Completing perfect complexes

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a proper foundation for completing a triangulated category.


Introduction
This note proposes a new method to complete a triangulated category, and we apply this to categories of perfect complexes [17]. For any category C, we introduce its sequential completion C, which is a categorical analogue of the construction of the real numbers from the rationals via equivalence classes of Cauchy sequences, following Cantor and Méray [5,27].
When a ring Λ is right coherent, then the category mod Λ of finitely presented modules is abelian and one can consider its bounded derived category D b (mod Λ), which contains the category of perfect complexes D per (Λ) as a full triangulated subcategory. The following theorem describes D b (mod Λ) as a completion of D per (Λ).

Theorem 1.1. For a right coherent ring Λ there is a canonical triangle equivalence
which sends a Cauchy sequence in D per (Λ) to its colimit.
The description of D b (mod Λ) as a completion extends to non-affine schemes. Thus for a noetherian scheme X there is a canonical triangle equivalence In particular, this provides a direct construction of the singularity category (in the sense of Buchweitz and Orlov [3,37]) as the Verdier quotient .
The completion C of a category C comes with an embedding C → C so that the objects in C are precisely the colimits of Cauchy sequences in C, and C identifies with a full subcategory of the ind-completion of C in the sense of Grothendieck and Verdier [14].
When C is triangulated, there is a natural finiteness condition such that C inherits a triangulated structure with exact triangles given as colimits of Cauchy sequences of exact triangles in C. This involves the notion of a phantom morphism and Milnor's exact sequence [28]. In order to explain this, let us assume for simplicity that C identifies with the full subcategory of compact objects of a compactly generated triangulated category T. Fix a class X of sequences X 0 → X 1 → X 2 → · · · in C that is stable under suspensions. We consider their homotopy colimits and have where Ph(U, V ) denotes the set of phantom morphisms U → V . It is this finiteness condition which is satisfied for categories of perfect complexes, and it enables us to establish a triangulated structure for the completion of C with respect to X.
The idea of completing a triangulated category C is not new; the method is always to identify the completion D with a category of certain cohomological functors C op → Ab. Note that the category of all cohomological functors is equivalent to the ind-completion of C. In most cases, C identifies with the category of compact objects of a compactly generated triangulated category T, and D is another triangulated subcategory of T. Let us mention the paper of Neeman [29] that addresses the question when a category of cohomological functors carries the structure of a triangulated category. In [38], Rouquier identifies various natural choices of cohomological functors C op → Ab. The recent work of Neeman [34,35] employs the notion of 'approximability'; it is crucial for understanding the case of non-affine schemes and recommended as an alternative approach via Cauchy sequences.
We also include a discussion of completions for abelian categories. Again, some finiteness condition is needed so that the completion is abelian. For instance, we show for a noetherian algebra Λ over a complete local ring that the completion of the category fl Λ of finite length modules identifies with the category of artinian Λ-modules. Using Matlis duality, this yields for Γ = Λ op triangle equivalences This paper has three appendices. The first one by Tobias Barthel discusses completions for stable homotopy categories. In particular, we see that Theorem 1.1 generalises to ring spectra.
The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. It is shown that the objects are precisely the colimits of Cauchy sequences of perfect complexes that satisfy an intrinsic boundedness condition.
In the final appendix, Bernhard Keller introduces the notion of a morphic enhancement of a triangulated category, following [19]. This allows us to capture the notion of standard triangle and of coherent morphism between standard triangles, generalising analogous approaches via stable model categories or stable derivators. Morphic enhancements provide a setting for turning a completion into a triangulated category. In fact, we see that in Theorem 1.1 the completion of the morphic enhancement of D per (Λ) identifies with the morphic enhancement of D b (mod Λ).
After completion of this work, Neeman published a survey [36] which discusses metrics in triangulated categories, following work of Lawvere from the 1970s. Completing with respect to such metrics yields an alternative method of completing triangulated categories; it does not depend on an enhancement and we recommend a comparison.
Acknowledgement. This work benefited from discussions at an Oberwolfach workshop in March 2018. Following the spirit of this workshop, it is intended as a contribution of potential common interest to stable homotopy theory, representation theory, and algebraic geometry.
I wish to thank Amnon Neeman for various helpful comments on this work, in particular for drawing my attention to the related notion of 'approximability', for providing the proof of Lemma B.5, and for pointing out problems in some previous versions of this manuscript. Also, the interest and comments of Greg Stevenson are very much appreciated.
I am grateful to Tobias Barthel and Bernhard Keller for many stimulating comments and for agreeing to include their ideas in form of an appendix.
Finally, I wish to thank an anonymous referee for various helpul comments concerning the exposition.

The sequential completion of a category
Let N = {0, 1, 2, . . .} denote the set of natural numbers, viewed as a category with a single morphism i → j if i ≤ j. Now fix a category C and consider the category Fun(N, C) of functors N → C. An object X is nothing but a sequence of morphisms X 0 → X 1 → X 2 → · · · in C, and the morphisms between functors are by definition the natural transformations. We call X a Cauchy sequence if for all C ∈ C the induced map Hom(C, X i ) → Hom(C, X i+1 ) is invertible for i ≫ 0. This means: Let Cauch(N, C) denote the full subcategory consisting of all Cauchy sequences.
Let S denote the class of eventually invertible morphisms in Cauch(N, C).
Definition 2.1. The sequential completion of C is the category that is obtained from the Cauchy sequences by formally inverting all eventually invertible morphisms, together with the canonical functor C → C that sends an object X in C to the constant sequence X A sequence X : N → C induces a functor and this yields a functor because the assignment X → X maps eventually invertible morphisms to isomorphisms. We will show that this functor is fully faithful. Let D be a category and S a class of morphisms in D. There is an explicit description of the localisation D[S −1 ] provided that the class S admits a calculus of left fractions in the sense of [12], that is, the following conditions are satisfied: (LF1) The identity morphism of each object is in S. The composition of two morphisms in S is again in S.
If S admits a calculus of left fractions, then the morphisms in D[S −1 ] are of the form σ −1 α given by a pair of morphisms X σ2 σ with σ in S; see [12]. We need some preparations for the proof of this lemma. Given functors f : N → N and X : A straightforward computation of filtered colimits in Set yields the following.
(1) Given a morphism φ : X → Y , there exists a cofinal f : N → N and a morphism α : Proof. (1) Fix n ∈ N and suppose f has been defined for all m < n. There exists n ′ ≥ max(n, f (n − 1)) and α n : X n → Y n ′ such that the composite maps id Xn to the image of α n under Hom(X n , Y n ′ ) → colim i Hom(X n , Y i ). We set f (n) = n ′ and can make this choice consistent such that the α n yield a morphism X → Y f .
(2) Fix n ∈ N and suppose f has been defined for all m < n. The fact that α and β induce the same map colim i Hom(X n , X i ) → colim i Hom(X n , Y i ) yields n ′ ≥ max(n, f (n − 1)) such that Hom(X n , Y n ) → Hom(X n , Y n ′ ) maps α n and β n to the same element. Then set f (n) = n ′ .
Let S denote the class of eventually invertible morphisms. If X : N → C is Cauchy and f : N → N is cofinal, then X f is Cauchy and the canonical morphism f X : X → X f is in S. Moreover, for any σ : X → Y in S there exists a cofinal f : N → N and a morphism σ ′ : Y → X f such that σ ′ σ = f X . This follows by applying Lemma 2.3 to σ −1 .
To check (LF3) fix a pair of morphisms α, β : X → Y . Let σ : X ′ → X in S such that ασ = βσ. This implies α = β, and it follows from Lemma 2.3 that there exists a cofinal f : Proposition 2.4. The canonical functor C → Fun(C op , Set) is fully faithful; it identifies C with the colimits of sequences of representable functors that correspond to Cauchy sequences in C. Also, the canonical functor C → C is fully faithful.
Proof. We use the fact that the class S of eventually invertible morphisms in Cauch(N, C) admits a calculus of left fractions. Then every morphism in C is of the form σ −1 α given by a pair of morphisms X Fix Cauchy sequences X, Y : N → C. We need to show that the canonical map is a bijection. The map is surjective, because Lemma 2.3 yields for any morphism with τ 1 , τ 2 ∈ S. Then we have τ 1 α 1 = τ 2 α 2 and there exists a cofinal f : Thus for X in Fun(N, C) the functor X is the colimit of the sequence in Fun(C op , Set). It follows that C identifies with the colimits of sequences of representable functors that correspond to Cauchy sequences in C.
Finally, the canonical functor C → C is fully faithful, since the composition with C → Fun(C op , Set) is fully faithful by Yoneda's lemma.
Corollary 2.5. For X, Y ∈ C we have a natural bijection Proof. Combining Proposition 2.4 and Yoneda's lemma, we have Call C sequentially complete if every Cauchy sequence in C has a colimit in C. Clearly, C is sequentially complete if and only if the canonical functor C → C is an equivalence. We do not know whether C is always sequentially complete.
Remark 2.6. From Proposition 2.4 it follows that the sequential completion of C identifies with a full subcategory of the ind-completion Ind(C) in the sense of [14, §8].
Remark 2.7. Let F : C → D be a functor.
(1) Suppose that F admits a left adjoint. Then F preserves Cauchy sequences and induces therefore a functor F : C → D such that (2) Suppose that D admits filtered colimits. Then F extends via (2.1) to a functor C → D.
(3) If (F, G) is an adjoint pair of functors that preserve Cauchy sequences, then ( F , G) is an adjoint pair since The notion of a Cauchy sequence goes back to work of Bolzano and Cauchy (providing a criterion for convergence), while the construction of the real numbers from the rationals via equivalence classes of Cauchy sequences is due to Cantor and Méray [5,27]. The sequential completion generalises this construction.
Example 2.8. View the rational numbers Q = (Q, ≤) with the usual ordering as a category and let R ∞ = (R ∪ {∞}, ≤). Taking a Cauchy sequence to its limit yields a functor For x ∈ R ∞ , there are precisely two isomorphism classes of objects in Q with limit x when x is rational (depending on whether the sequence is eventually constant or not); otherwise there is precisely one isomorphism class in Q with limit x. 1

Proof.
A Cauchy sequence x ∈ Cauch(N, Q) is by definition a sequence x 0 ≤ x 1 ≤ x 2 ≤ · · · of rational numbers that is either bounded, so converges tox ∈ R, or it is unbounded and we setx = ∞. Given a morphism x → y in Cauch(N, Q) that is eventually invertible, we havex =ȳ. Conversely, ifx =ȳ, then we define u ∈ Cauch(N, Q) by u i = min(x i , y i ) and have morphisms x ← u → y. It is easily checked that u → x is eventually invertible, except when x is eventually constant and y is not. Thus the assignment x →x yields the desired functor Q ։ R ∞ .
Let I = {0 < 1} denote the poset consisting of two elements. For any category C, the category of morphisms in C identifies with C I = Fun(I, C).  In order to see this, let φ : X → Y be a morphism in Fun(N, C) and α : C → D a morphism in C. Suppose X and Y are in Cauch(N, C). Thus there is n ∈ N such that Hom(C, On the other hand, if F (φ) is in Cauch(N, Fun(I, C)), then we choose α = id C and α : I → D (I the initial object in C) to see that X and Y are in Cauch(N, C). It is easily checked that F 0 induces a functor C I → C I , and we claim that it is an equivalence. Recall from Remark 2.6 that there is a canonical embedding C → Ind(C). So it remains to use the fact that Ind(C) I ∼ − → Ind(C I ), which follows from Propositions 8.8.2 and 8.8.5 in [14].
Let us generalise the definition of the completion C, because later on we need to modify the underlying choice of Cauchy sequences. We fix a class X of objects in Fun(N, C). The completion of C with respect to X is the category C X with class of objects X and It follows from the definition that the assignment X → colim i Hom(−, X i ) induces a fully faithful functor C X → Fun(C op , Set). Clearly, C identifies with C X when X equals the class of Cauchy sequences, by Corollary 2.5.
Example 2.10. Let C be an exact category and let X denote the class of sequences X such that each X i → X i+1 is an admissible monomorphism. Then C˜:= C X admits a canonical exact structure and is called countable envelope of C [18, Appendix B].

The sequential completion of an abelian category
Let C be an additive category.
Lemma 3.1. The sequential completion C is an additive category and the canonical functor C → C is additive.
Proof. The assertion follows from the fact that Cauch(N, C) is additive and that the eventually invertible morphisms admit a calculus of left fractions [12, I.3.3].
It follows that the assignment X → X yields a fully faithful additive functor C → Add(C op , Ab) into the category of additive functors C op → Ab.
Lemma 3.2. If C admits kernels, then C admits kernels and C → C is left exact.
Proof. A morphism X → Y in C is up to isomorphism given by a morphism φ : X → Y in Cauch(N, C). Then K := (Ker φ i ) i≥0 is a Cauchy sequence, and this yields the kernel in C, because the sequence 0 → K → X → Y is exact in Add(C op , Ab).
Let A be an abelian category. We write fl A for the full subcategory of objects having finite composition length, and art A denotes the full subcategory of artinian objects. For X ∈ A the socle soc X is the sum of all simple subobjects. One defines inductively soc n X ⊆ X for n ≥ 0 by setting soc 0 X = 0, and soc n+1 X is given by the exact sequence 0 −→ soc n X −→ soc n+1 X −→ soc(X/ soc n X) −→ 0. Proof. Set C = fl A. It is well known that a Λ-module X is artinian if and only if its socle has finite length. In that case the socle series (soc i X) i≥0 of X yields a Cauchy sequence in C with colim i (soc i X) = X.
Now let X ∈ C. The assignment X →X := colim i X i yields a fully faithful functor C → A. Let S denote the unique (up to isomorphism) simple object in A. Then socX has finite length, since socX ∼ = Hom(S,X) ∼ = colim i Hom(S, X i ). Thus X is artinian.
The preceding example suggests a general criterion such that the sequential completion of an abelian category is abelian.
Let us fix a length category C. Thus C is an abelian category and every object has finite length. We call C ind-artinian if (1) C has only finitely many isomorphism classes of simple objects, (2) C is right Ext-finite, that is, for every pair of simple objects S and T the End(S)-module Ext 1 (S, T ) has finite length, and (3) C satisfies the descending chain condition on subobjects of socle stable sequences in C. Here, we consider sequences

Proposition 3.4. Let A be a Grothendieck category with a fully faithful functor C ֒→ A that identifies C with the full subcategory of finite length objects in A.
Suppose that every object in A is the union of its finite length subobjects. Then the following are equivalent: (1) The category C is ind-artinian.
(2) The category C has only finitely many isomorphism classes of simple objects, and an object in A is artinian if its socle has finite length.

(3) The category A admits an artinian cogenerator. In this case an object in A is artinian if and only if it is the colimit of a Cauchy sequence in C.
Proof. Let us begin with the observation that for every artinian object X ∈ A the socle series (soc i X) i≥0 is a Cauchy sequence in C with colimit X. To see this, note that X i := soc i X ∈ C for all i since X i /X i−1 is semisimple and artinian, so of finite length. Furthermore, each object C ∈ C satisfies soc n C = C for some n, and then every morphism C → X factors through X n . Thus Hom(C, X i ) ∼ − → Hom(C, X i+1 ) for all i ≥ n, and X = i soc i X.
(1) ⇒ (2): Let X ∈ A and suppose that soc X has finite length. An injective envelope X → E induces an isomorphism soc X ∼ − → soc E. So we may assume that X is injective. Set X i := soc i X for i ≥ 0. The assumption on C implies that X n has finite length for all n > 0. This follows by induction from the defining exact sequence for soc n as follows. Let S = i S i be the direct sum of a representative set of simple objects. For n > 0 we have an isomorphism of End(S)-modules and their length equals the length of soc(X/X n ). Thus X n ∈ C for all n > 0.
The sequence (X i → X i+1 ) i≥0 is socle stable, and the subobjects U ⊆ X identify with subobjects of this socle stable sequence by taking U to the sequence ( (3) ⇒ (1): Choose an artinian cogenerator E of A, and we may assume E is injective since for each simple S the injective envelope E(S) is a direct summand of E. The number of isoclasses of simple objects is bounded by the length of soc E and is therefore finite. Fix simple objects S, T ∈ A and choose a monomorphism T → E. Then Hom(S, E/T ) ∼ = Ext 1 (S, T ) and the length of Ext 1 (S, T ) as End(S)-module is bounded by the length of soc(E/T ) which is finite since E/T is artinian. It follows that C is right ext-finite. Now fix a socle stable sequence (X i → X i+1 ) i≥0 in C and set X = colim i X i . Note that X i ∼ = soc i X for all i ≥ 0. Then X embeds into the injective envelope E(X 1 ) and is therefore artinian. Subobjects of (X i → X i+1 ) i≥0 correspond to subobjects of X by the first part of the proof. Thus C is ind-artinian.
It remains to establish the last assertion. We have already seen that any artinian object is the colimit of a Cauchy sequence in C. Conversely, let X ∈ A be the colimit of a Cauchy sequence (X i ) i≥0 in C and let n ∈ N such that for every simple object S we have Hom(S, X i ) ∼ − → Hom(S, X i+1 ) for all i ≥ n. Then soc X = soc X n is in C, and therefore X is artinian.
Recall that an abelian category satisfies the (AB5) condition if for every directed set of subobjects (A i ) i∈I of an object A and B ⊆ A one has Corollary 3.5. Let C be a length category and suppose C is ind-artinian. Then C is an abelian category with injective envelopes, satisfying the (AB5) condition, and every object is artinian. Moreover, the canonical functor C → C induces an equivalence C ∼ − → fl C.
Proof. We embed C into a Grothendieck category via the functor where Lex(C op , Ab) denotes the category of left exact functors C op → Ab; see [11]. Then C identifies with a subcategory of A via Proposition 2.4, and Proposition 3.4 implies that C = art A. From this the assertion follows.
Example 3.6. Let Λ be a noetherian algebra over a complete local ring and set A = Mod Λ. Then fl A is ind-artinian and fl A identifies with art A.
Proof. There are only finite many simple Λ-modules, up to isomorphism, and their injective envelopes are artinian. Thus fl A is ind-artinian by Proposition 3.4, and therefore fl A identifies with art A.
Example 3.7. Let Λ be a ring and C ⊆ Mod Λ a full subcategory of its module category that contains Λ. Then C is sequentially complete.

The sequential completion of a triangulated category
Let T be a triangulated category and suppose that countable coproducts exist in T. Let be a sequence of morphisms in T. A homotopy colimit of this sequence is by definition an object X that occurs in an exact triangle We write hocolim i X i for X and observe that a homotopy colimit is unique up to a (non-unique) isomorphism [2]. [7,8]. The phantom morphisms form an ideal and we write Ph(X, Y ) for the subgroup of all phantoms in Hom(X, Y ). Let us denote by T/ Ph the additive category which is obtained from T by annihilating all phantom morphisms.
Recall that for any sequence · · · → A 2 φ2 −→ A 1 φ1 −→ A 0 of maps between abelian groups the inverse limit and its first derived functor are given by the exact sequence The following result goes back to work of Milnor [28] and was later extended by several authors, for instance in [7,8].
Proof. Apply Hom(−, Y ) to the exact triangle defining hocolim i X i and use that a morphism X → Y is phantom if and only if it factors through the canonical morphism X → i≥0 ΣX i .
Let C ⊆ T be a full additive subcategory consisting of compact objects and consider the restricted Yoneda functor Note that for any sequence X 0 → X 1 → X 2 → · · · in C we have by Lemma 4.1 Proof. Using the preceding lemmas, we have Proposition 4.4. Let C ⊆ T be a full additive subcategory consisting of compact objects. Taking a sequence X 0 → X 1 → X 2 → · · · in C to its homotopy colimit induces a fully faithful functor C → T/ Ph.

Proof.
We have the functor Definition 4.5. Let C be a triangulated category and X a class of sequences (X i ) i≥0 in C that is stable under suspensions, i.e. (Σ n X i ) i≥0 is in X for all n ∈ Z. We say that X is phantomless if for any pair of sequences X, Y in X we have The following lemma justifies the term 'phantomless'.
Lemma 4.6. Let C ⊆ T be a full triangulated subcategory consisting of compact objects and X a class of sequences (X i ) i≥0 in C that is stable under suspensions. Consider the full subcategory Then the following are equivalent: The class X is phantomless. The final assertion follows from the identity (4.1) for any sequence X in C, since D identifies with a full subcategory of Add(C op , Ab).

Recall that a triangulated category is algebraic if it is triangle equivalent to the stable category St(A) of a Frobenius category
can be completed to a coherent morphism of exact triangles as above.
The following theorem establishes a triangulated structure for the sequential completion of a triangulated category C. Let us stress that we use a relative version of this result for our applications, as explained in Remark 4.10 below; it depends on the choice of a class X of sequences in C which is phantomless.
Theorem 4.7. Let C be an algebraic triangulated category, viewed as a full subcategory of its sequential completion C. Suppose that the class of Cauchy sequences is phantomless. Then C admits a unique triangulated structure such that the exact triangles are precisely the ones isomorphic to colimits of Cauchy sequences that are given by coherent morphisms of exact triangles in C.
Let us spell out the triangulated structure for C. Fix a sequence of coherent morphisms η 0 → η 1 → η 2 → · · · of exact triangles in C and suppose that it is also a sequence of morphisms X → Y → Z → ΣX of Cauchy sequences in C. This identifies with the sequence in C, and the exact triangles in C are precisely sequences of morphisms that are isomorphic to sequences of the above form.
Theorem C.8 provides a substantial generalisation, from algebraic triangulated categories to triangulated categories with a morphic enhancement. Moreover, in some interesting cases the morphic enhancement extends to a morphic enhancement of the completion; see Section C.8.
Proof. The proof is given in several steps. (1) The assumption on C to be algebraic implies that C identifies with the stable category St(A) of a Frobenius category A. Let A˜denote the countable envelope of A which is a Frobenius category containing A as a full exact subcategory; see Example 2.10. Then C identifies with a full triangulated subcategory of compact objects of T := St(A˜).
For any sequence of coherent morphisms η 0 → η 1 → η 2 → · · · of exact triangles Let us sketch the argument. We can lift the sequence (η i ) i≥0 to a sequence of exact sequencesη The vertical morphism are induced by the morphismsη i →η i+1 , and taking mapping cones of the vertical morphisms (given by cokernels in A˜) yields the desired exact triangle (4.3).
(2) The assumption on Cauchy sequences in C to be phantomless implies that the functor C → T taking a sequence to its homotopy colimit induces an equivalence This follows from Lemma 4.6. In particular, the homotopy colimit of a Cauchy sequence in C is actually a colimit in D.
(3) We claim that D is a triangulated subcategory of T and that the exact triangles in D are up to isomorphism the colimits of Cauchy sequences given by coherent morphisms of exact triangles in C.
For a Cauchy sequence given by coherent morphisms of exact triangles X i → Y i → Z i → ΣX i in C, we form in D its colimit and obtain an exact triangle (4.3); it does not depend on any choices.
Conversely, fix an exact triangle η :X →Ȳ →Z → ΣX in D that is given by X, Y ∈ Cauch(N, C) withX = colim i X i andȲ = colim i Y i . The morphism X →Ȳ is up to isomorphism given by a morphism φ : X → Y in Cauch(N, C), so of the form colim i φ i . Now complete the φ i : X i → Y i to a sequence of coherent morphisms between exact triangles X i → Y i → Z i → ΣX i in C. It is easily checked that (Z i ) i≥0 is a Cauchy sequence; setZ := colim i Z i . This yields an exact triangle η ′ :X →Ȳ →Z → ΣX in T, keeping in mind the above remark about homotopy colimits of exact triangles. It follows that D is closed under the formation of cones and therefore a triangulated subcategory of T. Clearly, η and η ′ are isomorphic triangles. Thus any exact triangle in D is up to isomorphism a colimit of exact triangles in C.
is a triangulated subcategory which is triangle equivalent to C. The exact triangles are precisely the ones isomorphic to colimits of Cauchy sequences given by coherent morphisms of exact triangles in C.
For a generalisation of Corollary 4.8 from algebraic triangulated categories to triangulated categories with a morphic enhancement, see Section C.7.
Remark 4.9. To be phantomless is a condition which can be checked for a specific of finite definition if it arises as the image of a map Y (D) → Y (C) for some morphism C → D in C; see [15]. The descending chain condition (dcc) for subgroups of finite definition implies that (4.2) holds for all sequences X in C, since it implies the Mittag-Leffler condition for The dcc for subgroups of finite definition is equivalent to Y being Σ-pure-injective when viewed as an object in Ind(C), by [9, §3.5]. On the other hand, when T is a compactly generated triangulated category, then Z ∈ T is pure-injective if and only if Ph(−, Z) = 0, by [22,Theorem 1.8].
Let C be a triangulated category and fix a cohomological functor H : C → A into an abelian category. Set H n := H • Σ n for n ∈ Z. We call a sequence (  More generally, fix a class X ⊆ Fun(N, C) that is closed under suspensions and cones. When X is phantomless, then the conclusion of Corollary 4.8 holds for For more details, cf. Section C.5.
Example 4.11. Let Λ be a quasi-Frobenius ring of finite representation type. Then the class of all sequences in the stable category St(mod Λ) is phantomless. In fact, this holds for any locally finite triangulated category [24,41] and can be deduced from Remark 4.9.
The following example is a continuation of our discussion in Section 3. For an abelian category A let D b (A) denote its bounded derived category. An object in A is locally finite if it is a directed union of finite length subobjects.
Example 4.12. Let k be a commutative ring and A a k-linear Grothendieck category such that Hom(X, Y ) is a finite length k-module for all X, Y ∈ fl A. Suppose that there are only finitely many isomorphism classes of simple objects and that the injective envelope of each simple object is locally finite and artinian. Then the class of Cauchy sequences in D b (fl A) is phantomless and we have triangle equivalences Proof. The first equivalence is clear from Proposition 3.4; so we focus on the second one. We may assume that all objects in A are locally finite. For all X, Y ∈ D b (fl A) the k-module Hom(X, Y ) has finite length, since Ext n (S, T ) has finite length for all simple S, T and n ≥ 0 by our assumptions on A. It follows that the class of Cauchy sequences in D b (fl A) is phantomless by the Mittag-Leffler condition. We wish 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 injective objects in A.
We claim that F induces an equivalence Any object M ∈ art A is the colimit of the Cauchy sequence (soc i M ) i≥0 in fl A by Proposition 3.4, and this yields a Cauchy sequence in D b (fl A). Thus F maps art A into D b , and therefore also D b (art A), since F is an exact functor and D b (art A) is generated by art A as a triangulated category. Conversely, letX = colim i X i be an object in D b . We may assume that the complexX is homotopically minimal, as in [23,Appendix B]. The Cauchy condition implies for each simple S ∈ A and n ∈ Z that Hom(S, Σ nX ) has finite length over k, so the degree n component ofX is artinian. ThusX belongs to K +,b (inj A), and this yields the claim.

Homologically perfect objects
Let T be a compactly generated triangulated category and denote by T c the full subcategory of compact objects. We fix a cohomological functor H : T → A into an abelian category. For n ∈ Z set H n := H • Σ n .
Definition 5.1. We say that an object X in T is homologically perfect (with respect to H) if X can be written as homotopy colimit of a sequence X 0 → X 1 → X 2 → · · · in T c such that the following holds: (HP1) The sequence (X i ) i≥0 is a Cauchy sequence in T c , that is, for every C ∈ T c Hom(C, X i ) ∼ − → Hom(C, X i+1 ) for i ≫ 0.
(HP2) For every n ∈ Z we have H n (X i ) ∼ − → H n (X i+1 ) for i ≫ 0. (HP3) For almost all n ∈ Z we have H n (X i ) = 0 for i ≫ 0.
When T is the derived category of an abelian category, then our choice of H is the natural one given by the degree zero cohomology of a complex, unless stated otherwise.
It is clear that the above definition depends on the choice of H; so a different choice of H may yield a different class of homologically perfect objects. However, in our applications there are natural choices for H, for instance when T is the derived category of an abelian category. It is a remarkable fact that in those cases there is an intrinsic description of homologically perfect objects that depends only on T. That means some choices of H are more natural than others. The following Lemma 5.3 makes this precise when the cohomological functor H is given by a compact generator. For noetherian schemes that are non-affine, the natural choice for H admits the same intrinsic description of homologically perfect objects, but the proof is more involved and we refer to Theorem B.1.
We begin with an elementary but useful observation.
Lemma 5.2. Let C be a triangulated category and G ∈ C an object that generates C, that is, C admits no proper thick subcategory containing G. Then a sequence X 0 → X 1 → X 2 → · · · in C is Cauchy if and only if for all n ∈ Z we have The following yields an intrinsic description of homologically perfect objects.
Lemma 5.3. Let G be a compact object in T that generates T c as a triangulated category. Then for X ∈ T the following are equivalent: (1) The object X is homologically perfect with respect to H = Hom(G, −).
(2) The object X can be written as homotopy colimit of a Cauchy sequence in T c , and for every C ∈ T c we have Hom(C, Σ n X) = 0 for |n| ≫ 0.
Proof. The assumption on H implies that the conditions (HP1) and (HP2) are equivalent, thanks to Lemma 5.2. Condition (HP3) means H n (X) = 0 for |n| ≫ 0, since H n (X) ∼ = colim i H n (X i ) by Lemma 4.1. Thus (HP3) is equivalent to the condition that for every C ∈ T c we have Hom(C, Σ n X) = 0 for |n| ≫ 0, since G generates T c . Now fix a ring Λ. We write D(Λ) for the derived category of the abelian category of all Λ-modules. Let D per (Λ) denote the full subcategory of perfect complexes, that is, objects isomorphic to bounded complexes of finitely generated projective modules. The triangulated category D(Λ) is compactly generated and the compact objects are precisely the perfect complexes.
Let mod Λ denote the category of finitely presented Λ-modules and proj Λ denotes the full subcategory of finitely generated projective modules. When Λ is a right coherent ring, then mod Λ is abelian and we consider its derived category D b (mod Λ) using the following identifications where the top row consists of categories of complexes of modules in proj Λ up to homotopy. Note that D per (Λ) = D b (mod Λ) if and only if every finitely presented Λ-module has finite projective dimension. We provide an intrinsic description of the objects from D b (mod Λ), which uses for any complex X the sequence of truncations · · · −→ σ ≥n+1 X −→ σ ≥n X −→ σ ≥n−1 X −→ · · · given by In the following lemma we use the notion of a homologically perfect object with respect to the functor that takes degree zero cohomology of a complex, keeping in mind Lemma 5.3.

Lemma 5.4. Let Λ be a right coherent ring. Then X in D(Λ) is homologically perfect if and only if X belongs to D b (mod Λ).
Proof. Let X be a complex in K −,b (proj Λ) = D b (mod Λ) and write X as homotopy colimit of its truncations X i = σ ≥−i X which lie in K b (proj Λ). It is clear that X is homologically perfect. In fact, D per (Λ) is generated by Λ; so it suffices to check the functor H n = Hom(Λ, Σ n −) for every n ∈ Z. We have H n (X i ) ∼ − → H n (X i+1 ) for i ≫ 0 and H n X = 0 for |n| ≫ 0. On the other hand, if X is homologically perfect, then H n X is finitely presented for all n, so X lies in D b (mod Λ).

The bounded derived category
Let Λ be a ring. We consider the category mod Λ of finitely presented Λmodules and its bounded derived category D b (mod Λ). Our aim is to identify D b (mod Λ) with a completion of D per (Λ) when Λ is right coherent; compare this with Rouquier's [38, Corollary 6.4].
Lemma 6.1. Let Λ be a ring and set P = proj Λ. Then the functor is fully faithful.
Proof. We view K −,b (P) as a subcategory of D(Λ). Let X, Y be objects in K −,b (P) and write X as homotopy colimit of its truncations and therefore Ph(X, Y ) = 0 by Lemma 4.2. From Lemma 4.3 we conclude that Let C be a triangulated category and fix a cohomological functor H : C → A. Recall that an object X in C is bounded if colim i H n (X i ) = 0 for |n| ≫ 0, and C b denotes the full subcategory of bounded objects in C. From Theorem 4.7 and Remark 4.10 we know that C b admits a canonical triangulated structure when C is algebraic and bounded Cauchy sequences are phantomless.

Theorem 6.2. For a right coherent ring Λ there is a canonical triangle equivalence
which sends a Cauchy sequence in D per (Λ) to its colimit.
Proof. We consider the functor which is fully faithful by Lemma 6.1. On the other hand, we have the functor

Proof. Matlis duality gives an equivalence (mod Λ) op
, and we have art Γ ∼ − → fl Γ by Example 3.6. This yields the first functor, and the second is from Example 4.12.

Pseudo-coherent objects
Let T be a triangulated category and H : T → A a cohomological functor into an abelian category. Set T ≤n := {X ∈ T | H i X = 0 for all i > n}. We suppose for all X, Y ∈ T and n ∈ Z the following: (TS1) There is an exact triangle with 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 is equiped with a t-structure [1].
We will use the following observation.
Proof. Note that for any object X the morphism X → τ >n X induces an isomor- Now suppose that T is compactly generated and write T c for the full subcategory of compact objects. Definition 7.2. An object X ∈ T is called pseudo-coherent (with respect to the chosen t-structure) if X can be written as homotopy colimit of a sequence X 0 → We say that X has bounded cohomology if H n X = 0 for |n| ≫ 0.
is fully faithful when restricted to pseudo-coherent objects with bounded cohomology.
Proof. Let X, Y be objects in T. Suppose that X = hocolim i X i is pseudo-coherent and H n Y = 0 for n ≪ 0. Let C i denote the cone of Proof. For X ∈ K − (proj Λ) and i ≥ 0 set Thus X is pseudo-coherent. The other implication is left to the reader.
The example shows that for a right coherent ring Λ and any object X in T = D(Λ) the following are equivalent: (PC) X is pseudo-coherent and has bounded cohomology.
(HP) X is homologically perfect. This seems to be a common phenomenon (cf. Propositions 8.1 and A.1) though we do not have a general proof.
Let C be a triangulated category and fix a cohomological functor H : C → A. Call a sequence (X i ) i≥0 in C strongly bounded if colim i H n (X i ) = 0 for |n| ≫ 0, and if for every n ∈ Z we have H n (X i ) ∼ − → H n (X i+1 ) for i ≫ 0. By abuse of notation, we write C b for the full subcategory of strongly bounded objects in C. 2 Lemma 7.5. Suppose that (PC) ⇔ (HP) for all X ∈ T, and set C := T c . Then the functor is fully faithful functor and identifies C b with the full subcategory of pseudo-coherent objects having bounded cohomology. When T admits a morphic enhancement, then F is a triangle functor.
Proof. The first assertion follows from Lemmas 4.6 and 7.3. For the second assertion, see Lemma C.10.

Noetherian schemes
We fix a noetherian scheme X. Let Qcoh X denote the category of quasi-coherent sheaves on X, and coh X denotes the full subcategory of coherent sheaves. We consider the derived categories The triangulated category D(Qcoh X) is compactly generated and the full subcategory of compact objects agrees with the category D per (X) of perfect complexes [31]. We use the standard t-structure and then the above notion of a pseudo-coherent object identifies with the usual one; see [17, §2.3], [40, §2.2], and [39,§0DJM]. A precise reference is [39, Lemma 0DJN], which uses approximations and builds on work of Lipman and Neeman [25].
We obtain the following description of the objects in D b (coh X). For a refinement, see Theorem B.1. We use the notion of a homologically perfect object with respect to the functor that takes degree zero cohomology of a complex. (1) X belongs to D b (coh X).
(2) X is pseudo-coherent and has bounded cohomology.
(3) X is homologically perfect. (2) ⇒ (3): Let X = hocolim i X i be pseudo-coherent and C a perfect complex. The argument in the proof of Lemma 7.3 shows that (X i ) i≥0 is a Cauchy sequence in D per (X). More precisely, the cone of X i → X i+1 belongs to T ≤−i , and therefore Hom(C, X i ) ∼ − → Hom(C, X i+1 ) for i ≫ 0; see [39, §09M2]. Also, Finally, for almost all n ∈ Z we have H n (X i ) ∼ = H n (X) = 0 for i ≫ 0, since X has bounded cohomology.
(3) ⇒ (1): Let X = hocolim i X i be homologically perfect and n ∈ Z. Then H n (X) ∼ = colim i H n (X i ) equals the cohomology of some perfect complex, so H n (X) is coherent. Also, H n (X) = 0 for |n| ≫ 0.
The following is now the analogue of Theorem 6.2 for schemes that are not necessarily affine. The proof is very similar; see also Lemma 7.5 for the general argument.

Theorem 8.2. For a noetherian scheme X there is a canonical triangle equivalence
which sends a Cauchy sequence in D per (X) to its colimit.
Proof. We consider the functor which is fully faithful by Lemma 7.3 and Proposition 8.1. On the other hand, we have the functor An immediate consequence is the following. Corollary 8.3. The singularity category of X (in the sense of Buchweitz and Orlov [3,37]) identifies with the Verdier quotient .
Appendix A. Homologically perfect objects in homotopy theory by Tobias Barthel Let R be an associative ring spectrum and let D R be the derived category of right R-module spectra as constructed for example in [10]; if no confusion is likely to arise, we will refer to an object in D R simply as an R-module. If R is connective, then D R inherits the standard t-structure from the stable homotopy category, and we denote by D b R ⊆ D R the full triangulated subcategory of bounded R-module spectra, i.e., those R-modules M with π i M finitely presented over π 0 R and π i M = 0 for |i| ≫ 0. As usual, D c R ⊆ D R is the full subcategory of compact R-modules or, equivalently, the thick subcategory generated by the right R-module R.
Throughout this appendix, we will employ homological grading, so for example the pseudo-coherence condition τ >−i M i ∼ − → τ >−i M introduced in Section 7 translates to π j M i ∼ − → π j M for all j < i. Furthermore, the notion of homologically perfect object used here will always be with respect to the homological functor π 0 .
Proposition A.1. Suppose R is a connective associative ring spectrum with π 0 R right coherent and π i R finitely presented over π 0 R for all i ≥ 0. The following conditions on M ∈ D R are equivalent: (1) M is pseudo-coherent and has bounded homotopy.
(3) M belongs to D b R . Proof. The implication (1) ⇒ (2) is proven as in Proposition 8.1[(2) ⇒ (3)]: indeed, it suffices to test against compact objects of the form C = Σ n R for all n ∈ Z, for which the claim is clear. In order to see that (2) ⇒ (3), we first observe that any compact R-module has finitely presented homotopy groups by assumption on R. Therefore, any homologically perfect M ∈ D R can only have finitely many nonzero homotopy groups, all of which must be finitely presented over π 0 R by the Cauchy condition. Thus, M is bounded. Now assume that M ∈ D b R , then M has bounded homotopy and it remains to show that M has to be pseudo-coherent. To this end, we use a mild variant of the cellular tower construction of [10, Thm. III.2.10] or [16,Prop. 2.3.1], in which we only attach R-cells of a fixed dimension in each step. Indeed, let M ∈ D R be a bounded below R-module with finitely presented homotopy groups and assume without loss of generality that the lowest nonzero homotopy group is in degree 0. Inductively, we construct a tower of R-modules (M k ) k≥0 under M with: where the direct sum is indexed by a set G(k) of generators of the finitely presented π 0 R-module π k M k . It follows by induction on k, the assumption on R, and the long exact sequence in homotopy that π * M k+1 is finitely presented over π 0 R in all degrees and zero below degree k + 1, which allows us to construct the map F k+1 → M k+1 and to proceed with the induction.
Set M k = fib(M → M k ). The octahedral axiom provides fiber sequences over M . The fiber sequences (A.1) imply that π i M k ∼ − → π i M k+1 and hence π i M k ∼ − → π i M for all i < k, which then also shows that the homotopy colimit over (M k ) k≥0 is equivalent to M by a connectivity argument. Moreover, because M k is built from finitely many R-cells, M k is compact for any k ≥ 0. It follows that M is pseudo-coherent as desired.
In light of Lemma 7.5, we obtain the following consequence.
Corollary A.2. With notation as in Proposition A.1, taking homotopy colimits R of triangulated categories. In particular, the corollary applied to the Eilenberg-Mac Lane ring spectrum HΛ of a right coherent ring Λ recovers Theorem 6.2.
Lemma A.3. Let R be as in Proposition A.1 and assume additionally that the right global dimension of π 0 R is finite, then D b R coincides with the thick subcategory of D b R generated by the Eilenberg-Mac Lane R-module Hπ 0 R.
Proof. Since π 0 R has finite right global dimension, any finitely presented π 0 Rmodule N admits a finite length resolution by finitely presented projective π 0 Rmodules. This implies that the R-module spectrum HN belongs to Thick(Hπ 0 R).
A Postnikov tower argument then shows that any M ∈ D b R is in Thick(Hπ 0 R). Conversely, since the R-module Hπ 0 R is bounded, so is any R-module that belongs to Thick(Hπ 0 R).
The stable homotopy category identifies with D R for the sphere spectrum R = S 0 and we obtain the following consequence, a variant of which has appeared independently in work of Neeman [36,Ex. 22].
Corollary A.4. For a spectrum X in the stable homotopy category the following conditions are equivalent: (1) X is pseudo-coherent and has bounded homotopy.
(3) X belongs to the thick subcategory generated by the spectrum HZ. In particular, a homologically perfect spectrum X is compact if and only if X = 0.
Proof. By the finite generation of the stable homotopy groups of spheres and because π 0 S 0 ∼ = Z, the sphere spectrum R = S 0 satisfies the conditions of the previous lemma, so we have D b S 0 = Thick(HZ). A theorem of Serre says that a finite spectrum must have infinitely many nonzero homotopy groups, so this result implies in particular that a homologically perfect spectrum X is compact if and only if X = 0.
Note that, in contrast to the perfect derived categories of right coherent rings or noetherian schemes, the category D b S 0 does not contain D c S 0 as a subcategory.
Appendix B. Homologically perfect objects on noetherian schemes by Tobias Barthel and Henning Krause Throughout this appendix, all schemes will be assumed to be separated and noetherian. For a scheme X, we write D(Qcoh X) for the derived category of quasicoherent sheaves on X and G X denotes a compact generator of D(Qcoh X), which exists by [4,Thm. 3.3.1]. The notion of homologically perfect object considered in this appendix is defined with respect to the cohomological functor H 0 , taking the degree zero cohomology of an object in D(Qcoh X). Our goal is to give an intrinsic description of the homologically perfect objects in D(Qcoh X) that does not depend on the chosen t-structure. For affine schemes, this has already been observed in Lemma 5.3, but the non-affine case is more complicated and relies crucially on Neeman's work on approximability [33,34]. The main result is: Theorem B.1. Let X be a separated noetherian scheme. An object X ∈ D(Qcoh X) belongs to D b (coh X) if and only if X is the homotopy colimit of a sequence X 0 → X 1 → X 2 → · · · of perfect complexes on X satisfying the following conditions 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 give the proof of the theorem at the end of this appendix, we record the following consequence, which is an immediate application of Proposition 8.1 and Theorem B.1. It shows in particular that the completion depends only on the triangulated structure of D per (X).
Corollary B.2. The homologically perfect objects on a separated noetherian scheme X with respect to the standard t-structure depend only on the triangulated structure of D(Qcoh X).
We will prepare for the proof of the theorem with three lemmata which make use of Neeman's study of strong generators and approximability for triangulated categories. Proof. First note that, because i is an open immersion and thus automatically quasi-compact, i * exhibits D(Qcoh U) as the essential image of a smashing Bousfield localisation on D(Qcoh X). Therefore, i * preserves homotopy colimits and compact object and it has a fully faithful right adjoint Ri * . In particular, i * G X is a compact generator of D(Qcoh U), which we will denote by G U . Moreover, it follows from the projection formula that for every Y ∈ D(Qcoh X) there is a canonical quasiisomorphism where O U is the structure sheaf of U.
In order to prove the lemma, it suffices by Lemma 5.2 to show that for every k ∈ Z there exists an s(k) such that for all s > s(k): Without loss of generality, we will demonstrate the existence of s(0); the remaining cases follow by an analogous argument applied to the shifts of G U . By adjunction, choice of G U = i * G X , and substituting (B.1), we thus have to show that there exists an integer s(0) such that for all s > s(0) and F = Ri * O U ∈ D(Qcoh X): Note that the class of objects F ∈ D(Qcoh X) for which (B.2) holds is closed under retracts and arbitrary direct sums as G X is compact and ⊗ L commutes with direct sums. The object O U is perfect on U , so we may invoke Neeman's result [33,Thm. 0.18]: , i.e., Ri * O U can be built from the collection (Σ k G X ) A0≤k≤B0 using direct sums, retracts, and at most N 0 extensions; we refer to [33] for the precise definition of G X A,B N . For F = Σ k G X and writing G ∨ X for the dual of G X , the Cauchy property applied to C = G X ⊗ L Σ −k G ∨ X provides an integer t k such that for all s > t k there is an isomorphism 2) holds for all s > f (1, A, B) and We will proceed by induction on N , proving the following claim: for any N ≥ 1 and any A ≤ B there exists an integer f (N, A, B) such that for all s > f (N, A, B) and F ∈ G X A,B N there is an isomorphism This will then imply the existence of s(0) := f (N 0 , A 0 , B 0 ). We have just checked that the claim holds for N = 1 and arbitrary A ≤ B. Assume the claim has been proven for N ≥ 1 and let F ∈ G X A,B N +1 , i.e., F is a Proof of Theorem B.1. First assume that X belongs to D b (coh X). Then the implication (1) ⇒ (3) of Proposition 8.1 shows that X can be written as the homotopy colimit of a Cauchy sequence in D per (X), keeping in mind the comments about pseudo-coherent objects before Proposition 8.1. Condition (2) follows from Lemma B.5.
Conversely, suppose X is the homotopy colimit of a Cauchy sequence in D per (X) satisfying Condition (2). It follows from Lemma B.5 that X has bounded cohomology, while Lemma B.4 guarantees that X has coherent cohomology sheaves, hence X belongs to D b (coh X).

Appendix C. Morphic enhancements and triangle completions by Bernhard Keller
For a category C, we denote by MC the category of morphisms f : X 1 → X 0 of C. Let A be an abelian category and DA its derived category. We have the cone functor DMA → DA taking a morphism of complexes f to its mapping cone. It allows us to capture the notion of standard triangle and of coherent morphism between standard triangles. In this appendix, we recall the axiomatization of the links between DMA and DA given in Section 6 of [19] and apply it to the construction of triangulated completions of triangulated categories. Our treatment is slightly different from that of [19] because we work with triangulated categories instead of suspended categories.
C.1. Morphic enhancements. Let T and T 1 be triangulated categories and Q 0 : T → T 1 a fully faithful triangle functor admitting a left adjoint P 0 and a right adjoint P 1 . We define an additive functor M : T 1 → MT , X → M X = (αX : P 1 X → P 0 X) by requiring that Q 0 αX equals the composition of the adjunction morphisms For example, with the above notations, we can take T = DA and T 1 = DMA. We identify the objects of DMA with morphisms of complexes. With this convention, the functor Q 0 takes a complex X to its identity morphism and the functors P i take a morphism of complexes f : X 1 → X 0 to X i , i = 0, 1. Then the functor M : DMA → MDA takes a morphism of complexes f : X 1 → X 0 to its image in the category of morphisms of the derived category DA (each morphism of complexes yields a morphism in the derived category).
Recall that the triple (P 0 , Q 0 , P 1 ) yields a canonical recollement [1] (C.1) Recall that a functor is an epivalence if it is conservative (i.e. it detects isomorphisms), full and essentially surjective. Proof of Theorem C.1. We prove the implication from (i) to (ii). We start by constructing a right adjoint P 1ρ of P 1 . Let Y be an object of T. Since M is an epivalence, we can find an object P 1ρ Y and an isomorphism in MT Its image under M is the morphism of triangles Since there are no nonzero morphisms from Ker P 1 = Im Q ′ 2 to Ker P 0 = Im Q ′ 1 , it follows that we have a bijection . Thus, the functor P 1 admits the right adjoint Q 1 = Q ′ 1 (P 1 Q ′ 1 ) −1 . Since P 1 is a localization functor, Q 1 is fully faithful. We have Im Q 1 = Im Q ′ 1 = Ker P 0 . We prepare for the proof of the fullness of M . Let X be in T 1 . We form the triangle over the morphism whose components are the adjunction morphisms. The adjunctions yield a canonical isomorphism Using this we see that the image of the triangle (C.2) under M is the morphism of triangles The middle and the right hand term lift respectively to Q 1 X 1 ⊕ Q 0 X 0 and Q 1 X 0 . Since M is full, the morphism between the middle and the right hand term lifts to a morphism g : C.2. Examples. As we have already seen, if A is abelian, then the derived category DMA yields a morphic enhancement of DA.
More generally, if T is an algebraic triangulated category, i.e. triangle equivalent to the stable category E of a Frobenius category E, then it admits a morphic enhancement given by the functor from E to the stable category IE of the category IE of inflations X 1 → X 0 of E endowed with the class of short exact sequences inducing conflations in the two components and in the cokernel, cf. Example b) of Section 6.1 in [19].
More generally, if T is the base category of a (strong) stable derivator D in the sense of [13], then the value of D on the index category {0 < 1} yields a morphic enhancement of T. Since the homotopy category of each combinatorial stable model category is the base of a stable derivator (cf. [6] [13]), the triangulated categories arising commonly in algebra and topology admit morphic enhancements.
More generally, if T is the base of an epivalent tower of triangulated categories in the sense of [19], then T has a morphic enhancement given by the first floor of the tower. C.3. Properties. Let Q 0 : T → T 1 be a morphic enhancement in the sense of Section C.1. Part b) of the following proposition shows that the underlying additive categories of T and T 1 and the additive functor Q 0 : T → T 1 determine the triangulated structure of T.
We have natural isomorphisms ΣQ n ∼ − → Q n+3 and ΣP n ∼ − → P n−3 for all integers n. Each Q n : T → T 1 is a morphic enhancement with associated recollement For each X ∈ T 1 , there is a functorial triangle where the first two morphisms are given by the adjunctions and the third one is the composition Each triangle of T is isomorphic to a triangle of this form. c) The kernel of M : T 1 → MT is an ideal of square zero. For X, Y ∈ T 1 , there is a bifunctorial exact sequence Proof. a) In the proof of Theorem C.1, we have constructed the adjoint Q 1 as Q ′ 1 (P 1 Q ′ 1 ) −1 . Since P 1 Q ′ 1 is an equivalence and Q ′ 1 admits the right adjoint P ′ 2 , the functor Q 1 admits a right adjoint P 2 . We have Im Q 1 = Ker P 0 . Whence a short exact sequence We need to show that Q 1 : T → T 1 is a morphic enhancement. We have Ker P 2 = Im Q 0 and for X ∈ T and Y ∈ Ker P 1 , we have Thus, we have Ker P 1 ⊆ (Ker P 2 ) ⊥ . We need to check that P 2 Q −1 is an equivalence. For X ∈ T, we have the triangle By applying P 1 to this triangle we find the triangle and therefore we have X ∼ − → ΣP 2 Q −1 X. Therefore, the functor P 2 Q −1 : T → T is an equivalence isomorphic to Σ −1 . Thus, the functor Q 1 : T → T 1 is a morphic enhancement. By induction, we get a sequence of adjoints Since our assumption is self-dual, we also get a sequence of adjoints We have already constructed an isomorphism P 2 Q −1 ∼ − → Σ −1 and we have Σ −1 ∼ − → P 2 Σ −1 Q 2 . Whence an isomorphism P 2 ΣQ −1 ∼ − → P 2 Q 2 . We have Im Q −1 = Ker P 1 ⊆ (Ker P 2 ) ⊥ . By the recollement we also have Im Q 2 = Ker P 1 ⊆ (Ker P 2 ) ⊥ . Since the restriction of P 2 to (Ker P 2 ) ⊥ is an equivalence, we get an isomorphism ΣQ −1 ∼ − → Q 2 . By induction, we get ΣQ n ∼ − → Q n+3 and by adjunction Σ −1 P n ∼ − → P n+3 for all integers n. b) We have seen in the proof of a) that there is a canonical isomorphism P 2 Q −1 ∼ − → Σ −1 . By passing to the left adjoints we get an isomorphism Q 1 P −1 ∼ − → Σ. As it follows from the proof of a), we have a recollement Thus, for X ∈ T 1 , we have the functorial triangle Q 0 P 1 X / / X / / Q −1 P −1 X / / ΣQ 0 P 1 X.
By applying P 0 to this triangle we get the functorial triangle P 1 X / / P 0 X / / P −1 X / / ΣP 1 X.
Since M is essentially surjective, each triangle of T is isomorphic to a triangle of this form.
One checks easily that T X ′ is canonically isomorphic to IT X and that the morphism X → X ′ whose components are the adjunction morphisms induces the morphism i T X : T X → IT X. Thus, the morphism f : X → Y ⊕ X ′ induces the morphism T f iT X : T X → T Y ⊕ IT X. Notice that this is a componentwise split monomorphism whose cokernel is canonically isomorphic to the mapping cone over T f : T X → T Y . Now the triangle / / T 1 commutative up to given isomorphism such that the canonical morphisms P 0 F 1 → F P 0 and F P 1 → P 1 F 1 are invertible. Examples of morphic functors are provided by morphisms of epivalent towers of triangulated categories [19] and by morphisms of stable derivators [13].
Lemma C.6. If F : S → T is a morphic functor, we have canonical isomorphisms P n F 1 ∼ − → F P n and Q n F ∼ − → F 1 Q n for all integers n.
Proof. Since Q −1 : S → S 1 and Q −1 : T → T 1 are again morphic enhancements, it is enough to show the claim for Q −1 and P −1 . Indeed, by induction it will then follow for Q n and P n for all n < 0 and by duality for all n ≥ 0. The image of the canonical morphism Q −1 F → F 1 Q −1 under P 0 fits into the commutative square Thus, it is invertible. The image of Q −1 F → F 1 Q −1 under P 1 is the identity of the zero object. Since M is conservative, it follows that the morphism Q −1 F → F 1 Q −1 is invertible. Now consider the canonical morphism For each X of S 1 , it fits into a morphism of triangles / / F P 0 X / / F P −1 X / / ΣF P 1 X. C.7. Completion inside a compactly generated category. Let T be a compactly generated triangulated category with a morphic enhancement Q 0 : T → T 1 . By Lemma C.7, we have an induced morphic enhancement Q 0 : T c → T c 1 between the subcategories of compact objects and the inclusion T c → T is morphic. Let X be a class of sequences of T c satisfying the hypotheses of Section C.5. For a sequence X in X, we define hocolim p X p as in Section 4. As we have seen there, the facts that each sequence X ∈ X is formed by compact objects and that X is phantomless and stable under Σ imply that for X, Y in X, we have a canonical bijection Thus we have a fully faithful functor F : T c X → T taking LX to hocolim X. Clearly, F is endowed with a canonical isomorphism F Σ ∼ − → ΣF .
Lemma C.10. F is a triangle functor.
Proof. Let (C.4) LP 1 X / / LP 0 X / / LP −1 X / / ΣLP 0 X be the standard triangle associated with a sequence X of T c 1 such that P 0 X and P 1 X belong to X. Put Y = hocolim p X p in T 1 . Using the fact that P 1 , P 0 and P −1 commute with coproducts, it is easy to see that the standard triangle is isomorphic to the image of the triangle (C.4) under F . C.8. Completions of morphic enhancements. Let T be a triangulated category with a morphic enhancement Q 0 : T → T 1 and let X be a class of sequences of T satisfying the assumptions of Section C.5. It is natural to ask whether the triangulated category T X admits a morphic enhancement given by a completion of T 1 . Clearly, the class of sequences X 1 of T 1 needed for this is formed by the sequences X such that P 1 X and P 0 X belong to X. Let Q j 0 : T 1 → T 2 , j = 1, 2, be morphic enhancements of T 1 such that the categories T, T 1 , T 2 together with the given functors and their needed adjoints satisfy the axioms for the first three floors of an epivalent tower of triangulated categories [19]. Then it is not hard to show that X 1 satisfies assumptions a), b), and c) of Section C.5. We cannot expect that X 1 is phantomless in general but this is the case in many examples. Indeed, if T is the perfect derived category of a right coherent ring Λ, then the canonical morphic enhancement for T is the perfect derived category T 1 of the ring Λ 1 of upper triangular 2 × 2 matrices over Λ. Notice that Λ 1 is still right coherent. If X is the class of bounded Cauchy sequences in T, then X 1 is easily seen to be the class of bounded Cauchy sequences of T 1 . So in this example, X 1 is still phantomless. We can iterate this process to see that the epivalent tower associated with the bounded derived category of mod Λ is the bounded Cauchy completion of the tower associated with the perfect derived category of Λ. An analogous statement holds for the stable derivators, defined on the 2-category of finite directed categories, associated with the bounded derived category of mod Λ and with the perfect derived category of Λ, cf. the appendix [21] to [26] for these derivators.
Let MX be the class of sequences (X p1 → X p0 ) of morphisms of T such that (X p1 ) and (X p0 ) belong to X.
Lemma C.11. X 1 is phantomless in T 1 if and only if MX is phantomless in MT.
Proof. Let X and Y be in X 1 . By part c) of Lemma C.3, for all p, q ≥ 0, we have an exact sequence We pass to the colimit over q to get an inverse system of exact sequences Since lim 1 is right exact, it induces an exact sequence Now since X is phantomless, the group lim 1 A p vanishes.