hocolm X, s fully fathful functor and dentfes Ĉb wth the full subcategory of pseudo-coherent objects havng bounded cohomology. When T admts a morphc enhancement, then F s a trangle functor. Proof. The frst asserton follows from Lemmas 4.6 and 7.2. For the second asserton, see Lemma C Noetheran schemes We fx a noetheran scheme X. Let Qcoh X denote the category of quas-coherent sheaves on X, and coh X denotes the full subcategory of coherent sheaves. We consder the derved categores D per (X) D b (coh X) D(Qcoh X). The trangulated category D(Qcoh X) s compactly generated and the full subcategory of compact objects agrees wth the category D per (X) of perfect complexes [31]. We use the standard t-structure and then the above noton of a pseudo-coherent object dentfes wth the usual one; see [17, 2.3], [39, 2.2], and [38, 0DJM]. A precse reference s [38, Lemma 0DJN], whch uses approxmatons and bulds on work of Lpman and Neeman [25]. We obtan the followng descrpton of the objects n D b (coh X). For a refnement, see Theorem B.1. Proposton 8.1. For an object X n D(Qcoh X) the followng are equvalent: (1) X belongs to D b (coh X). (2) X s pseudo-coherent and has bounded cohomology. (3) X s homologcally perfect. Proof. (1) (2): See [39, Example 2.2.8]. (2) (3): Let X = hocolm X be pseudo-coherent and C a perfect complex. The argument n the proof of Lemma 7.2 shows that (X ) N s a Cauchy sequence n D per (X). More precsely, the cone of X X +1 belongs to T, and therefore Hom(C, X ) Hom(C, X +1 ) for 0; see [38, 09M2]. Also, H n (X ) H n (X +1 ) for all > n. Fnally, for almost all n Z we have H n (X ) = H n (X) = 0 for 0, snce X has bounded cohomology. (3) (1): Let X = hocolm X be homologcally perfect and n Z. Then H n (X) = colm H n (X ) equals the cohomology of some perfect complex, so H n (X) s coherent. Also, H n (X) = 0 for n 0. The followng s now the analogue of Theorem 6.2 for schemes that are not necessarly affne. The proof s very smlar; see also Lemma 7.4 for the general argument. 1 The condton H n (X ) H n (X +1 ) for 0 s automatc for a Cauchy sequence X when H = Hom(C, ) for an object C C.

18 18 HENNING KRAUSE Theorem 8.2. For a noetheran scheme X there s a canoncal trangle equvalence D per (X) b D b (coh X) whch sends a Cauchy sequence n D per (X) to ts colmt. Proof. We consder the functor D b (coh X) Add(D per (X) op, Ab), X Hom(, X) D per (X), whch s fully fathful by Lemma 7.2 and Proposton 8.1. On the other hand, we have the functor D per (X) b Add(D per (X) op, Ab), X X, whch s fully fathful by Proposton 2.4. Both functors have the same essental mage by Proposton 8.1, because we can dentfy ths wth a full subcategory of D(Qcoh X) by Lemma 4.6. Ths yelds a trangle equvalence, snce the trangulated structures of both categores dentfy wth the one from D(Qcoh X); see Corollary 4.8 plus Remark An mmedate consequence s the followng. Corollary 8.3. The sngularty category of X (n the sense of Buchwetz and Orlov [3, 36]) dentfes wth the Verder quotent D per (X) b D per (X). Appendx A. Homologcally perfect objects n homotopy theory by Tobas Barthel Let R be an assocatve rng spectrum and let D R be the derved category of rght R-module spectra as constructed for example n [10]; f no confuson s lkely to arse, we wll refer to an object n D R smply as an R-module. If R s connectve, then D R nherts the standard t-structure from the stable homotopy category, and we denote by D b R D R the full trangulated subcategory of bounded R-module spectra,.e., those R-modules M wth π M fntely presented over π 0 R and π M = 0 for 0. As usual, D c R D R s the full subcategory of compact R-modules or, equvalently, the thck subcategory generated by the rght R-module R. Throughout ths appendx, we wll employ homologcal gradng, so for example the pseudo-coherence condton τ > M τ > M ntroduced n Secton 7 translates to π j M π j M for all j <. Proposton A.1. Suppose R s a connectve assocatve rng spectrum wth π 0 R rght coherent and π R fntely presented over π 0 R for all 0. The followng condtons on M D R are equvalent: (1) M s pseudo-coherent and has bounded homotopy. (2) M s homologcally perfect. (3) M belongs to D b R. Proof. The mplcaton (1) (2) s proven as n Proposton 8.1 testng aganst free R-modules C = Σ n R for all n Z. In order to see that (2) (3), we frst observe that any compact R-module has fntely presented homotopy groups by assumpton on R. Therefore, any homologcally perfect M D R can only have fntely many nonzero homotopy groups, all of whch must be fntely presented over π 0 R by the Cauchy condton. Thus, M s bounded. Now assume that M D b R, then M has bounded homotopy and t remans to show that M has to be pseudo-coherent. To ths end, we use a mld varant of

19 COMPLETING PERFECT COMPLEXES 19 the cellular tower constructon of [10, Thm. III.2.10] or [16, Prop ], n whch we only attach R-cells of a fxed dmenson n each step. Indeed, let M D R be a bounded below R-module wth fntely presented homotopy groups and assume wthout loss of generalty that the lowest nonzero homotopy group s n degree 0. Inductvely, we construct a tower of R-modules (M k ) k 0 under M wth: () M 0 = M. () For all k 0, there s a cofber sequence F k = G(k) Σ k R M k M k+1, where the drect sum s ndexed by a set G(k) of generators of the fntely presented π 0 R-module π k M k. It follows by nducton on k, the assumpton on R, and the long exact sequence n homotopy that π M k+1 s fntely presented over π 0 R n all degrees and zero below degree k + 1, whch allows us to construct the map F k+1 M k+1 and to proceed wth the nducton. Set M k = fb(m M k ). The octahedral axom provdes fber sequences (A.1) and a sequence of R-modules M k M k+1 F k M 0 M 1 M 2... over M. The fber sequences (A.1) mply that π M k π M k+1 and hence π M k π M for all < k, whch then also shows that the homotopy colmt over (M k ) k 0 s equvalent to M by a connectvty argument. Moreover, because M k s bult from fntely many R-cells, M k s compact for any k 0. It follows that M s pseudo-coherent as desred. In lght of Lemma 7.4, we obtan the followng consequence. Corollary A.2. Wth notaton as n Proposton A.1, takng homotopy colmts nduces an equvalence D c Rb D b R of trangulated categores. In partcular, the corollary appled to the Elenberg Mac Lane rng spectrum HΛ of a rght coherent rng Λ recovers Theorem 6.2. Lemma A.3. Let R be as n Proposton A.1 and assume addtonally that the rght global dmenson of π 0 R s fnte, then D b R concdes wth the thck subcategory of D b R generated by the Elenberg Mac Lane R-module Hπ 0R. Proof. Snce π 0 R has fnte rght global dmenson, any fntely presented π 0 R- module N admts a fnte length resoluton by fntely presented projectve π 0 R- modules. Ths mples that the R-module spectrum HN belongs to Thck(Hπ 0 R). A Postnkov tower argument then shows that any M D b R s n Thck(Hπ 0R). Conversely, snce the R-module Hπ 0 R s bounded, so s any R-module that belongs to Thck(Hπ 0 R). The stable homotopy category dentfes wth D R for the sphere spectrum R = S 0 and we obtan the followng consequence. Corollary A.4. For a spectrum X n the stable homotopy category the followng condtons are equvalent: (1) X s pseudo-coherent and has bounded homotopy. (2) X s homologcally perfect. (3) X belongs to the thck subcategory generated by the spectrum HZ. In partcular, a homologcally perfect spectrum X s compact f and only f X = 0.

20 20 HENNING KRAUSE Proof. By the fnte generaton of the stable homotopy groups of spheres and because π 0 S 0 = Z, the sphere spectrum R = S 0 satsfes the condtons of the prevous lemma, so we have D b S = Thck(HZ). 0 A theorem of Serre says that a fnte spectrum must have nfntely many nonzero homotopy groups, so ths result mples n partcular that a homologcally perfect spectrum X s compact f and only f X = 0. Note that, n contrast to the perfect derved categores of rght coherent rngs or noetheran schemes, the category D b S 0 does not contan D c S 0 as a subcategory. Appendx B. Homologcally perfect objects on noetheran schemes by Tobas Barthel and Hennng Krause Throughout ths appendx, all schemes wll be assumed to be separated and noetheran. For a scheme X, we wrte D(Qcoh X) for the derved category of quascoherent sheaves on X and G X denotes a compact generator of D(Qcoh X), whch exsts by [4, Theorem 3.3.1]. Our goal s to gve an ntrnsc descrpton of the homologcally perfect objects n D(Qcoh X) that does not depend one the chosen t-structure. For affne schemes, ths has already been observed n the man text, but the non-affne case s more complcated and reles crucally on Neeman s work on approxmablty [33, 34]. The man result s: Theorem B.1. Let X be a separated noetheran scheme. An object X D(Qcoh X) belongs to D b (coh X) f and only f X s the homotopy colmt of a sequence X 0 X 1 X 2 of perfect complexes on X satsfyng the followng condtons for every C D per (X): (1) Hom(C, X s ) Hom(C, X s+1 ) for s 0, and (2) Hom(C, Σ n X) = 0 for n 0. Before we gve the proof of the theorem at the end of ths appendx, we record the followng consequence, whch s an mmedate applcaton of Proposton 8.1 and Theorem B.1. It shows n partcular that the completon D per (X) b D b (coh X) depends only on the trangulated structure of D per (X). Corollary B.2. The homologcally perfect objects on a separated noetheran scheme X wth respect to the standard t-structure depend only on the trangulated structure of D(Qcoh X). We wll prepare for the proof of the theorem wth three lemmata whch make use of Neeman s study of strong generators and approxmablty for trangulated categores. Lemma B.3. Let : U X be an open mmerson. If (X s ) s s a Cauchy sequence n D per (X), then ( X s ) s s a Cauchy sequence n D per (U). Proof. Frst note that, because s an open mmerson and thus automatcally quas-compact, exhbts D(Qcoh U) as the essental mage of a smashng Bousfeld localsaton on D(Qcoh X). Therefore, preserves homotopy colmts and compact object and t has a fully fathful rght adjont R. In partcular, G X s a compact generator of D(Qcoh U), whch we wll denote by G U. Moreover, t follows from the projecton formula that for every Y D(Qcoh X) there s a canoncal quassomorphsm (B.1) R Y (R O U ) L Y, where O U s the structure sheaf of U.

21 COMPLETING PERFECT COMPLEXES 21 In order to prove the lemma, t suffces by Lemma 5.2 to show that for every k Z there exsts an s(k) such that for all s > s(k): Hom(Σ k G U, X s ) Hom(Σ k G U, X s+1 ). Wthout loss of generalty, we wll demonstrate the exstence of s(0); the remanng cases follow by an analogous argument appled to the shfts of G U. By adjuncton, choce of G U = G X, and substtutng (B.1), we thus have to show that there exsts an nteger s(0) such that for all s > s(0) and F = R O U D(Qcoh X): (B.2) Hom(G X, F L X s ) Hom(G X, F L X s+1 ). Note that the class of objects F D(Qcoh X) for whch (B.2) holds s closed under retracts and arbtrary drect sums as G X s compact and L commutes wth drect sums. The object O U s perfect, so we may nvoke Neeman s result [33, Thm. 0.18]: there exst A 0, B 0, N 0 wth A 0 B 0 such that R O U G X A0,B0 N 0,.e., R O U can be bult from the collecton (Σ k G X ) A0 k B 0 usng drect sums, retracts, and at most N 0 extensons; we refer to [33] for the precse defnton of G X A,B N. For F = Σ k G X and wrtng G X for the dual of G X, the Cauchy property appled to C = G X L Σ k G X provdes an nteger t k such that for all s > t k there s an somorphsm Hom(G X L Σ k G X, X s ) Hom(G X L Σ k G X, X s+1 ). Set f(1, A, B) = max{t k k [A, B]}, then (B.2) holds for all s > f(1, A, B) and F G X A,B 1. We wll proceed by nducton on N, provng the followng clam: for any N 1 and any A B there exsts an nteger f(n, A, B) such that for all s > f(n, A, B) and F G X A,B N there s an somorphsm Hom(G X, F L X s ) Hom(G X, F L X s+1 ). Ths wll then mply the exstence of s(0) := f(n 0, A 0, B 0 ). We have just checked that the clam holds for N = 1 and arbtrary A B. Assume the clam has been proven for N 1 and let F G X A,B N+1,.e., F s a retract of an object F whch fts n a trangle E F G ΣE wth E G X A,B 1 and G G X A,B N. We thus obtan a morphsm of exact sequences Hom(G X, Σ 1 G L X s) Hom(G X, E L X s) Hom(G X, F L X s) Hom(G X, G L X s) Hom(G X, ΣE L X s) α1 α2 α3 α4 α5 Hom(G X, Σ 1 G L X s+1) Hom(G X, E L X s+1) Hom(G X, F L X s+1) Hom(G X, G L X s+1) Hom(G X, ΣE L X s+1). Note that Σ 1 G G X A 1,B 1 N and ΣE G X A+1,B+1 1. Therefore, f we set f(n +1, A, B) = max{f(1, A, B), f(1, A+1, B +1), f(n, A, B), f(n, A 1, B 1)}, then the morphsms α 1, α 2, α 4, α 5 n the above dagram are somorphsms, hence so s α 3 by the fve lemma. Consequently, f(n + 1, A, B) has the desred propertes, and we conclude by nducton. Lemma B.4. If X D(Qcoh X) s the homotopy colmt of a Cauchy sequence (X s ) s n D per (X), then X has coherent cohomology sheaves.