STABLE EQUIVALENCES AND STABLE GROTHENDIECK GROUPS

Apostolos Beligiannis

Department of Mathematics, University of the Aegean,

83200 Karlovassi, Samos, Greece. Email: abeligia@aegean.gr.

1. Introduction

It is well–known that one of the most basic discrete invariants of an abelian, resp. triangulated, category is its Grothendieck group. This useful characteristic is defined as the free abelian group with basis the isoclasses of objects modulo the relations given by the exact sequences, resp. triangles, so it is plainly invariant under exact equivalences. Our main purpose in this paper is to study the invariance of the Grothendieck group of a triangulated category under equivalences which are not necessarily exact. More generally we search for criteria such that a functor between triangulated or abelian categories induces an isomorphism on the level of Grothendieck groups. We investigate also the invariance under various types of equivalences of some related groups which reflect the complexity of the structure of the exact sequences or the triangles.

Most of the interesting triangulated categories which occur in practice arise from a model, in the sense that there is another category, the model, which is equipped with a distinguished class of morphisms, the weak equivalences, such that the triangulated category is obtained from the model by formally inverting the weak equivalences. For instance this is the case for the stable category of an exact Frobenius category and in particular for the derived category. Now even if two triangulated categories admit such models, a functor (usually an equivalence) which induces an isomorphism on the level of Grothendieck groups is not defined in general on the level of models. Hence it is useful to develop the theory in the setting of arbitrary triangulated categories. Actually in this paper we consider one-sided triangulated categories which include for instance many interesting classes of stable categories.

The article is organized as follows.

In section 2 we recall the definition and the elementary properties of the Grothendieck group K_{0}(C) of a left triangulated category C and we give necessary and sufficient conditions such that an equivalence between two left triangulated categories induces an isomorphism on the level of Grothendieck groups. This is done by an analysis of the cone of a morphism and its behaviour under the given equivalence. Our main result in this section shows that the Grothendieck group of a Krull–Schmidt left triangulated category is invariant under a grade equivalence, i.e. an equivalence which commutes functorially with the loop functors.

In section 3 we concentrate on stable categories induced by pairs (C, X) consisting of a Krull–Schmidt category C and a contravariantly finite subcategory X of C in the sense of Auslander-Smalø. Then under a mild condition on C the stable category C/X admits the structure of a left triangulated category. Its Grothedieck group K_{0}(C/X) is called the stable Grothendieck group. We construct some useful free presentations of K_{0}(C/X) and we show that the Grothendieck group K_{0}(mod(C/X)) of the abelian category mod(C/X) of finitely presented functors (C/X)^{op} ⟶ Ab can be indentified with the kernel E(X) of the natural map K_{0}(C, ⊕) ↠ K_{0}(C, X), where the latter is the Grothendieck group of C modulo the relations given by the X −exact sequences. It follows that the group E(X) is invariant under stable equivalences. If C is abelian with enough projectives and P is the full subcategory of projectives of C, then the stable Grothendieck group K_{0}(C/P) is a classical object of study, appearing as the cokernel of the Cartan map K_{0}(P, ⊕) ⟶ K_{0}(C). In this situation we can improve the main result of section 2, by removing the functoriallity of the commutation of a given stable equivalence with the loop functors.

In section 4 we interpret the Grothendieck group of a stable category as a suitable Waldhausen group and we indicate some consequences of this interpretation on the invariance of the stable Grothendieck group. We do this by establishing a bijective correspondence between pairs (C, X) where C is an additive category and X is a contravariantly finite subcategory such that any X −epic admits a kernel, and additive Waldhausen categories of a special form.

In Section 5 we study other types of equivalences and their effect on Grothendieck groups and we apply the results of the previous sections to the stable category of finitely presented modules modulo projectives over a ring (usually an Artin algebra), generalizing some results of the literature. For instance we show that the stable Grothendieck group is invariant under derived equivalences, in particular under tilting, under stable equivalences which commute (not necessarily in a functorial way) with the loop functors (e.g. under general stable equivalences if the involved algebras are self-injective which are indecomposable or of Lowey length ≥ 2), and with the Auslander-Reiten operators in the representation-finite case.

A general convention used in the paper is that the composition of morphisms in a given category is meant in the diagrammatic order, i.e. the composition of f : A ⟶ B, g : B ⟶ C is denoted by f ∘ g. There is one exception: we use antidiagrammatic order when we apply elements to morphisms in concrete categories.

2. Left Triangulated Categories and their Grothendieck Groups

Troughout this section we fix a left triangulated category C = (C, Ω, Δ) and we assume that C is skeletally small, i.e. the collection Iso(C) of isoclasses of objects of C is a set.

We recall that a left triangulated category is a triple (C, Ω, Δ), where C is an additive category, Ω : C ⟶ C is an additive endofunctor, and Δ is a class of diagrams in C of the form (T) : Ω(C) ⟶ A ⟶ B ⟶ C, satisfying all the axioms of a triangulated category [23], except that Ω is not necessarily an equivalence. The functor Ω is called the loop functor and the class of diagrams Δ is called the left triangulation of C, see [5], [14] for more details. A diagram (T) as above is called

split, if it is isomorphic to the diagram Ω(C) ⟶^{0} A ⟶^{(1_{A},0)} A ⊕ C ⟶^{^{t}(0,1_{C})} C. Since the latter is always in the left triangulation Δ, it follows that the class Δ_{0} consisting of all split triangles, is a full subcategory of Δ.

If U is a class of objects in C, then we denote by ℤ(U), the free abelian group with basis the set Iso(U) and by ( ) : U ⟶ ℤ(U) the canonical map. Let E be a class of left triangles in Δ, closed under isomorphisms and such that Δ_{0} ⊆ E ⊆ Δ. We denote by < E >, the subgroup of ℤ(C) generated by all elements of the form (A) − (B) + (C), whenever there exists a triangle Ω(C) ⟶ A ⟶ B ⟶ C in E.

Definition 2.1. The Grothendieck group of C with respect to the class of left triangles E is defined by: K_{0}(C, E) = ℤ(C)/ < E >.

Choosing E to be the class Δ of all left triangles, we obtain the Grothendieck group K_{0}(C, Δ) := K_{0}(C). Choosing E to be the class Δ_{0} of split triangles, we obtain the split Grothendieck group K_{0}(C, Δ_{0}) := K_{0}(C, ⊕), i.e. K_{0}(C, ⊕) is the Grothendieck group of the monoidal category (C, ⊕). Observe that the inclusions Δ_{0} ⊆ E ⊆ Δ induce canonical epimorphisms K_{0}(C, ⊕) ↠ K_{0}(C, E) ↠ K_{0}(C). Their composition induces a short exact sequence

0 ⟶ D(C) ⟶^{i_{C}} K_{0}(C, ⊕) ⟶^{p_{C}} K_{0}(C) ⟶ 0 (2.1)

in Ab. We denote by | | : ℤ(C) ⟶ K_{0}(C, ⊕) and by [ ] : ℤ(C) ⟶ K_{0}(C) the canonical morphisms. Plainly D(C) is the subgroup of K_{0}(C, ⊕) generated by all elements of the form |A| − |B| + |C|, where Ω(C) ⟶ A ⟶ B ⟶ C is a left triangle in Δ. Observe that ∀A, B ∈ C: [A] + [B] = [A ⊕ B], since the split triangle Ω(B) ⟶ A ⟶ A ⊕ B ⟶ B is in Δ. Parts (i), (ii), (iii) of the next observation follow trivially from [6] and part (iv) is proved as in the case of abelian categories.

Lemma 2.2. (i) ∀C ∈ C, ∀n ∈ ℕ: [Ω^{n}(C)] = (−1)^{n}[C]. (ii) Any element of K_{0}(C) is of the form [C], for C ∈ C. (iii) K_{0}(C) = 0, if ∀C ∈ C, there exists n_{C} ∈ ℕ: Ω^{n_{C}} (C) = 0. (iv) [C_{1}] = [C_{2}] in K_{0}(C) iff there are triangles Ω(A′′) ⟶ A′ ⟶ A_{1} ⟶ A′′ and

Ω(A′′) ⟶ A′ ⟶ A_{2} ⟶ A′′ in C such that: C_{1} ⊕ A_{1} ≅ C_{2} ⊕ A_{2}.

2.1. Cones and Finitely Presented Functors. It is easy to see that if f : B ⟶

C is a morphism in C, and if Ω(C) ⟶^{h} A ⟶^{g} B ⟶^{f} C, Ω(C) ⟶^{h′} A′ ⟶^{g′} B ⟶^{f} C are left triangles in Δ, then there exists an isomorphism α : A ⟶ A′ such that α ∘ g′ = g and h ∘ α = h′. In particular |A| = |A′| in K_{0}(C, ⊕) and [A] = [A′] in K_{0}(C). However α is not uniquely determined in general. We call A the cone of f and we denote it by Cone(f), bearing in mind that Cone(f) is uniquely determined up to a non-unique isomorphism.

We recall that an additive functor F : C^{op} ⟶ Ab is called finitely presented,

if there exists an exact sequence of functors C(−, A) ⟶^{C(−,f)} C(−, B) ⟶ F ⟶ 0. Such a finitely presented functor is denoted always by F_{f}. We denote by mod(C) the category of finitely presented functors C^{op} ⟶ Ab and by 𝕐 : C ↪ mod(C) the Yoneda embedding, defined by 𝕐(A) = C(−, A) and 𝕐(f) = C(−, f). It is well known that if idempotents split in C, then 𝕐 induces an equivalence between C and the category Proj(mod(C)) of finitely generated projective functors.

Recall that a weak kernel of a morphism f : B ⟶ C in C is a morphism g : A ⟶ B such that any morphism h : X ⟶ B with h ∘ f = 0, factors through g, in a not nessecarily unique way. Equivalently the sequence of func-

tors 𝕐(A) ⟶^{𝕐(g)} 𝕐(B) ⟶^{𝕐(f)} 𝕐(C) is exact. A weak-kernel sequence in C, is a

diagram ⋯ ⟶ A_{n−1} ⟶^{f_{n−1}} A_{n} ⟶^{f_{n}} A_{n+1} ⟶ ⋯, in which f_{n−1} is a weak kernel of

f_{n}, ∀n. By the axioms of a left triangulated category, it follows that any morphism in C admits a weak kernel. Then it is well–known and easy to see that mod(C) is abelian. If f : B ⟶ C is any morphism in C then by embedding f in a left triangle Ω(C) ⟶ A ⟶ B ⟶^{f} C and developing the latter to the left, we obtain a weak

kernel-sequence ⋯ ⟶ Ω(A) ⟶ Ω(B) ⟶^{−Ω(f)} Ω(C) ⟶ A ⟶ B ⟶^{f} C. Applying to this sequence the Yoneda embedding 𝕐, we obtain a projective resolution of F_{f}:

⋯ ⟶ 𝕐Ω(B) ⟶ 𝕐Ω(C) ⟶ 𝕐(A) ⟶ 𝕐(B) ⟶^{𝕐(f)} 𝕐(C) ⟶ F_{f} ⟶ 0 (2.2)

Definition 2.3. Let P be a full additive subcategory of an additive category F. We say that F admits cancelation with respect to P, if ∀F ∈ F, ∀P ∈ P: F ⊕ P ≅ F ⇒ P = 0, equivalently if the canonical map K_{0}(P, ⊕) ⟶ K_{0}(F, ⊕) defined by |P | ↦ |P | is a monomorphism.

Proposition 2.4. If Ω(B) ⟶^{Ω(f)} Ω(C) ⟶^{α} D ⟶^{β} B ⟶^{f} C is a weak-kernel sequence in C, then: |𝕐(D)| = |𝕐(Cone(f))| in K_{0}(mod(C), ⊕).

If the category mod(C) admits cancelation with respect to Proj(mod(C)), then: |D| = |Cone(f)| in K_{0}(C, ⊕) and [D] = [Cone(f)] in K_{0}(C).

Proof. Let Ω(C) ⟶^{h} A ⟶^{g} B ⟶^{f} C be a left triangle in C. Then we have the following exact sequences in mod(C):

𝕐Ω(B) ⟶^{𝕐Ω(f)} 𝕐Ω(C) ⟶^{𝕐(h)} 𝕐(A) ⟶^{𝕐(g)} 𝕐(B) ⟶^{𝕐(f)} 𝕐(C) ⟶ F_{f} ⟶ 0,

𝕐Ω(B) ⟶^{𝕐Ω(f)} 𝕐Ω(C) ⟶^{𝕐(α)} 𝕐(D) ⟶^{𝕐(β)} 𝕐(B) ⟶^{𝕐(f)} 𝕐(C) ⟶ F_{f} ⟶ 0.

The above exact sequences induce the following exact commutative diagram

0 ⟶ Coker(𝕐Ω(f)) ⟶ 𝕐(A) ⟶ Ker(𝕐(f)) ⟶ 0
κ↓  ^{λ}↓  ^{∥}↓
0 ⟶ Coker(𝕐Ω(f)) ⟶ 𝕐(D) ⟶ Ker(𝕐(f)) ⟶ 0

By Schanuel’s lemma, it follows that Coker(𝕐Ω(f)) ⊕ 𝕐(A) ≅ Coker(𝕐Ω(f)) ⊕ 𝕐(D). Hence in the Grothendieck group K_{0}(mod(C), ⊕) we have |Coker(𝕐Ω(f))|+ |𝕐(A)| = |Coker(𝕐Ω(f))| + |𝕐(D)| ⇒ |𝕐(A)| = |𝕐(D)|. If mod(C) admits cancelation with respect to Proj(mod(C)), then |𝕐(A)| = |𝕐(D)| in K_{0}(Proj(mod(C)), ⊕). The Yoneda functor 𝕐 : C ↪ mod(C) induces a morphism 𝕐_{∗} : K_{0}(C, ⊕) ⟶ K_{0}(Proj(mod(C)), ⊕), defined by 𝕐_{∗}(|A|) = |𝕐(A)|. Then 𝕐_{∗}(|A|) = 𝕐_{∗}(|D|). Now the cofinality principle ensures that the inclusion of an additive category into its indempotent (or Karoubian) completion induces a monomorphism on the level of the split Grothendieck groups [21]. Since Proj(mod(C)) is the idempotent completion of C ≈ 𝕐(C), the morphism 𝕐_{∗} : K_{0}(C, ⊕) ⟶ K_{0}(Proj(mod(C)), ⊕) is a monomorphism. Then we have |Cone(f)| = |A| = |D| in K_{0}(C, ⊕) and [Cone(f)] = p_{C}(|Cone(f)|) = p_{C}(|D|) = [D] in K_{0}(C).

We recall that an additive category F is called a Krull-Schmidt category, if any object of F is a finite coproduct of indecomposable objects with local endomorphism ring. It is well-known that for a skeletally small Krull-Schmidt category F, the group K_{0}(F, ⊕) is free with basis the set of isoclasses of indecomposable objects.

Lemma 2.5. If C is a Krull-Schmidt category, then mod(C) admits cancelation with respect to Proj(mod(C)).

Proof. Let F be a finitely presented functor in mod(C) and assume that F ⊕𝕐(C) ≅ F, where C ∈ C. Since C is a Krull-Schmidt category, it is well–known that any finitely generated functor admits a projective cover. If ρ : 𝕐(A) ↠ F is a projective cover, then so is ρ ⊕ 1_{𝕐(C)} : 𝕐(A) ⊕ 𝕐(C) ⟶ F ⊕ 𝕐(C). Since F ⊕ 𝕐(C) ≅ F, by the uniquenness of the projective covers we infer that there exists an isomorphism 𝕐(A) ⊕ 𝕐(C) ≅ 𝕐(A) ⇒ 𝕐(A ⊕ C) ≅ 𝕐(A) ⇒ A ⊕ C ≅ A. Since C is Krull-Schmidt this implies trivially that C = 0, i.e. 𝕐(C) = 0.

Remark 2.6. Lemma 2.5 holds in the more general case in which the endomorphism ring End_{C}(C) of any object C ∈ C is F −semiperfect, i.e. End_{C}(C)/r is Von Neumann regular and idempotents can be lifted modulo the Jacobson radical r.

Corollary 2.7. If C is a Krull-Schmidt category and Ω(B) ⟶^{Ω(f)} Ω(C) ⟶^{α} D ⟶^{β} B ⟶^{f} C is a weak kernel sequence in C, then: D ≅ Cone(f).

Proof. By Proposition 2.4 and Lemma 2.5, we have |D| = |Cone(f)| in K_{0}(C, ⊕). Since C is Krull-Schmidt, this implies trivially that Cone(f) ≅ D.

In general the Yoneda embedding 𝕐 : C ⟶ mod(C) does not induces a group homomorphism 𝕐_{∗} : K_{0}(C) ⟶ K_{0}(mod(C)), except if any triangle in C splits. We correct this situation as follows. Let G be the subgroup of K_{0}(mod(C)), generated by all elements of the form [F_{f}] + [F_{Ω(f)}], where f is a morphism in C and set K̃_{0}(mod(C)) = K_{0}(mod(C))/G. We call K̃_{0}(mod(C)) the reduced Grothendieck group of mod(C) and we denote by ∥ ∥: K_{0}(mod(C)) ⟶ K̃_{0}(mod(C)) the natural map. It is easily checked that ||F_{Ω^{n}f}|| = (−1)^{n}||F_{f}||. Let Ω(C) ⟶ A ⟶ B ⟶ C be a triangle in C. Then the resolution (2.2) shows that [F_{f}] = [𝕐(C)] − [𝕐(B)] + [𝕐(A)] − [F_{Ω(f)}] in K_{0}(mod(C)). Hence [F_{f}] + [F_{Ω(f)}] = [𝕐(C)] − [𝕐(B)] + [𝕐(A)]. It follows that [𝕐(C)] − [𝕐(B)] + [𝕐(A)] = 0 in K̃_{0}(mod(C)), so the assignement [C] ↦ ||𝕐(C)|| defines a group homomorphism ||𝕐(·)|| : K_{0}(C) ⟶ K̃_{0}(mod(C)).

2.2. Equivalences. If C = (C, Ω, Δ) and C′ = (C′, Ω′, Δ′) are left triangulated categories, then a grade functor F : C ⟶ C′ is a pair (F, ω) : C ⟶ C′ where F : C ⟶ C′ is an additive functor and ω : FΩ ⟶ Ω′F is a natural isomorphism. An exact functor F : C ⟶ C′ is a grade functor (F, ω) such that for any left triangle

Ω(C) ⟶^{h} A ⟶^{g} B ⟶^{f} C in Δ, the diagram Ω′F(C) ⟶^{ω_{C}^{−1}∘F(h)} F(A) ⟶^{F(g)} F(B) ⟶^{F(f)}

F(C) is a left triangle in Δ′. A grade, resp. exact, functor (F, ω) : C ⟶ C′ is called a grade, resp. triangle, equivalence, if F is an equivalence of categories.

It is trivial to see that if F : C ⟶ C′ is an exact functor between left triangulated categories, then F induces a morphism K_{0}(F) : K_{0}(C) ⟶ K_{0}(C′) defined by K_{0}(F)([A]) = [F(A)] and a morphism K̃_{0}(mod(C)) ⟶ K̃_{0}(mod(C′)). For instance the loop functor Ω : C ⟶ C is exact and K_{0}(Ω) = −Id. Plainly if F is a triangle equivalence, then K_{0}(F) is invertible. However our main purpose is to study when F induces an isomorphism K_{0}(F), in case F is not necessarily an exact equivalence.

If F : C ⟶ C′ is an additive functor, then we have an induced morphism F̃ : K_{0}(C, ⊕) ⟶ K_{0}(C′, ⊕), defined by F̃(|C|) = |F(C)|, and an exact diagram:

0 ⟶ D(C) ⟶^{i} K_{0}(C, ⊕) ⟶^{p} K_{0}(C) ⟶ 0
(†)  _{F}↓
0 ⟶ D(C′) ⟶^{i′} K_{0}(C′, ⊕) ⟶^{p′} K_{0}(C′) ⟶ 0

Trivially F̃ is invertible, if F is an equivalence.

Theorem 2.8. Let F : C ⟶ C′ be an equivalence between the left triangulated categories C, C′, with quasi-inverse G. Then the following statements are equivalent.

(i) The assignement [A] ↦ [F(A)] induces an isomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′). (ii) The assignement |A| ↦ |F(A)| induces an isomorphism F_{∗} : D_{0}(C) ⟶ D_{0}(C′). (iii) For any morphism f in C: [F(Cone(f))] = [Cone(F(f))] in K_{0}(C′) and for

any morphism g in C′: [G(Cone(g))] = [Cone(G(g))] in K_{0}(C).

Proof. (iii) ⇒ (i) Consider the group homomorphism ϕ : ℤ(C) ⟶ K_{0}(C′) defined by ϕ((C)) = [F(C)], uniquely induced by the map Iso(C) ⟶ K_{0}(C′) sending the class of C to [F(C)]. If (A) − (B) + (C) ∈ ℤ(C), where Ω(C) ⟶ A ⟶ B ⟶^{f} C is a triangle in C, then ϕ((A)−(B)+(C)) = [F(A)]−[F(B)]+[F(C)] = [F(Cone(f))]− [F(B)] + [F(C)] = [Cone(F(f))] − [F(B)] + [F(C)] = 0, since in C′ we have the

triangle Ω′F(C) ⟶ Cone(F(f)) ⟶ F(B) ⟶^{F(f)} F(C). Hence there exists a unique homomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′) defined by F^{∗}([C]) = [F(C)]. Similarly using the quasi-inverse G, we see that there exists a unique homomorphism G^{∗} : K_{0}(C′) ⟶ K_{0}(C) defined by G^{∗}([A]) = [G(A)]. Plainly F^{∗} is the inverse of G^{∗}.

(i) ⇒ (iii) Let Ω(C) ⟶ A ⟶ B ⟶^{f} C be a left triangle in Δ. Then we have a relation [B] = [A] + [C] = [Cone(f)] + [C] in K_{0}(C). Since F^{∗} is a group morphism, we have [F(B)] = [F(A)] + [F(C)] = [F(Cone(f))] + [F(C)] in

K_{0}(C′). Let Ω′F(C) ⟶ D ⟶ F(B) ⟶^{F(f)} F(C) be a left triangle in Δ′. Since we have also [F(B)] = [D] + [F(C)] = [Cone(F(f))] + [F(C)], it follows that [Cone(F(f))] = [F(Cone(f))]. The proof of the remaining assertion is similar.

(i) ⇒ (ii) The isomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′) completes the right square of the diagram (†) above in a unique way. Hence there exists a unique isomorphism F_{∗} : D(C) ⟶ D(C′) making the above diagram commutative. Obviously F_{∗} is defined by F_{∗}(|A|) = |F(A)|. The direction (ii) ⇒ (i) is similar.

The following result is a useful refinement of the above Theorem.

Proposition 2.9. Let F : C ⟶ C′ be an equivalence between the left triangulated categories (C, Ω, Δ) and (C′, Ω′, Δ′). If for any morphism f in C it holds: |F(Cone(f))| = |Cone(F(f))| in K_{0}(C′, ⊕), then the assignement [A] ↦ [F(A)] induces an isomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′).

Proof. The hypothesis implies that [Cone(F(f))] = [F(Cone(f))], for any morphism f : B ⟶ C in C. As in the above Theorem, this implies that F̃ induces a morphism F_{∗} : D(C) ⟶ D(C′) making the left square of the diagram (†), commutative. Obviously F_{∗} is a monomorphism. We show that F_{∗} is also an epimorphism. To prove this it is sufficient to prove that F_{∗} is surjective on generators. So let

|A′| − |B′| + |C′| be a generator of D(C′), where Ω′(C′) ⟶ A′ ⟶ B′ ⟶^{f′} C′

is a left triangle in C′. Let B, C ∈ C and let f : B ⟶ C be a morphism in C such that F(B) = B′, F(C) = C′, F(f) = f′. Then in K_{0}(C′, ⊕) we have |F(Cone(f))| = |Cone(F(f))| = |Cone(f′)| = |A′|. Consider the element |Cone(f)| − |B| + |C| ∈ K_{0}(C, ⊕), which lies in D(C), since we have a trian-

gle Ω(C) ⟶ Cone(f) ⟶ B ⟶^{f} C in C. Then F_{∗}(|Cone(f)| − |B| + |C|) =

|F(Cone(f))| − |F(B)| + F(C)| = |A′| − |B′| + |C′|. We infer that F_{∗} is surjective, hence an isomorphism. By the diagram (†), it follows that there exists a unique isomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′) such that the whole diagram commutes. It is easy to see that indeed F^{∗} is given by F^{∗}([A]) = [F(A)].

The following main result of this section shows that the Grothendieck group of a Krull-Schmidt left triangulated category is invariant under grade equivalences.

Theorem 2.10. Let F : C ⟶ C′ be a grade equivalence between the left triangulated categories C, C′. If C′ is Krull-Schmidt, then the assignement [A] ↦ [F(A)] induces an isomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′) and an isomorphism F_{∗} : D(C) ⟶ D(C′). Further F induces an isomorphism K̃_{0}(mod(C)) ⟶^{≅} K̃_{0}(mod(C′)).

Proof. Let f : B ⟶ C be a morphism in C and let Ω(C) ⟶ A ⟶ B ⟶^{f} C be a left

triangle in Δ. Consider the sequence FΩ(B) ⟶^{FΩ(f)} FΩ(C) ⟶ F(A) ⟶ F(B) ⟶^{F(f)} F(C) in C′. Since F commutes functorially with the loop functors Ω, Ω′, the above

sequence is isomorphic to Ω′F(B) ⟶^{Ω′F(f)} Ω′F(C) ⟶ F(A) ⟶ F(B) ⟶^{F(f)} F(C). The last diagram is a weak kernel sequence in C′, since F as an equivalence preserves weak kernels. Then by Corollary 2.7, |Cone(F(f))| = |F(Cone(f))| = |F(A)| in K_{0}(C′, ⊕) and by Proposition 2.9 we infer that the assignement [C] ↦ [F(C)] induces an isomorphism F^{∗} : K_{0}(C) ⟶ K_{0}(C′). Finally for any morphism f in C, we have: [F_{F(f)}] + [F_{Ω′F(f)}] = [F_{F(f)}] + [F_{FΩ(f)}] in K_{0}(mod(C′)). This implies that the

isomorphism K_{0}(mod(C)) ⟶^{≅} K_{0}(mod(C′)) induced by F, induces an isomorphism K̃_{0}(mod(C)) ⟶^{≅} K̃_{0}(mod(C′)) in the reduced Grothendieck groups.

We shall see in the next section that in some specific situations it is possible to remove the functoriality of the isomorphism FΩ(C) ≅ Ω′F(C), ∀C ∈ C.

3. Grothendieck Groups of Stable Categories and Stable Equivalences

Throughout this section we fix a skeletally small additive category C with split idempotents. Let X ⊆ C be a full additive subcategory of C closed under direct summands and isomorphisms. We denote by C/X the induced stable category. We recall that the objects of C/X are the objects of C. The morphisms of C/X are equivalence classes of morphisms, two morphisms being equivalent if their difference factorizes through an object of X. We denote by \underline{C} the object C ∈ C viewed as an object in C/X and by \underline{f}, the equivalence class of a morphism f in C. Setting π(C) = \underline{C} and π(f) = \underline{f}, we obtain the additive projection functor π : C ⟶ C/X.

3.1. Stable Categories as left triangulated Categories. We call a morphism f : B ⟶ C in C, an X −epic, if the morphism C(X , f) : C(X , B) ⟶ C(X , C) is surjective in Ab. We recall [4] that X is called contravariantly finite in C, if for any object C ∈ C, there exists an X −epic χ_{C} : X_{C} ⟶ C with X_{C} ∈ X; such a morphism χ_{C} is called a right X −approximation of C. The pair (C, X) is called a left homotopy pair [7], if X is contravariantly finite in C and any X −epic in C has a kernel. We entrust to the reader to formulate the dual notions of X −monic, covariantly finite subcategory and right homotopy pair.

Throughout this section we fix a left homotopy pair (C, X). By [5], the stable category C/X is equipped in a natural way with a left triangulated structure (C/X , Ω_{X} , Δ_{X}). We recall briefly the definition of the loop functor Ω_{X} and the left triangulation Δ_{X}, and we refer to [5] for details. Let C, D ∈ C and let 0 ⟶ K_{C} ⟶^{g_{C}} X_{C} ⟶^{χ_{C}} C, 0 ⟶ K_{D} ⟶^{g_{D}} X_{D} ⟶^{χ_{D}} D be left exact sequences in C, where χ_{C}, resp. χ_{D}, is a right X −approximation of C, resp. D. Then any morphism α : C ⟶ D induces a commutative diagram

0 ⟶ K_{C} ⟶^{g_{C}} X_{C} ⟶^{χ_{C}} C
κ_{α}↓  ^{χ_{α}}↓  ^{α}↓
0 ⟶ K_{D} ⟶^{g_{D}} X_{D} ⟶^{χ_{D}} D

Setting Ω_{X} (\underline{C}) = \underline{K}_{C}, and Ω_{X} (\underline{α}) = \underline{κ}_{α}, we obtain the well–defined additive loop functor Ω_{X} : C/X ⟶ C/X. The triangulation Δ_{X} of the pair (C/X , Ω_{X}) is defined as follows. Let 0 ⟶ A ⟶^{g} B ⟶^{f} C be a left exact sequence in C, where f is an X −epic. Then there exists an exact commutative diagram

0 ⟶ K_{C} ⟶^{g_{C}} X_{C} ⟶^{χ_{C}} C
h↓  ^{γ}↓  ^{∥}↓
0 ⟶ A ⟶^{g} B ⟶^{f} C

which induces a diagram Ω_{X} (\underline{C}) ⟶^{\underline{h}} \underline{A} ⟶^{\underline{g}} \underline{B} ⟶^{\underline{f}} \underline{C} in C/X. Then Δ_{X} is defined to be the collection of all such diagrams together with their isomorphic copies.

Any morphism f : B ⟶ C in C can be made to be an X −epic as follows. Let χ_{C} : X_{C} ⟶ C be a right X −approximation and set Cyl(f) = B ⊕ X_{C}. Then ^{t}(f, χ_{C}) : Cyl(f) ⟶ C is an X −epic, hence it induces a left exact sequence

0 ⟶ Cone(f) ⟶ Cyl(f) ⟶^{^{t}(f,χ_{C})} C in C, where now Cone(f) denotes the pull-back of f and χ_{C}:

0 ⟶ K_{C} ⟶^{h} Cone(f) ⟶^{g} B
∥↓  ^{λ}↓  ^{f}↓
0 ⟶ K_{C} ⟶^{κ_{C}} X_{C}  ⟶^{χ_{C}} C

Plainly \underline{Cyl(f)} = \underline{B} in C/X and we have a triangle Ω_{X} (\underline{C}) ⟶ \underline{Cone(f)} ⟶ \underline{B} ⟶^{\underline{f}} \underline{C} in Δ_{X}. Hence for any morphism f in C we have: \underline{Cone(f)} ≅ Cone(\underline{f}).

Remark 3.1. (1) If f : B ⟶ C is a morphism in C and χ_{Cone(f)} : X_{Cone(f)} ⟶ Cone(f) is a right X −approximation of Cone(f), then the morphisms χ_{Cone(f)} ∘g : X_{Cone(f)} ⟶ B and χ_{Cone(f)} ∘ g ∘ f : X_{Cone(f)} ⟶ C are right X −approximations of B, C respectively. Indeed this follows easily from the above pull-back diagram.

(2) If the morphism f is an X −epic with kernel A, then the pull-back property forces λ to be also an X −epic. Since X_{C} is in X, it follows that λ splits and we have an isomorphism Cone(f) ≅ A ⊕ X_{C}. Hence: \underline{Cone(f)} ≅ Cone(\underline{f}) ≅ \underline{A}.

(3) ∀B, C ∈ C: \underline{B} ≅ \underline{C} ⇔ B ⊕ X_{C} ≅ C ⊕ X_{B}.

3.2. Presentations of Grothendieck Groups. We denote by K_{0}(C/X) the Grothendieck group of the left triangulated category C/X, and we call it the stable Grothendieck group of C. There is another group which is closely related to the stable Grothendieck group. We define K_{0}(C, X) to be the quotient of ℤ(C) modulo the subgroup generated by all elements of the form (A)−(B)+(C), whenever there exists a left exact sequence 0 ⟶ A ⟶ B ⟶^{f} C in C with f an X −epic. From now on such a sequence is called an X −exact sequence. Then it is proved in [6] that there exists an exact sequence

(3.1) 0 ⟶ Ker(c_{X}) ⟶ K_{0}(X , ⊕) ⟶^{c_{X}} K_{0}(C, X) ⟶ K_{0}(C/X) ⟶ 0

where c_{X} is the Cartan morphism defined by c_{X} (|X|) = [X]. If any object C ∈ C admits a finite X −resolution 0 ⟶ X_{n} ⟶ ⋯ ⟶ X_{1} ⟶ X_{0} ⟶ C ⟶ 0 in the sense that X_{t} ∈ X , ∀t ≥ 0 and the sequence remains exact after the application of C(X, −), ∀X ∈ X, then by [6], c_{X} is invertible and K_{0}(C/X) = 0.

Consider the naturally induced short exact sequences

0 ⟶ E(X) ⟶ K_{0}(C, ⊕) ⟶^{ϖ_{X}} K_{0}(C, X) ⟶ 0 (3.2)

0 ⟶ D(X) ⟶ K_{0}(C/X , ⊕) ⟶^{π_{X}} K_{0}(C/X) ⟶ 0 (3.3)

The following exact commutative diagram of short exact sequences:

0 ⟶ Ker(c_{X}) ⟶ K_{0}(X , ⊕) ⟶ Im(c_{X})) ⟶ 0
↓  ↓  ↓
(†)  0 ⟶ E(X) ⟶ K_{0}(C, ⊕) ⟶ K_{0}(C, X) ⟶ 0
↓  ↓  ↓
0 ⟶ D(X) ⟶ K_{0}(C/X , ⊕) ⟶ K_{0}(C/X) ⟶ 0

includes all the above information and it induces an exact sequence in Ab:

0 ⟶ Ker(c_{X}) ⟶ E(X) ⟶^{ω_{X}} K_{0}(C/X , ⊕) ⟶ K_{0}(C/X) ⟶ 0 (3.4)

where ω_{X} is the composition E(X) ⟶ D(X) ⟶ K_{0}(C/X , ⊕). Clearly E(X) is the subgroup of K_{0}(C, ⊕) generated by all elements of the form |A|−|B|+|C| ∈ K_{0}(C, ⊕) for which there exists an X −exact sequence 0 ⟶ A ⟶^{g} B ⟶^{f} C in C. Similarly D(X) is the subgroup of K_{0}(C/X , ⊕) generated by all elements of the form |\underline{A}|−|\underline{B}|+|\underline{C}| ∈ K_{0}(C/X , ⊕) for which there exists a left triangle Ω_{X} (\underline{C}) ⟶ \underline{A} ⟶ \underline{B} ⟶ \underline{C} in Δ_{X}.

Remark 3.2. If C is Krull-Schmidt, then the categories X , C/X also enjoy the Krull-Schmidt property. It follows in this case that the “split” groups K_{0}(C, ⊕), K_{0}(X , ⊕), K_{0}(C/X , ⊕) are free abelian on the set of isoclasses of their indecomposable objects and moreover K_{0}(C, ⊕) ≅ K_{0}(X , ⊕) ⊕ K_{0}(C/X , ⊕). It follows that E(X), D(X), Ker(c_{X}) are free abelian, the exact sequence (3.4) is a free resolution of the stable group K_{0}(C/X) and: E(X) ≅ D(X) ⊕ Ker(c_{X}).

From now on we denote by A(X) the free group with basis the subset

{|A| − |B| + |C| ∈ K_{0}(C, ⊕) | whenever 0 ⟶ A ⟶ B ⟶ C is X − exact in C}.

Consider the canonical epimorphism A(X) ↠ E(X). Composing this morphism with the composition E(X) ↠ D(X) ↪ K_{0}(C/X , ⊕), we obtain a morphism α_{X} : A(X) ⟶ K_{0}(C/X) and an exact sequence

A(X) ⟶^{α_{X}} K_{0}(C/X , ⊕) ⟶ K_{0}(C/X) ⟶ 0 (3.5)

If C is a Krull-Schmidt category, then (3.5) is a free presentation of the stable Grothendieck group, which is some cases it is more useful to consider.

3.3. Stable Equivalences. We fix two left homotopy pairs (C, X) and (D, Y) and we regard the stable categories C/X, D/Y as left triangulated categories. We fix also a stable equivalence F : C/X ⟶^{≈} D/Y, which is not necessarily exact. The next result shows that the group A(X) is invariant under stable equivalence.

Proposition 3.3. The functor F induces an isomorphism F_{2} : A(X) ⟶^{≅} A(Y).

Proof. Let |A| − |B| + |C| be a generator of A(X), which comes from an X −exact sequence 0 ⟶ A ⟶^{g} B ⟶^{f} C in C. Let A′, B′ ∈ D such that F(\underline{B}) = \underline{B}′, F(\underline{C}) = \underline{C}′ and let f′ : B′ ⟶ C′ be a morphism in D such that F(\underline{f}) = \underline{f}′. Then f′ induces a

Y−exact sequence (1) 0 ⟶ Cone(f′) ⟶ Cyl(f′) ⟶^{^{t}(f′,ψ_{C′})} C′ in D, so the element α(f′) := |Cone(f′)|−|Cyl(f′)|+|C′| is in A(Y). We set F_{2}(|A|−|B|+|C|) = α(f′) and we show that this element is independent of the choices we made. Let A′′, B′′ ∈ D be such that F(\underline{B}) = \underline{B}′′, F(\underline{C}) = \underline{C}′′ and let f′′ : B′′ ⟶ C′′ be a morphism in D such that F(\underline{f}) = \underline{f}′′. Consider the Y−exact sequence (2) 0 ⟶ Cone(f′′) ⟶

Cyl(f′′) ⟶^{^{t}(f′′,ψ_{C′′})} C′′ in D, which defines the element α(f′′) ∈ A(Y). Then in D/Y we have: \underline{Cyl(f′)} = \underline{Cyl(f′′)}, \underline{C}′ = \underline{C}′′. Since (1), (2) induce isomorphic triangles in Δ_{Y}, we have \underline{Cone(f′)} ≅ \underline{Cone(f′′)}. By Remark 3.1, we have isomorphisms:

C′ ⊕ Y_{C′′} ≅ C′′ ⊕ Y_{C′}, B′ ⊕ Y_{B′′} ≅ B′′ ⊕ Y_{B′},
Cone(f′) ⊕ Y_{Cone(f′′)} ≅ Cone(f′′) ⊕ Y_{Cone(f′)}.

By Remark 3.1, the right Y−approximations Y_{C′}, Y_{B′} of C′, B′ and Y_{C′′}, Y_{B′′} of Y_{C′′}, Y_{B′′} can be chosen to be the right Y−approximations of Cone(f′), Cone(f′′). Then the first of the above isomorphisms take the following form:

C′ ⊕ Y_{Cone(f′′)} ≅ C′′ ⊕ Y_{Cone(f′)}, B′ ⊕ Y_{Cone(f′′)} ≅ B′′ ⊕ Y_{Cone(f′)}.

Hence in K_{0}(D, ⊕) we have:

|C′| + |Y_{Cone(f′′)}| = |C′′| + |Y_{Cone(f′)}|, |B′| + |Y_{Cone(f′′)}| = |B′′| + |Y_{Cone(f′)}|,

|Cone(f′)| + |Y_{Cone(f′′)}| = |Cone(f′′)| + |Y_{Cone(f′)}|.

It follows that |Cyl(f′′)| − |Cyl(f′)| = |B′′ ⊕ Y_{B′′}| − |B′ ⊕ Y_{B′}| = |B′′| + |Y_{B′′}| − |B′|−|Y_{B′}| = |B′′|+|Y_{Cone(f′′)}|−|B′|−|Y_{Cone(f′)}|. By the above equalities we infer easily that α(f′) = α(f′′), so F_{2} is well-defined. Since A(X) is free on all elements |A| − |B| + |C| of K_{0}(C, ⊕), for any X −exact sequence 0 ⟶ A ⟶^{g} B ⟶^{f} C in C, it follows that F_{2} extends to a morphism F_{2} : A(X) ⟶ A(Y) and dually the quasiinverse G of F, induces a morphism G_{2} : A(Y) ⟶ A(X). We claim that G_{2}F_{2} acts identically on the generator |A| − |B| + |C| of A(X). Indeed it is easy to see that G_{2}F_{2}(|A| − |B| + |C|) = |Cone(f)| − |Cyl(f)| + |C| = α(f), which comes from the

X −exact sequence 0 ⟶ Cone(f) ⟶ Cyl(f) ⟶^{^{t}(f,χ_{C})} C. Since f is an X −epic, by Remark 3.1, we have Cone(f) ≅ A⊕X_{C}. Then α(f) = |A|+|X_{C}|−|B|−|X_{C}|+|C| = |A|−|B|+|C|. Hence G_{2}F_{2}(|A|−|B|+|C|) = |A|−|B|+|C| in A(X). Interchanging the roles of F_{2}, G_{2}, we have F_{2}G_{2}(|A| − |B| + |C|) = |A| − |B| + |C| in A(Y). We conclude that G_{2} is an inverse of F_{2}.

3.4. Krull-Schmidt Categories. Assume throughout this subsection that in the left homotopy pair (C, X), C is a skeletally small Krull-Schmidt additive category with weak kernels and split idempotents. Since X is contravariantly finite, it follows easily that X admits weak kernels. Since the stable category C/X is left triangulated, C/X also admits weak kernels. It follows that the categories mod(C), mod(X), mod(C/X) are abelian with enough projectives.

We denote by mod_{X} (C) the full subcategory of mod(C) consisting of all finitely presented functors of the form F_{f}, where f is an X −epic in C, i.e. F ∈ mod_{X} (C)

iff there exists a presentation 𝕐(B) ⟶^{𝕐(f)} 𝕐(C) ⟶ F ⟶ 0, where f : B ⟶ C is an X −epic in C. It is easy to see that the functor H_{X} : mod_{X} (C) ⟶ mod(C/X) defined by H_{X} (F_{f}) = F\underline{f} is an equivalence of categories with inverse the functor

K_{X} : mod(C/X) ⟶ mod_{X} (C) defined as follows. If C/X (−, \underline{B}) ⟶^{𝕐(\underline{f})} C/X (−, \underline{C}) ⟶ F\underline{f} ⟶ 0 is a finite presentation of F\underline{f} in mod(C/X), then K_{X} (F\underline{f}) = F_{f^{∗}}, where f^{∗} := ^{t}(χ_{C}, f) : Cyl(f) = X_{C} ⊕ B ⟶ C and χ_{C} is a right X −approximation of C. Usually we view the above equivalences as identifications.

Since X is contravariantly finite in C, the restriction functor R_{X} : C ⟶ Mod(X) defined by R_{X} = C(−, A)|_{X}, has its image in mod(X). Moreover R_{X} induces an equivalence between X and Proj(mod(X)) and sends an X −exact sequence 0 ⟶ A ⟶ B ⟶ C in C to an exact sequence 0 ⟶ R_{X} (A) ⟶ R_{X} (B) ⟶ R_{X} (C) ⟶ 0 in mod(X). Hence it induces a morphism s_{X} : K_{0}(C, X) ⟶ K_{0}(mod(X)). The functor R_{X} extends uniquely, via the Yoneda embedding 𝕐 : C ↪ mod(C), to a right exact functor R_{X} : mod(C) ⟶ mod(X). Obviously R_{X} coincides with the restriction functor R_{X} (F) = F |_{X}. In particular R_{X} is exact and induces a morphism r_{X} : K_{0}(mod(C)) ⟶ K_{0}(mod(X)). Using the identification mod_{X} (C) = mod(C/X), it is not difficult to see that R_{X} induces a short exact sequence of abelian categories

0 ⟶ mod(C/X) ⟶ mod(C) ⟶^{R_{X}} mod(X) ⟶ 0.

By well-known results of Quillen [18], [2], there exist a long exact sequence

⋯ ⟶ K_{1}(mod(X)) ⟶ K_{0}(mod(C/X)) ⟶^{j_{X}} K_{0}(mod(C)) ⟶^{r_{X}} K_{0}(mod(X)) ⟶ 0

The morphisms s_{X} , j_{X} , r_{X} are embedded in the exact commutative diagram

0 ⟶  E(X)  ⟶^{i_{X}} K_{0}(C, ⊕) ⟶^{ϖ_{X}} K_{0}(C, X) ⟶ 0
∃! ω_{X}^{∗} ↓  ^{c}↓  ^{s_{X}} ↓
⋯ ⟶ K_{0}(mod(C/X)) ⟶^{j_{X}} K_{0}(mod(C)) ⟶^{r_{X}} K_{0}(mod(X)) ⟶ 0

where c : K_{0}(C, ⊕) ⟶ K_{0}(mod(C)) is the Cartan map induced by the Yoneda embedding 𝕐 : C ↪ mod(C), i.e. it is defined by sending |C| to [𝕐(C)]. The morphism ω_{X}^{∗} is constructed as follows. Let j_{X} = κ ∘ λ : K_{0}(mod(C/X)) ⟶^{κ} Im(j_{X}) ⟶^{λ} K_{0}(mod(C)) be the canonical factorization of j_{X}. The above diagram induces a unique morphism ω_{X} : E(X) ⟶ Im(j_{X}) such that ω_{X} ∘ λ = i_{X} ∘ c. Since C is Krull-Schmidt, the Grothendieck group K_{0}(C, ⊕) is free with basis the set Ind(C) of isoclasses of indecomposable objects of C. Hence E(X) is free. It follows that there exists a unique morphism ω_{X}^{∗} : E(X) ⟶ K_{0}(mod(C/X)) such that ω_{X}^{∗} ∘ κ = ω_{X}.

Theorem 3.4. The morphism ω_{X}^{∗} : E(X) ⟶ K_{0}(mod(C/X)) is invertible. In particular K_{0}(mod(C/X)) is free.

Proof. We examine first the action of the morphism ω_{X}^{∗} on the generators |A| −

|B| + |C| of E(X), where 0 ⟶ A ⟶^{g} B ⟶^{f} C is an X −exact sequence in C. Since by construction ω_{X} is the restriction of c on E(X), it follows that ω_{X} (|A|−|B|+|C|) = c(|A| − |B| + |C|) = [𝕐(A)] − [𝕐(B)] + [𝕐(C)]. Then we have a projective resolution 0 ⟶ 𝕐(A) ⟶ 𝕐(B) ⟶ 𝕐(C) ⟶ F_{f} ⟶ 0 of the functor F_{f} = Coker(−, f) in mod(C). Then in K_{0}(mod(C)) we have [F_{f}] = [𝕐(A)] − [𝕐(B)] + [𝕐(C)]. It follows that ω_{X} (|A| − |B| + |C|) = [F_{f}]. Since f is X −epic, the functor F_{f} lies in mod(C/X). Then by the construction of ω_{X}^{∗} we have ω_{X}^{∗} (|A| − |B| + |C|) = [F_{f}] ∈ K_{0}(mod(C/X)). We shall construct an inverse ζ_{X}^{∗} of ω_{X}^{∗}. Let ζ_{X} : Iso(mod(C/X)) ⟶ E(X) be the map defined as follows. If F ∈ mod(C/X), then there exists an X −epic f : B ⟶ C in C such that F = F_{f} = Coker(−, f). We denote the isomorphism class of F by (F). Consider the X −exact sequence

0 ⟶ A ⟶ B ⟶^{f} C in C. We set ζ_{X} ((F_{f})) = |A| − |B| + |C| ∈ E(X). Let

0 ⟶ F_{1} ⟶ F_{2} ⟶ F_{3} ⟶ 0 be a short exact sequence in mod(C/X). Then there are X −epics f_{i} : B_{i} ⟶ C_{i} in C such that F_{i} = F_{f_{i}}, i = 1, 3. Consider the

X −exact sequences 0 ⟶ A_{i} ⟶^{g_{i}} B_{1} ⟶^{f_{i}} C_{i}, i = 1, 3, which induce projective

resolutions 0 ⟶ 𝕐(A_{i}) ⟶^{𝕐(g_{i})} 𝕐(B_{i}) ⟶^{𝕐(f_{i})} 𝕐(C_{i}) ⟶ F_{i} ⟶ 0 in mod(C). Using Horseshoe Lemma, it follows that there exists a projective resolution 0 ⟶

𝕐(A_{1} ⊕ A_{3}) ⟶^{𝕐(g_{2})} 𝕐(B_{1} ⊕ B_{3}) ⟶^{𝕐(f_{2})} 𝕐(C_{1} ⊕ C_{3}) ⟶ F_{2} ⟶ 0 of F_{2} in mod(C),

i.e. F_{2} = F_{f_{2}}. It is easy to check that the morphism f_{2} is an X −epic in C, hence setting A_{2} = A_{1} ⊕ A_{3}, B_{2} = B_{1} ⊕ B_{3}, C_{2} = C_{1} ⊕ C_{3}, we have an X −exact sequence

0 ⟶ A_{2} ⟶^{g_{2}} B_{2} ⟶^{f_{2}} C_{2} in C. Then ζ_{X} ((F_{f_{1}})) − ζ_{X} ((F_{f_{2}})) + ζ_{X} ((F_{f_{3}})) = (|A_{1}| −

|B_{1}|+|C_{1}|)−(|A_{2}|−|B_{2}|+|C_{2}|)+(|A_{3}|−|B_{3}|+|C_{3}|) = (|A_{1}|−|B_{1}|+|C_{1}|)−(|A_{1}|+ |A_{3}|−|B_{1}|−|B_{3}|+|C_{1}|+|C_{3}|)+(|A_{3}|−|B_{3}|+|C_{3}|) = 0. By the universal property of K_{0}(mod(C/X)), there exists a unique morphism ζ_{X}^{∗} : K_{0}(mod(C/X)) ⟶ E(X) such that ζ_{X}^{∗} ([F_{f}]) = ζ_{X} ((F_{f})) = |A| − |B| + |C|, where f : B ⟶ C is an X −epic with kernel A. By construction it follows easily that ζ_{X}^{∗} is an inverse of ω_{X}^{∗}.

Recall that an abelian category is called regular, if any object of C has finite projective dimension. For instance if C has kernels, then mod(C) has global dimension ≤ 2. If for an additive category A, the category mod(A) is abelian, then the stable category modulo projectives of the latter is denoted by \underline{mod}(A).

Corollary 3.5. Assume that mod(C) is regular. Then the morphism s_{X} : K_{0}(C, X) ⟶ K_{0}(mod(X)) is invertible and we have a free presentation K_{0}(mod(C/X)) ↣ K_{0}(mod(C)) ↠ K_{0}(mod(X)). Moreover the functor R_{X} : C ⟶ mod(X) induces an isomorphism R_{X}^{∗} : K_{0}(C/X) ⟶ K_{0}(\underline{mod}(X)).

Proof. Since mod(C) is regular, the Cartan map c : K_{0}(C, ⊕) ⟶ K_{0}(mod(C)) is invertible, so by Theorem 3.4 the morphism s_{X} : K_{0}(C, X) ⟶ K_{0}(mod(X)) is invertible. The last claim follows from the commutativity of the following diagram

K_{0}(X , ⊕) ⟶^{c_{X}} K_{0}(C, X) ⟶ K_{0}(C/X) ⟶ 0
||↓  ^{s_{X}} ↓  ^{R_{X}^{∗}} ↓
K_{0}(X , ⊕) ⟶ K_{0}(mod(X)) ⟶ K_{0}(\underline{mod}(X)) ⟶ 0

Now let (D, Y) be another left homotopy pair, where D is skeletally small Krull-Schmidt category with weak kernels and split idempotents, and we fix a stable equivalence F : C/X ⟶^{≈} D/Y. The following consequence shows that the abelian group E(X) is invariant under stable equivalences.

Corollary 3.6. The functor F induces an isomorphism F^{!} : E(X) ⟶^{≅} E(Y).

Proof. Obviously the stable equivalence F extends to an equivalence F̂ : mod(C/X) ⟶^{≈} mod(D/Y). It follows that F induces an isomorphism K_{0}(mod(C/X)) ⟶^{≅} K_{0}(mod(D/Y)) and the assertion follows from Theorem 3.4.

Corollary 3.7. Assume that mod(C) and mod(D) are regular. Let F : C ⟶ D be an additive functor which preserves weak kernels (this happens for instance if F admits a left adjoint) and such that F(X) ⊆ Y. If the induced functor F̂ : mod(X) ⟶ mod(Y) induces a stable equivalence ˆ \underline{F} : \underline{mod}(X) ⟶ \underline{mod}(Y), then the induced functor \underline{F} : C/X ⟶ D/Y induces an isomorphism F^{∗} : K_{0}(C/X) ⟶^{≅} K_{0}(D/Y).

Proof. By our assumption on F it follows easily that the induced functor F_{1} is exact and preserves projectives. Hence the induced functor \underline{F}_{1} is a triangle equivalence. Then the assertion follows from Corollary 3.5.

It is useful to indicate the action of the isomorphism F^{!} : E(X) ⟶ E(Y) constructed above, which is the composition of the isomorphisms

E(X) ⟶^{ω_{X}^{∗}} K_{0}(mod_{X} (C)) ⟶^{K_{0}(H_{X})} K_{0}(mod(C/X)) ⟶^{K_{0}(_{F})}

K_{0}(mod(D/Y)) ⟶^{K_{0}(K_{Y})} K_{0}(mod_{Y}(D)) ⟶^{(ω_{Y}^{∗})^{−1}} E(Y)

where the three middle isomorphism are induced by the equivalences H_{X} : mod_{X} (C) ⟶^{≈} mod(C/X), F̂ : mod(C/X) ⟶^{≈} mod(D/Y) and K_{Y} : mod(D/Y) ⟶^{≈} mod_{Y}(D) respectively. If |A|−|B|+|C| ∈ E(X) is a generator, then F^{!}(|A|−|B|+|C|) = |C′|− |Cyl(f′)| + |Cone(f′)| ∈ E(Y), where C′, B′ ∈ D such that \underline{C}′ = F(\underline{C}), \underline{B}′ = F(\underline{B}) and f′ : B′ ⟶ C′ is a morphism in D such that F(\underline{f}) = \underline{f}′.

The free presentation (3.4) for the left homotopy pairs (C, X) and (D, Y) of subsection 3.2, induces the following exact diagram with invertible vertical morphisms:

0 ⟶ Ker(c_{X}) ⟶ E(X) ⟶^{i_{X}} K_{0}(C/X , ⊕) ⟶^{π_{X}} K_{0}(C/X) ⟶ 0
F^{!}↓  ^{F_{∗}}↓
0 ⟶ Ker(c_{Y}) ⟶ E(Y) ⟶^{i_{Y}} K_{0}(D/Y, ⊕) ⟶^{π_{Y}} K_{0}(D/Y) ⟶ 0

Lemma 3.8. The central square of the above diagram commutes iff for any morphism f in C, we have: |F(Cone(f))| = |Cone(F(f))| in K_{0}(D/Y, ⊕).

Proof. Let f : B ⟶ C be a morphism in C and consider the element α(f) = |C| − |Cyl(f)| + |Cone(f)| ∈ E(X) induced by the X −exact sequence 0 ⟶ Cone(f) ⟶

Cyl(f) ⟶^{f^{∗}} C in C. If the central square commutes, then: i_{Y}F^{!}(α(f)) = i_{Y}(α(f′)) = |\underline{C}′|−|\underline{Y}_{C′} ⊕\underline{B}′|+|\underline{Cone(f′)}| = |\underline{C}′|−|\underline{B}′|+|\underline{Cone(f′)}| = |\underline{C}′|−|\underline{B}′|+|Cone(F(\underline{f}))|, where C′, B′ ∈ D such that F(\underline{C}) = \underline{C}′, F(\underline{B}) = \underline{B}′, ψ : Y_{C′} ⟶ C′ is a right Y−approximation of C′ and f′ : B′ ⟶ C′ is a morphism in D such that \underline{f}′ = F(\underline{f}). On the other hand F_{∗}i_{X} (α(f)) = |F(\underline{C})| − |F(\underline{Cyl(f)}| + |F(\underline{Cone(f)}))| = |F(\underline{C})| − |F(\underline{B})| + |F(\underline{Cone(f)})| = |\underline{C}′| − |\underline{B}′|+ |F(\underline{Cone(f)})|. Since \underline{Cone(f)} = Cone(\underline{f}), we have |\underline{C}′| − |\underline{B}′| + |F(Cone(\underline{f}))| = |\underline{C}′| − |\underline{B}′| + |Cone(F(\underline{f})| ⇒ |F(Cone(\underline{f}))| = |Cone(F(\underline{f}))|. Conversely if the last relation holds for any morphism f in C, then the above analysis shows that the central square in the above diagram commutes.

We have the following consequence of Lemma 3.8 and Theorem 2.10.

Corollary 3.9. Let F : C/X ⟶ D/Y be a stable equivalence and assume that either: (α) for any morphism f in C we have: |F(Cone(f))| = |Cone(F(f))| in K_{0}(D/Y, ⊕) or (β) F commutes functorially with the loop functors Ω_{X} , Ω_{Y}. Then the stable equivalence F induces isomorphisms:

(i) F^{∗} : K_{0}(C/X) ⟶^{≅} K_{0}(D/Y).

(ii) F_{∗} : Ker(c_{X}) ⟶^{≅} Ker(c_{Y}).

(iii) D(X) ⟶^{≅} D(Y).

Observe that a direct consequence of condition (α) above is that F commutes with the loop functors Ω_{X} , Ω_{Y}, not necessarily in a functorial way.

3.5. Abelian Categories. In this subsection we assume that C is a Krull-Schmidt abelian category with enough projective and injective objects. If P(C) is the full subcategory of projective objects of C, then (C, P(C)) is a left homotopy pair and the Grothendieck Group K_{0}(C, P(C)) is identified with the usual Grothendieck group K_{0}(C). The stable Grothendieck group K_{0}(C/P(C)) of the left triangulated category C/P(C) is identified with Coker(c_{C}), where c_{C} : K_{0}(P(C), ⊕) ⟶ K_{0}(C) is the induced Cartan map, and as before admits the free presentation 0 ⟶ D(P(C)) ⟶ K_{0}(C/P(C), ⊕) ⟶ K_{0}(C/P(C)) ⟶ 0. The loop functor of C/P(C) is denoted by Ω_{C}.

Let F : C/P(C) ⟶ D/P(D) be a stable equivalence, where D is a Krull-Schmidt abelian category with enough projective and injective objects. The following shows that in this case we can remove the functoriallity of the isomorphism FΩ_{C}(\underline{C}) ≅ Ω_{D}F(\underline{C}) in Corollary 3.9.

Proposition 3.10. If ∀C ∈ C: FΩ_{C}(\underline{C}) ≅ Ω_{D}F(\underline{C}), then F induces isomorphisms F^{∗} : K_{0}(C/P(C)) ⟶ K_{0}(D/P(D)) and F_{∗} : Ker(c_{C}) ⟶ Ker(c_{D}).

Proof. By our previous results it suffices to show that for any generator |\underline{A}| − |\underline{B}| +

|\underline{C}| of D(P(C)) which comes from a left triangle Ω_{C}(\underline{C}) ⟶^{\underline{h}} \underline{A} ⟶^{\underline{g}} \underline{B} ⟶^{\underline{f}} \underline{C} in C/P(C), we have: [F(\underline{A})] − [F(\underline{B})] + [F(\underline{C})] = 0 in K_{0}(D/P(D)).

First we assume that \underline{g} = 0. Then the triangle Ω_{C}(\underline{A}) ⟶ Ω_{C}(\underline{B}) ⟶ Ω_{C}(\underline{C}) ⟶ \underline{A} is split, hence Ω_{C}(\underline{C}) ≅ \underline{A}⊕Ω_{C}(\underline{B}). Then FΩ_{C}(\underline{C}) ≅ F(\underline{A})⊕FΩ_{C}(\underline{B}) ⇒ Ω_{D}F(\underline{C})) ≅ F(\underline{A}) ⊕ Ω_{D}F(\underline{B}). Hence [Ω_{D}F(\underline{C}))] = [F(\underline{A})] + [Ω_{D}F(\underline{B})] ⇒ −[F(\underline{C})] = [F(\underline{A})] − [F(\underline{B})] ⇒ [F(\underline{A})] − [F(\underline{B})] + [F(\underline{C})] = 0.

Assume now that \underline{g} ≠ 0. We fix a short exact sequence (E) : 0 ⟶ A′ ⟶ B′ ⟶ C′ ⟶ 0 in C which induces the given triangle in C/P(C). If \underline{A} = \underline{A}_{1} ⊕\underline{A}_{2}, then since \underline{A} ≅ \underline{A}′, there are projective objects P, Q and an isomorphism A′⊕Q ≅ A_{1}⊕A_{2}⊕P. Consider the push-out diagram:

0 ⟶ A′ ⊕ Q ⟶ B′ ⊕ Q ⟶ C′ ⊕ Q ⟶ 0
↓  ↓  ^{||}↓
0 ⟶ A_{2} ⊕ P ⟶ E ⟶ C′ ⊕ Q ⟶ 0

which induces a triangle Ω_{C}(\underline{E}) ⟶ \underline{A}_{1} ⟶ \underline{B} ⟶ \underline{E} and a split triangle Ω_{C}(\underline{A}_{2}) ⟶ \underline{A}_{1} ⟶ \underline{A} ⟶ \underline{A}_{2}. Then |\underline{A}| − |\underline{B}| + |\underline{C}| = (|\underline{A}_{1}| − |\underline{B}| + |\underline{E}|) + (|\underline{A}_{2}| − |\underline{E}| + |\underline{C}|). Since C, C/P(C) are Krull-Schmidt, using induction it follows directly that any element |\underline{A}| − |\underline{B}| + |\underline{C}| as above is a sum of elements of the form |\underline{A}′| − |\underline{B}′| + |\underline{C}′|, where Ω(\underline{C}′) ⟶ \underline{A}′ ⟶ \underline{B}′ ⟶ \underline{C}′ is a triangle in C/P(C) and A′ is indecomposable. Hence it

suffices to show that [F(\underline{A})] − [F(\underline{B})] + [F(\underline{C})] = 0, for any triangle (T) : Ω(\underline{C}) ⟶^{\underline{h}} \underline{A} ⟶^{\underline{g}} \underline{B} ⟶^{\underline{f}} \underline{C} in C, where A, hence \underline{A}, is indecomposable and \underline{g} ≠ 0. Since F preserves split triangles, we can assume that the sequence (E) which induces (T) has

no split exact summands. Then since A is indecomposable, the sequence 𝕐(\underline{A}) ⟶^{𝕐(\underline{g})}

𝕐(\underline{B}) ⟶^{𝕐(\underline{f})} 𝕐(C) ⟶ F\underline{f} ⟶ 0 is the start of a minimal projective resolution of F\underline{f} in

mod(C/P(C)). Since F is an equivalence, 𝕐F(\underline{A}) ⟶^{𝕐F(\underline{g})} 𝕐F(\underline{B}) ⟶^{𝕐F(\underline{f})} 𝕐F(C) ⟶ F_{F(\underline{f})} ⟶ 0 is the start of a minimal projective resolution of F_{F(\underline{f})} in D/P(D). Since C has injective envelopes and A is indecomposable, the functor Ext^{1}(−, A) lies in mod(C/P(C)) and is indecomposable. Following the methods of Auslander-Reiten

in [1], we infer that there exists an exact sequence 0 ⟶ A′′ ⟶ B′′ ⟶^{f′′} C′′ ⟶ 0 wihout split exact summands, such that \underline{A}′′ ≅ F(\underline{A}), \underline{B}′′ ≅ F(\underline{B}), \underline{C}′′ ≅ F(\underline{C}) and \underline{f}′′ = F(\underline{f}). Then [F(\underline{A})] − [F(B)] + [F(C)] = [\underline{A}′′] − [F(B)] + [F(C)] which is zero since the above short exact sequence induces a triangle Ω_{D}F(\underline{C}) ⟶ \underline{A}′′ ⟶

F(B) ⟶^{F(\underline{f})} F(C) in D/P(D). We conclude that [F(\underline{A})] − [F(\underline{B})] + [F(\underline{C})] = 0.

4. Stable Grothendieck Groups as Waldhausen Groups

We recall that an (additive) Waldhausen category is a triple (C, Cof, Weq), where C is an additive category and Cof, resp. Weq, is a class of morphisms in C, called cofibrations, resp. weak equivalences, subject to certain axioms, see [24] for details. An easy consequence of the axioms is that any cofibration f : A ⟶ B admits a cokernel g, and then the resulting sequence A ⟶^{f} B ⟶^{g} C is called a cofibration sequence. A morphism f is a trivial cofibration, if f is a cofibration and a weak equivalence. An object C is called cofibrant, resp. acyclic, if 0 ⟶ C is a cofibration, resp. weak equivalence. An object C is called fibrant, if any trivial cofibration f : C ⟶ R is split monic. The full subcategory of acyclic objects is denoted by Ac(C). A Waldhausen category is called strict, if any of its objects is fibrant and cofibrant. A Waldhausen category satisfies the factorization property, if any morphism f : A ⟶ B admits a factorization f = g ∘ h, where g is a cofibration and h is a weak equivalence. Finally a Waldhausen category is called saturated, if whenever f ∘ g is a weak equivalence, f is a weak equivalence iff g is.

Now let (C, X) be a right homotopy pair. We denote by Cof_{X} (C) the class of X −monics in C and by Weq_{X} (C) the class of stable equivalences, i.e. the morphisms f in C such that \underline{f} is invertible in C/X. In [7] it is shown that (left and right) homotopy pairs correspond bijectively with special closed model categories in the sense of Quillen. The following result is a one-sided version of this correspondence.

Proposition 4.1. The map (C, X) ↦ (C, Cof_{X} (C), Weq_{X} (C)) is a bijection between right homotopy pairs and strict saturated Waldhausen categories satisfying the factorization property. The inverse map is given by (C, Cof, Weq) ↦ (C, Ac(C)). Moreover the homotopy category Ho(C), defined by formally inverting the weak equivalences, is equivalent to the stable category C/Ac(C).

Proof. This is similar to the proof of Theorem 4.5 in [7] and is left to the reader.

We recall [24] that the Waldhausen group K_{0}(C) of a skeletally small additive Waldhausen category (C, Cof, Weq) is defined as follows. K_{0}(C) has one generator [C] for each object of C, subject to the relations: [A] = [C], if there exists a weak equivalence A ⟶ C and [B] = [A] + [C], if there exists a cofibration sequence A ⟶ B ⟶ C. Now let (C, X) be a right homotopy pair, where C is skeletally small. It is clear that the relative Grothendieck group K_{0}(C, X) is isomorphic to the Waldhausen group of the Waldhausen category (C, Cof_{X} (C), Iso), where Iso is the class of isomorphisms in C and the split group K_{0}(X , ⊕) is the Waldhausen group of the Waldhausen subcategory (X , Cof_{X} , Iso).

Corollary 4.2. The Grothendieck group K_{0}(C/X) of the stable left triangulated category C/X is isomorphic to the Waldhausen group K_{0}(C) of the Waldhausen category (C, Cof_{X} (C), Weq_{X} (C)).

Proof. Define a morphism α : K_{0}(C) ⟶ K_{0}(C/X) by α([C]) = [\underline{C}]. If [C] = [A], then there exists a weak equivalence C ⟶ A, i.e. an isomorphism \underline{C} ⟶ \underline{A}, hence [\underline{C}] = [\underline{A}]. If [B] = [A]+[C] in K_{0}(C), then there exists an X −exact (or cofibration) sequence A ⟶ B ⟶ C ⟶ 0 in C, hence a triangle (T) : \underline{A} ⟶ \underline{B} ⟶ \underline{C} ⟶ Σ_{X} (\underline{A}) in C/X, where Σ_{X} is the suspension functor of C/X [5]. Then [\underline{B}] = [\underline{A}] + [\underline{C}]. Hence α is well-defined. Define a morphism β : K_{0}(C/X) ⟶ K_{0}(C) as follows. If [\underline{C}] ∈ K_{0}(C/X), then β([\underline{C}] = [C]. If [\underline{C}] = [\underline{A}], then by the dual of Remark 3.1, C ⊕ X_{A} ≅ A ⊕ X_{C}, where X_{A}, X_{C} are left X −approximations of A, C. Then [C] + [X_{A}] = [A] + [X_{C}]. Since the objects of X are weakly equivalent to 0, it follows that [C] = [A] in K_{0}(C). If [\underline{B}] = [\underline{A}] + [\underline{C}], then there exists a triangle (T) as above. By [5], there exists an X −exact sequence A′ ⟶ B′ ⟶ C′ ⟶ 0 in C such that the induced right triangle \underline{A}′ ⟶ \underline{B}′ ⟶ \underline{C}′ ⟶ Σ_{X} (\underline{A}′) is isomorphic to (T) in C/X. This implies that [B] = [A] + [C] in K_{0}(C). It follows that β is a well–defined morphism which plainly is the inverse of α.

Note that the exact sequence K_{0}(X , ⊕) ⟶^{c_{X}} K_{0}(C, X) ⟶ K_{0}(C/X) ⟶ 0, where c_{X} is the Cartan map, coincides with the exact sequence induced by the exact functors (X , Cof_{X} (C), Iso) ⟶ (C, Cof_{X} (C), Iso) ⟶ (C, Cof_{X} (C), Weq_{X} (C)), using for instance the Localization Theorem [24].

Proposition 4.3. Let F : C ⟶ D be an additive functor, where (C, X) and (D, Y) are right homotopy pairs, such that F(X) ⊆ Y. Assume that F satisfies the following:

(i) F sends X −monics to Y−monics and F preserves cokernels of X −monics. (ii) If \underline{F(f)} is invertible, then so is \underline{f}. (iii) For any morphism f : F(A) ⟶ B in D, there exists an X −monic g : A ⟶ A′

and a stable equivalence f′ : F(A′) ⟶ B such that f = F(g) ∘ f′. Then F induces an exact functor F : C/X ⟶ D/Y and an isomorphism

K_{0}(F) : K_{0}(C/X) ⟶^{≅} K_{0}(D/Y).

Proof. The imposed assumptions imply that F is an exact functor between the associated Waldhausen categories (C, Cof_{X} (C), Weq_{X} (C)), (D, Cof_{Y}(D), Weq_{Y}(D)). Since by [5] any triangle in the stable category comes from an X −exact sequence, it follows that F induces an exact functor F : C/X ⟶ D/Y between the associated right triangulated categories. Since trivially F satisfies the conditions of the Approximation Theorem [24], it follows that K_{0}(F) is invertible.

Given a skeletally small Waldhausen category C, Waldhausen generalizing work of Quillen [18], defines in [24] a sequence of Waldhausen groups K_{n}(C), ∀n ≥ 0, satisfying most of the nice properties which are known to be true for the Quillen K−groups. If (C, X) is a right homotopy pair, where C is skeletally small, then since K_{0}(C) ≅ K_{0}(C/X), it is reasonable to define the higher stable Grothendieck groups K_{n}(C/X), ∀n ≥ 0, of the right triangulated category C/X as the Waldhausen groups K_{n}(C), of the associated Waldhausen category (C, Cof_{X} (C), Weq_{X} (C)). Then most of the above results are true in the higher theory. We don’t know if it is possible for a general right triangulated category R, to define a higher K-theory K′_{∗}(R), for instance using the methods of Neeman [16], such that K′_{∗}(R) = K_{∗}(C/X), if R is the stable category C/X induced by a right homotopy pair (C, X).

5. Other Types of Equivalences

5.1. Stabilizations. We recall from [14], [8], that for any left triangulated category C = (C, Ω, Δ), there exists a triangulated category S(C), the stabilization of C, and an exact functor S : C ⟶ S(C), the stabilization functor, which is universal for exact functors out of C to triangulated categories. We need the following useful result which shows that the Grothendieck group is invariant under stabilization.

Proposition 5.1. [8] The stabilization functor S : C ⟶ S(C) induces an isomorphism K_{0}(S) : K_{0}(C) ⟶^{≅} K_{0}(S(C)).

5.2. Complexes. Throughout this subsection we fix an additive category D with split idempotents. Let C(D) be the category of complexes over D. The full subcategory of C(D) consisting of right bounded, resp. bounded, complexes is denoted by C^{−}(D), resp. C^{b}(D). Let X be the full subcategory of C(D) consisting of all contractible complexes. It is easy to see that the pair (C(D), X) is a homotopy pair and the stable category C(D)/X coincides with the triangulated homotopy category H(D). Similarly we obtain the triangulated homotopy categories H^{−}(D), H^{b}(D).

We denote by S : D ⟶ H(D) the functor sending an object D to the stalk complex D^{•}[0] concentrated in degree zero. The following is well–known.

Lemma 5.2. The functor S induces an isomorphism S_{∗} : K_{0}(D, ⊕) ⟶ K_{0}(H^{b}(D)) defined by S_{∗}(|D|) = [S(D)].

Proof. Any triangle in H^{b}(D) is induced by a sequence A^{•} ↣ B^{•} ↠ C^{•} of bounded complexes such that A^{n} ↣ B^{n} ↠ C^{n} is split exact in D, ∀n ∈ ℤ. It is easy to see that defining j : K_{0}(H^{b}(D)) ⟶ K_{0}(D, ⊕) by j([D^{•}]) = ∑_{i∈ℤ}(−1)^{i}|D^{i}|, we obtain a well–defined morphism which plainly is the inverse of S_{∗}.

Assume now that D is an exact category with enough projectives and let P be the full subcategory of projectives. Then (D, P) is a left homotopy pair, and the Grothendieck group K_{0}(D, P) coincides with the group K_{0}(D) of the exact category D as defined by Quillen [18]. The stable category D/P is left triangulated and we have the exact sequence, where c is the induced Cartan morphism:

0 ⟶ Ker(c) ⟶ K_{0}(P, ⊕) ⟶^{c} K_{0}(D) ⟶ K_{0}(D/P) ⟶ 0

Let H^{−,b}(P) be the full subcategory of H^{−}(P) consisting of all complexes which are acyclic [14] everywhere, except of a finite number of degrees. Then H^{−,b}(P) is a full triangulated subcategory of H^{−}(P) and contains as a thick subcategory, the homotopy category H^{b}(P) of bounded complexes over P. As in [12] the Verdier quotient H^{−,b}(P)/H^{b}(P) is denoted by D_{P}. By a well–known result of Grothendieck [10] the short exact sequence of triangulated categories H^{b}(P) ↪ H^{−,b}(P) ↠ D_{P}

induces an exact sequence K_{0}(H^{b}(P)) ⟶^{c^{•}} K_{0}(H^{−,b}(P)) ⟶ K_{0}(D_{P}) ⟶ 0, first considered by Happel [12]. Note that if C is abelian then as is well–known, H^{−,b}(P) is identified with the bounded derived category D^{b}(C) of C.

Let P : D ⟶ H^{−,b}(P) be the functor sending an object D ∈ D to its deleted projective resolution. The composite functor D ⟶^{P} H^{−,b}(P) ⟶^{Q} D_{P}, where Q is the quotient functor, kills projective objects and it induces in a natural way an exact functor T : D/P ⟶ D_{P}.

Proposition 5.3. The functor T induce an isomorphism

T_{∗} : K_{0}(D/P) ⟶^{≅} K_{0}(D_{P})

such that the following diagram commutes:

K_{0}(P, ⊕) ⟶^{c} K_{0}(D) ⟶ K_{0}(D/P) ⟶ 0
S_{∗}↓  ^{P_{∗}}↓  ^{T_{∗}}↓
K_{0}(H^{b}(P)) ⟶^{c^{•}} K_{0}(H^{−,b}(P)) ⟶ K_{0}(D_{P}) ⟶ 0.

If D is abelian, then P_{∗} : K_{0}(D) ⟶^{≅} K_{0}(D^{b}(D)).

Proof. Trivially the left square of the above diagram commutes and by Lemma 5.2, S_{∗} is invertible. By [14], [8], T is isomorphic to the stabilization functor. Hence by Proposition 5.1, the induced morphism T_{∗} is invertible. Then by the diagram, P_{∗} is surjective. If D is abelian, then H^{−,b}(P) ≈ D^{b}(D) and the last assertion follows by a well–known result of Grothendieck. For completeness we include the argument. Define a morphism H_{∗} : K_{0}(D^{b}(D)) ⟶ K_{0}(D) by H_{∗}([A^{•}]) = ∑_{i∈ℤ}(−1)^{i}[H^{i}(A^{•})], where H^{i}(A^{•}) is the i−th cohomology object. Plainly H_{∗}P_{∗}([A]) = [A] and this shows that P_{∗} is injective. Hence P_{∗} is invertible.

5.3. Resolving Subcategories. Let C be an abelian category with enough projectives and let P be the full subcategory of projective objects of C. We recall that a full additive subcategory A ⊆ C is called resolving, if A is closed under extensions, kernels of epimorphisms and contains the projectives. If A is resolving in C, then trivially (C, P), (A, P) are left homotopy pairs. Let K_{0}(C/P), K_{0}(A/P) be the Grothendieck groups of the stable left triangulated categories C/P, A/P. The relative Grothendieck groups K_{0}(C, P), K_{0}(A, P) are identified with the Grothendick groups K_{0}(C), K_{0}(A), the latter viewing A as an exact category. Trivially the exact inclusions A ↪ C and A/P ↪ C/P induce morphisms α : K_{0}(A) ⟶ K_{0}(C) and β : K_{0}(A/P) ⟶ K_{0}(C/P) making the following exact diagram commutative:

K(P, ⊕) ⟶ K_{0}(A) ⟶ K_{0}(A/P) ⟶ 0
||↓  ^{α}↓  ^{β}↓
K(P, ⊕) ⟶^{c} K_{0}(C) ⟶ K_{0}(C/P) ⟶ 0

Corollary 5.4. If Â = C, i.e. if any object of C admits a finite resolution by objects from A, then the morphisms α, β above are invertible.

Proof. Since A is closed under kernels of epimorphisms, and any object of C admits a resolution by objects from A, by the resolution Theorem [21], it follows that α is invertible. Then the above diagram implies that β is also invertible.

If Â = C and Ext_{C}^{t}(A, P) = 0, ∀t ≥ 1, then it can be shown that the inclusion A/P ↪ C/P admits a right adjoint S which induces the inverse of β in K_{0}. Actually S is the stabilization functor of C/P, in case P is a cogenerator of A, see [8].

5.4. Derived Equivalences and Gorenstein Rings. If Λ is an associative ring, then we denote by Mod(Λ) the category of right Λ−modules and by mod(Λ) the category of finitely presented right Λ−modules. The category of finitely generated projective right Λ−modules is denoted by P_{Λ}. If Λ is right coherent, then mod(Λ) is abelian, (mod(Λ), P_{Λ}) is a left homotopy pair and the stable category mod(Λ)/P_{Λ} := \underline{mod}(Λ) is left triangulated.

Recall that two associative rings Λ, Γ are derived equivalent, if there exists a triangle equivalence D^{b}(Mod(Λ)) ⟶^{≈} D^{b}(Mod(Γ)). The following shows that for right coherent rings the stable Grothendieck group is invariant under derived equivalence.

Corollary 5.5. Let Λ, Γ be derived equivalent right coherent rings. Then the stable Grothendieck groups K_{0}(\underline{mod}(Λ)), K_{0}(\underline{mod}(Γ)) are isomorphic.

Proof. By [20], a given derived equivalence restricts to triangle equivalences H^{b}(P_{Λ}) ≈ H^{b}(P_{Γ}) and H^{−,b}(P_{Λ}) ≈ H^{−,b}(P_{Γ}). Hence it induces a triangle equivalence D_{P_{Λ}} ≈ D_{P_{Γ}}. Then the assertion follows from Proposition 5.3.

Recall that a Noetherian ring Λ is called Gorenstein, if Λ has finite self-injective dimension from both sides. If Λ is a Gorenstein ring, then the category CM(Λ) of Cohen-Macaulay modules is defined by CM(Λ) = {C ∈ mod(Λ) | Ext_{Λ}^{n}(C, Λ) = 0, ∀n ≥ 1}. Trivially CM(Λ) is a resolving subcategory of mod(Λ), so its stable category modulo projectives, denoted by \underline{CM}(Λ), is left triangulated.

Corollary 5.6. If Λ is a Gorenstein ring, then there are isomorphisms K_{0}(CM(Λ)) ⟶^{≅} K_{0}(mod(Λ)) and K_{0}(\underline{CM}(Λ)) ⟶^{≅} K_{0}(\underline{mod}(Λ)). Moreover if Γ is a Noetherian ring derived equivalent to Λ, then there are isomorphisms K_{0}(CM(Λ)) ⟶^{≅} K_{0}(CM(Γ)) and K_{0}(\underline{CM}(Λ)) ⟶^{≅} K_{0}(\underline{CM}(Γ)).

Proof. By [8], [12], the stable category \underline{CM}(Λ) is the stabilization of \underline{mod}(Λ), so the first claim follows by Propositions 5.1, 5.3. If Γ is a Noetherian ring derived equivalent to Λ, then by [8], Γ is also Gorenstein and the second claim follows.

5.5. Artin Algebras. Throughout we fix an Artin algebra Λ and consider the exact sequence

(5.1) 0 ⟶ Ker(c_{Λ}) ⟶ K_{0}(P_{Λ}, ⊕) ⟶^{c_{Λ}} K_{0}(mod(Λ)) ⟶ K_{0}(\underline{mod}(Λ)) ⟶ 0

induced by the Cartan map c_{Λ}. We recall that the abelian groups K_{0}(P_{Λ}, ⊕), K_{0}(mod(Λ)) are free of rank n, where n is the number of non-isomorphic simple Λ−modules. In particular Ker(c_{Λ}) is free of rank s ≤ n.

Lemma 5.7. (i) K_{0}(\underline{mod}(Λ)) = 0 iff the Cartan map c_{Λ} is invertible iff Det(c_{Λ})

= ±1. This happens if gl.dimΛ < ∞. (ii) K_{0}(\underline{mod}(Λ)) is finite iff the Cartan map c_{Λ} is a monomorphism iff Det(c_{Λ}) ≠

0. In this case: |K_{0}(\underline{mod}(Λ))| = |Det(c_{Λ})|. (iii) If K_{0}(\underline{mod}(Λ)) is free, then Det(c_{Λ}) = 0.

Proof. (i) Obviously c_{Λ} is invertible ⇔ Det(c_{Λ}) = ±1 ⇒ K_{0}(\underline{mod}(Λ)) = 0. If the last condition holds, the above exact sequence induces an isomorphism K_{0}(P_{Λ}, ⊕) ≅ K_{0}(mod(Λ)) ⊕ Ker(c_{Λ}). Since K_{0}(mod(Λ)), K_{0}(P_{Λ}, ⊕) are free of the same rank, it follows that Ker(c_{Λ}) = 0. Hence c_{Λ} is invertible. If gl.dimΛ = n < ∞, then ∀C ∈ mod(Λ), Ω^{n}(\underline{C}) = 0. Hence (−1)^{n}[\underline{C}] = [Ω^{n}(\underline{C})] = 0 ⇒ K_{0}(mod(Λ)) = 0.

(ii) If Det(c_{Λ}) ≠ 0, ±1, then it is well-known that |Coker(c_{Λ})| = |K_{0}(\underline{mod}(Λ))| = |Det(c_{Λ})|. Hence in this case K_{0}(\underline{mod}(Λ)) is finite. If this is true, tensoring with ℚ the exact sequence (5.1), we have an exact sequence 0 ⟶ Ker(c_{Λ}) ⊗_{ℤ} ℚ ⟶ ℚ^{n} ⟶ ℚ^{n} ⟶ 0. This implies that Ker(c_{Λ}) ⊗_{ℤ} ℚ = 0. Since Ker(c_{Λ}) ≅ ℤ^{s}, this implies that s = 0, hence Ker(c_{Λ}) = 0, so c_{Λ} is a monomorphism. Finally if c_{Λ} is a monomorphism, then arguing as above we infer that K_{0}(\underline{mod}(Λ)) ⊗_{ℤ} ℚ = 0, i.e. c_{Λ} ⊗_{ℤ} ℚ is invertible. This implies trivially that Det(c_{Λ}) ≠ 0.

(iii) If K_{0}(\underline{mod}(Λ)) is free, then by (ii), Det(c_{Λ}) = 0.

Example 5.8. Let Λ be an algebra given by a Brauer tree with e edges and multiplicity m of the exceptional vertex. It is well–known that Det(c_{Λ}) = e^{2}m + 1. Hence |K_{0}(\underline{mod}(Λ))| = e^{2}m+1. By a classical result of Brauer, for any finite group G and any field k of characteristic p, |Det(c_{kG})|, hence the order of K_{0}(\underline{mod}(kG)), is a power of p. On the other hand there are algebras of infinite global dimension, even self-injective, with trivial stable Grothendieck group, see [12].

Trivially the stable Grothendieck group is invariant under Morita equivalence. The next result summarizes our knowledge about the invariance of the stable Grothendieck group under stable, derived or tilting equivalence. First we need the following consequence of Corollary 3.6, first observed by Geigle [9] with a different proof, which shows that, with the notation of section 3, the group E(Λ) := E(P_{Λ}) which measures the complexity of the structure of the short exact sequences in mod(Λ), is always invariant under a stable equivalence.

Proposition 5.9. Any stable equivalence F : \underline{mod}(Λ) ⟶^{≈} \underline{mod}(Γ), induces an isomorphism F^{!} : E(Λ) ⟶^{≅} E(Γ).

Theorem 5.10. Assume that one of the conditions in (i) holds or the condition (ii) holds.

(i) There is a stable equivalence F : \underline{mod}(Λ) ⟶^{≈} \underline{mod}(Γ), such that: (a) F commutes (not necessarily functorially) with the loop functors Ω_{Λ}, Ω_{Γ}.

(b) Λ or Γ is representation-finite and F commutes with the equivalences DTr_{Λ}, DTr_{Γ}.

(c) Λ, Γ are symmetric, e.g. group algebras.

(d) Λ, Γ are self-injective with Lowey length ≥ 3

(e) Λ, Γ are indecomposable self-injective.

(f) Λ, Γ are without nodes in the sense of [15]. (ii) Λ, Γ are derived equivalent, in particular if Λ, Γ are tilting-cotilting equivalent. Then in any of the above cases the equivalence functor induces the following isomorphisms and equality:

Ker(c_{Λ}) ⟶^{≅} Ker(c_{Γ}), K_{0}(\underline{mod}(Λ)) ⟶^{≅} K_{0}(\underline{mod}(Γ)) and |Det(c_{Λ})| = |Det(c_{Γ})|.

Proof. (i) If F commutes with the loop functors Ω_{Λ}, Ω_{Γ}, then the assertion follows from Proposition 3.10. Assume now that Λ is representation-finite and F commutes with the equivalences DTr_{Λ}, DTr_{Γ}. By Butler-Auslander Theorem [3], the set of elements |DTr_{Λ}(C)| − |E| + |C|, where 0 ⟶ DTr(C) ⟶^{g} E ⟶^{f} C ⟶ 0 is an almost split sequence in mod(Λ) is a basis of E(Λ). Since trivially Γ is also representationfinite, the same is true for E(Γ). By Proposition 5.9, F induces an isomorphism F^{!} : E(Λ) ⟶^{≅} E(Γ), so by Proposition 2.9, to prove the assertion, it is sufficient to prove that Cone(F(\underline{f})) ≅ F(Cone(\underline{f})) holds for any minimal right almost split morphism f, with non-projective target. Let |DTr_{Λ}(C)|−|E|+|C| be a basis element of E(Λ) as above. If \underline{f} ≠ 0, then by [3], there exists an almost split sequence 0 ⟶

DTr_{Γ}(C′) ⟶ E′ ⟶^{f′} C′ ⟶ 0 in mod(Γ), where \underline{C}′ = F(\underline{C}), \underline{E}′ = F(C), F(\underline{f}) = \underline{f}′ and then DTr_{Γ}(\underline{C}′) = DTr_{Γ}F(\underline{C}) = FDTr_{Λ}(\underline{C}). Since DTr_{Λ}(\underline{C}′) = Cone(F(\underline{f})) and DTr_{Λ}(\underline{C}) = Cone(\underline{f}), we infer that F(Cone(\underline{f})) = Cone(F(\underline{f})). If \underline{g} = 0, then

we have a triangle Ω_{Λ}(\underline{C}) ⟶^{\underline{h}} DTr_{Λ}(\underline{C}) ⟶^{0} \underline{E} ⟶^{\underline{f}} \underline{C} and a weak kernel sequence

FΩ_{Λ}(\underline{C}) ⟶^{F(\underline{h})} FDTr_{Λ}(\underline{C}) ⟶^{0} F(\underline{E}) ⟶^{F(\underline{f})} F(\underline{C}), so F(\underline{f}) is monic. Using that \underline{f} is minimal right almost split and F preserves such maps, it is not difficult to see that Cone(F(\underline{f})) ≅ DTr_{Γ}(F(\underline{C})). We infer again that Cone(F(\underline{f})) ≅ F(Cone(\underline{f})).

If (c), (d) or (f) holds, then by [3] and a simple modification of the arguments of [15] respectively, we infer that F commutes with the loop functors , so the assertion follows from part (a).

Assume now that Λ, Γ are indecomposable selfinjective. If LL denotes Loewy length, then it is easy to see that LL(Λ) ≤ 2 iff LL(Γ) ≤ 2 in which case LL(Λ) = LL(Γ). If LL(Λ) = 1, then the assertion is trivial since \underline{mod}(Λ) = 0 = \underline{mod}(Γ). If LL(Λ) = 2 = LL(Γ), then Λ, Γ are Nakayama algebras and it is easy to see that the stable equivalence can be lifted to an equivalence mod(Λ) ≈ mod(Γ). We infer that in both cases the stable Grothendieck groups are isomorphic. If LL(Λ), LL(Γ) ≥ 3, then the assertion follows by part (d).

(ii) Follows by Corollary 5.5.

Remark 5.11. The above result contains some results of [2], [15], [17] and can be applied to more general situations. For instance using Proposition 3.10, if the condition (i)(a) holds, we obtain the results of Solberg [22]. Let R and S be commutative 2-dimensional integrally closed domains with an algebraically closed residue field. If R and S are stably equivalent by an equivalence which commutes with the loop functors, then R and S have isomorphic ideal class groups: C(R) ≅ C(S). This follows from the fact that in this situation, the stable group K_{0}(\underline{mod}(R)) is isomorphic to the ideal class group C(R) of R, see [2]. We refer to [19] for further examples, mainly of blocks of group algebras, of stable equivalences commuting with the loop functors.

The following example shows that in general a stable equivalence F does not induces an isomorphism on the level of stable Grothedieck groups, even if F is induced by a functor F : mod(Λ) ⟶ mod(Γ) on the level of module categories.

Example 5.12. Let Λ be an Artin Algebra with radical square zero and with infinite global dimension, e.g. Λ = k[x]/(x^{2}), where k is a field. It is well-known [3] that Λ is stably equivalent to a hereditary Artin algebra Γ. More precisely there exists a functor F : mod(Λ) ⟶ mod(Γ) which induces a stable equivalence F : \underline{mod}(Λ) ⟶^{≈} \underline{mod}(Γ). Since Γ is hereditary, it follows that K_{0}(\underline{mod}(Γ)) = 0. However K_{0}(\underline{mod}(Λ)) ≠ 0 in general. For instance if Λ = k[x]/(x^{2}), then it is easy to see that K_{0}(\underline{mod}(Λ)) ≅ ℤ_{2}. Of course in this example we have that F does not commutes with the loop functors Ω_{Λ}, Ω_{Γ} or the equivalences DTr_{Λ}, DTr_{Γ}.

Example 5.13. Let T ∈ mod(Λ) and let Γ := End_{Λ}(T). Since X := add(T) is contravariantly finite, the stable category mod(Λ)/X, henceforth denoted by \underline{mod}_{T} (Λ), is left triangulated. Then by Corollary 3.5 we have K_{0}(mod(Λ), X) ≅ K_{0}(mod(Γ)) and K_{0}(\underline{mod}_{T} (Λ)) ≅ K_{0}(\underline{mod}(Γ)). If T is a (generalized) tilting module, then by [11] Λ, Γ are derived equivalent. Hence by Corollary 5.5 we infer that: K_{0}(mod(Λ), X) ≅ K_{0}(mod(Λ)) and K_{0}(\underline{mod}_{T} (Λ)) ≅ K_{0}(\underline{mod}(Λ)). If T is a generator, then the isomorphism K_{0}(mod(Λ), X) ≅ K_{0}(mod(Γ)) above shows that K_{0}(mod(Λ), X) is free with rank equal to the number of non-isomorphic indecomposable direct summands of T. It is an open question (equivalent to the generalized Nakayama conjecture [13]) whether the canonical epimorphism K_{0}(mod(Λ), X) ↠ K_{0}(mod(Λ)) is invertible, if T is a generator without selfextensions.

Acknowledgement. The author thanks the referee for his/her useful comments and suggestions which resulted in an improvement of the paper.

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