GROTHENDIECK GROUPS ARISING FROM CONTRAVARIANTLY FINITE SUBCATEGORIES

Apostolos Beligiannis and Nikolaos Marmaridis

Department of Mathematics, University of Ioannina 45110 Ioannina, Greece

To the memory of Maurice Auslander

Abstract

The subject of the paper is the study of the relative homological properties of a given additive category C in relation to a given contravariantly finite subcategory X in C under the assumption that any X –epic has a kernel in C. We introduce the notion of the Grothendieck group relative to the pair (C, X) and also that of the Cartan map c_{X} relative to (C, X) and we show that the cokernel of c_{X} is isomorphic to the corresponding Grothendieck group of the stable category C/J_{X}. We also show that if the right X –dimension of C is finite, then c_{X} is an isomorphism. In case C is a finite dimensional k–additive Krull– Schmidt category, we introduce the notion of the X –dimension vector of an object of C. We give criteria for when an indecomposable object is determined, up to isomorphism, by its X –dimension vector.

1 Introduction

The notion of contravariantly finite subcategory was introduced in [2] by Auslander–Smalø for studying the problem of which subcategories of an artin algebra have almost split sequences. Since then, contravariantly finite subcategories turned to be very fruitful stimulating a long list of papers with substantial results on different aspects of the representation theory of finite dimensional algebras as also on relative homological algebra, cf. [3].

In [5], we exhibited how a contravariantly finite subcategory X of an additive category C induces on the stable category C/J_{X} a left triangulated structure assuming that any X –epic has a kernel in C. In particular this applies to the following case: Suppose that mod Λ is the category of finitely generated right Λ–modules over an artin algebra Λ and P_{Λ}^{f} is the full subcategory of the projective Λ– modules, then the stable category mod Λ/P_{Λ}^{f} has a left triangulated structure.

In [1] Auslander–Reiten showed that the cokernel of the Cartan map is isomorphic to the Grothendieck group of the stable category mod Λ/P_{Λ}^{f}. The first task of the present paper is to extend this result. Namely, given an additive category C and a contravariantly finite subcategory X such that any X –epic has a kernel in C, we define a Cartan map relative to the pair (C, X) and we show that its cokernel is isomorphic to the Grothendieck group of the left triangulated category C/J_{X}. We also show that if the relative to X right homological dimension of C is finite, then the Cartan map is an isomorphism. This result is analogous to the classical “reduction by resolution” result of Grothendieck for admissible subcategories of an abelian category, cf. [4], Theorem 4.2.

The next task of the paper is to give the relative to X homological interpretation of the Euler characteristic [9], Section 2.4., in case where C is a k–additive Krull–Schmidt category and X has finitely many indecomposable objects.

Finally, by introducing the notion of the X –dimension vector for any object of C we give a criterion for deciding when an indecomposable object is determined, up to isomorphism, by its X –dimension vector.

The contents of the paper section by section are as follows:

In Section 2, we introduce the notion of the right X –resolution for any object of C and show that any such resolution is obtained in fact in a “unique way”, see Lemma 2.5 ii. We prove the relative to X homological versions of “Schanuel’s Lemma”, (2.8) and “Horseshoe Lemma”, (2.11). We define the notion of the right X –dimension of C and we introduce the X –relative Extfunctors.

In Section 3, we introduce the notion of the Grothendieck group and of the Cartan map c_{X} relative to the pair(C, X). We also reveal the notion of the Grothendieck group of a left triangulated category, a notion already introduced by Grothendieck cf. [7] for triangulated categories. We show that in case where we deal with the left triangulated structure of the stable category C/J_{X}, its Grothendieck group is isomorphic to the cokernel of the Cartan map c_{X}, Theorem 3.19. We also show that if the X –dimension of C is finite then c_{X} has to be an isomorphism, Theorem 3.20.

In Section 4, we deal with the case where C is a finite dimensional k– additive Krull–Schmidt category and X has finitely many non–isomorphic indecomposable objects. We define for any object in C its X –dimension vector and also the Cartan matrix relative to the pair (C, X). In case this Cartan matrix is ℤ–invertible, we introduce the X –Euler characteristic of (C, X) and show its connection to the relative Ext–functors, Proposition 4.4. Moreover, we show that if the X –dimension of C is finite, then this Cartan matrix has to be ℤ–invertible, Proposition 4.7.

In Section 5, we deal with the question when an indecomposable object of C is determined by its X –dimension vector. By using the main result of [6] we show that a particular full subcategory D_{X} of C admits this property, Theorem 5.2. By applying this result in the case of mod Λ, we obtain, Theorem 5.4, the relative to X analogon to the main result of [6].

2 Relative Homological Algebra and Contravariantly Finite Subcategories

In this paper we assume, unless otherwise stated, that all considered categories are skeletally small and additive. We also assume that their subcategories are full, additive, closed under isomorphisms and direct summands. Functors between additive categories are supposed to be additive. In the following, we shall denote by AB the category of abelian groups.

Given a category C, we denote by C(A, B) the group of morphisms Mor_{C}(A, B) from A to B. Given a morphism α : A ⟶ B and an object C, we denote by α^{C} (resp. by α_{C}) the induced group homomorphism C(C, α) : C(C, A) ⟶C(C, B) (resp. C(α, C) : C(B, C) ⟶C(A, C)) in AB.

Let X be a subcategory of C and

A : ⋯ ⟶A_{i+1} ⟶^{α_{i+1}} A_{i} ⟶^{α_{i}} A_{i−1} ⟶⋯

be a complex of C.

Definition 2.1. The complex A is said to be right X –exact at the integer i, if the complex

A^{X} : ⋯ ⟶C(X, A_{i+1}) ⟶^{α_{i+1}^{X}} C(X, A_{i}) ⟶^{α_{i}^{X}} C(X, A_{i−1}) ⟶⋯

is exact at the integer i, for any object X of X. The complex A is said to be right X –exact, if it is right X –exact ∀i ∈ ℤ.

Definition 2.2. A morphism α : A ⟶ B is said to be an X –epic, if the complex A ⟶^{α} B ⟶ 0 is right X –exact. If A is an object of X, then an X –epic α is said to be a right X –approximation of B.

The above definitions lead to the notion of contravariantly finite subcategory introduced in [2] by M. Auslander and S.O. Smalø, recalled below.

Definition 2.3. A subcategory X of C is said to be contravariantly finite in C, if any object A of C has a right X –approximation .

Definition 2.4. A right X –exact complex of the form

X : ⋯ ⟶X_{i+1} ⟶^{x_{i+1}} X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{0} ⟶^{x_{0}} A ⟶0,

with X_{i} in X , ∀i ∈ ℕ ∪ {0}, is said to be a right X –resolution of A.

In the remaining part of the paper we assume that

(i) X is a subcategory contravariantly finite in C and

(ii) any X –epic has a kernel.

As the next Lemma shows, the assumptions above imply that any object A of C has a right X –resolution.

Lemma 2.5. (i) Any object A of C has a right X –resolution.

(ii) If

X : ⋯ ⟶X_{i+1} ⟶^{x_{i+1}} X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{0} ⟶^{x_{0}} A ⟶0,

is a right X –resolution of A, then x_{i} = g_{i} ∘ h_{i}, ∀i ∈ ℕ, where g_{i} : K_{i} ⟶ X_{i−1} is a kernel of x_{i−1} and h_{i} : X_{i} ⟶ K_{i} is a right X – approximation of K_{i}.

Proof. (i) Let x_{0} : X_{0} ⟶A be a right X –approximation of A, g_{1} : K_{1} ⟶X_{0} be the kernel of x_{0} and h_{1} : X_{1} ⟶ K_{1} be a right X –approximation of K_{1}. Since the sequences

0 ⟶C(Y, K_{1}) ⟶^{g_{1}^{Y}} C(Y, X_{0}) ⟶^{x_{0}^{Y}} C(Y, A) ⟶0 and C(Y, X_{1}) ⟶^{h_{1}^{Y}} C(Y, K_{1}) ⟶0

are exact in AB, the complex X_{1} ⟶^{g_{1}∘h_{1}} X_{0} ⟶^{x_{0}} A ⟶0 is right X –exact.

Let x_{1} denote the composition g_{1} ∘ h_{1}. We consider the kernel g_{2} : K_{2} ⟶ X_{1} of x_{1} and a right X –approximation h_{2} of K_{2} and continue inductively repeating the above construction.

(ii). Let g_{1} : K_{1} ⟶ X_{0} be the kernel of x_{0} and h_{1} : X_{1} ⟶ K_{1} the (unique) morphism with g_{1} ∘ h_{1} = x_{1}. We shall show that h_{1} is a right X –approximation of K_{1}.

For any object Y of X, the commutative diagram

X_{1} ⟶^{x_{1}} X_{0} ⟶^{x_{0}} A
h_{1}↓  ^{id_{X_{0}}} ↓  ^{id_{A}}↓
K_{1} ⟶^{g_{1}} X_{0} ⟶^{x_{0}} A

yields, the commutative diagram

C(Y, X_{1}) ⟶^{x_{1}^{Y}} C(Y, X_{0}) ⟶^{x_{0}^{Y}} C(X, A) ⟶ 0
h_{1}^{Y} ↓  ∥  ∥
0 ⟶ C(Y, K_{1}) ⟶^{g_{1}^{Y}} C(Y, X_{0}) ⟶^{x_{0}^{Y}} C(Y, A) ⟶ 0

in AB with exact rows.

By the Snake Lemma, h_{1}^{Y} is an epimorphism. Hence h_{1} is a right X – approximation of K_{1}. We repeat the above procedure using the kernel g_{2} : K_{2} ⟶X_{1} of x_{1} and continue inductively.

Definition 2.6. An object A of C is said to be of finite right X –dimension, if there is a right X –resolution

X : 0 ⟶X_{n} ⟶^{x_{n}} X_{n−1} ⟶⋯ ⟶X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{0} ⟶^{x_{0}} A ⟶0.

of length n.

In this case we write rt. dim_{X} A ≤ n. The least such integer n is called the right X –dimension of A and is denoted by rt. dim_{X} A = n. Otherwise, A is said to have infinite right X –dimension and we write rt. dim_{X} A = ∞.

Definition 2.7. The category C is said to be right X –regular, if rt. dim_{X} A < ∞ for any object A of C.

If there is some fixed n ∈ ℕ ∪ {0} with rt. dim_{X} A ≤ n for all objects A of C, then C is said to have finite right X –dimension. The least integer n ≥ 0 with this property is said to be the right X –dimension of the category C. In this case we write rt. dim_{X} C = n.

The next Lemma is the relative to X version of “Schanue’ls Lemma”:

Lemma 2.8. Let A and B two isomorphic objects of C of finite right X – dimension.

If

0 ⟶X_{n} ⟶^{x_{n}} X_{n−1} ⟶⋯ ⟶X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{1} ⟶^{x_{1}} X_{0} ⟶^{x_{0}} A ⟶0

and

0 ⟶Y_{n} ⟶^{y_{n}} Y_{n−1} ⟶⋯ ⟶Y_{i} ⟶^{y_{i}} Y_{i−1} ⟶⋯ ⟶Y_{1} ⟶^{y_{1}} Y_{0} ⟶^{y_{0}} B ⟶0

are two right X -resolutions, then

X_{0} ⊕ Y_{1} ⊕ X_{2} ⊕ ⋯ ≅ Y_{0} ⊕ X_{1} ⊕ Y_{2} ⊕ · · · .

Proof. By Lemma 2.5, x_{i} = g_{i} ∘ h_{i}, i ∈ ℕ, (resp. y_{i} = ℓ_{i} ∘ m_{i}, i ∈ ℕ) where g_{i} : K_{i} ⟶X_{i−1} (resp. ℓ_{i} : L_{i} ⟶Y_{i−1}) is the kernel of x_{i−1} (resp. of y_{i−1}) and h_{i} : X_{i} ⟶K_{i} (resp. m_{i} : Y_{i} ⟶L_{i}) is a right X –approximation of K_{i} (resp. of L_{i}). Let σ_{0} denote the isomorphism from A to B.

Since x_{0} is a right X –approximation of A and ℓ_{1} is the kernel of y_{0}, there is the commutative diagram:

K_{1} ⟶^{g_{1}} X_{0} ⟶^{x_{0}} A
s_{1}↓  ^{t_{0}}↓  ^{σ_{0}}↓
L_{1} ⟶^{ℓ_{1}} Y_{0} ⟶^{y_{0}} B

One checks easily that (t_{0}, ℓ_{1}) : X_{0} ⊕ L_{1} ⟶ Y_{0} is an X –epic with kernel the

morphism ( \begin{matrix}g_{1} \\ −s_{1} \end{matrix} ) : K_{1} ⟶ X_{0} ⊕ L_{1}. Since Y_{0} is an object of X, there is a

morphism r : Y_{0} ⟶X_{0} ⊕ L_{1}, satisfying (t_{0}, ℓ_{1}) ∘ r = id_{Y_{0}}.

Hence, Y_{0} ⊕ K_{1} ≅^{σ_{1}} X_{0} ⊕ L_{1}.

The morphism ( \begin{matrix}id_{Y_{0}} & 0 \\ 0 & h_{1} \end{matrix} ) : Y_{0} ⊕ X_{1} ⟶ Y_{0} ⊕ K_{1} (resp. ( \begin{matrix}id_{X_{0}} & 0 \\ 0 & m_{1} \end{matrix} ) :

X_{0} ⊕Y_{1} ⟶X_{0} ⊕L_{1} ) is a right X –approximation of Y_{0} ⊕K_{1} (resp.of X_{0} ⊕L_{1})

with kernel ( \begin{matrix}0 \\ g_{2} \end{matrix} ) : K_{2} ⟶Y_{0} ⊕ K_{1} (resp. ( \begin{matrix}0 \\ ℓ_{2} \end{matrix} ) : L_{2} ⟶X_{0} ⊕ Y_{1} ).

We consider the right X –resolutions:

0 ⟶X_{n} ⟶⋯ ⟶X_{2} \underrightarrow{( \begin{matrix}0 \\ x_{2} \end{matrix} )} Y_{0} ⊕ X_{1} \underrightarrow{( \begin{matrix}id_{Y_{0}} & 0 \\ 0 & h_{1} \end{matrix} )} Y_{0} ⊕ K_{1} ⟶0

and

0 ⟶Y_{n} ⟶⋯ ⟶Y_{2} \underrightarrow{( \begin{matrix}0 \\ y_{2} \end{matrix} )} X_{0} ⊕ Y_{1} \underrightarrow{( \begin{matrix}id_{X_{0}} & 0 \\ 0 & m_{1} \end{matrix} )} X_{0} ⊕ L_{1} ⟶0

and we conclude, as before, that X_{0} ⊕ Y_{1} ⊕ K_{2} ≅^{σ_{2}} Y_{0} ⊕ X_{1} ⊕ L_{2}.

Hence, the Lemma follows by induction.

Lemma 2.9. Let

\begin{matrix}K & ⟶^{k} & D & ⟶^{d} & A \\ ^{η}↓ & & ^{δ}↓ & & ↓^{α} \\ L & ⟶^{ℓ} & F & ⟶^{f} & B \end{matrix} (2.1)

be a commutative diagram where d and f are X –epics with kernels k and ℓ respectively. If α and η are X –epics, then also δ is an X –epic.

Proof. For any object X of X, the diagram (2.1) yields a commutative diagram in AB with exact rows:

0 ⟶ C(X, K) ⟶^{k^{X}} C(X, D) ⟶^{d^{X}} C(X, A) ⟶ 0
η^{X} ↓  δ^{X} ↓  α^{X} ↓
0 ⟶ C(X, L) ⟶^{ℓ^{X}} C(X, F) ⟶^{f^{X}} C(X, B) ⟶ 0

Since η^{X} and α^{X} are epimorphisms, δ^{X} is also an epimorphism. Hence δ is an X –epic.

The previous Lemma allows us to complete the diagram (2.1) with the kernels μ, ϵ and γ of the morphisms η, δ and α respectively. In this way we obtain the fully commutative diagram:

\begin{matrix}M & ⟶^{m} & E & ⟶^{e} & C \\ ^{μ}↓ & & ^{ϵ}↓ & & ^{γ}↓ \\ K & ⟶^{k} & D & ⟶^{d} & A \\ ^{η}↓ & & ^{δ}↓ & & ↓^{α} \\ L & ⟶^{ℓ} & F & ⟶^{f} & B \end{matrix} (2.2)

where the morphisms m and e are obtained by the kernel property. We leave for the reader the proof of the following:

Lemma 2.10. In the diagram (2.2) the morphism m is the kernel of e and the morphism e is an X –epic.

Now, we are able to show the relative to X version of the “Horseshoe Lemma”.

Lemma 2.11. Given a sequence A ⟶^{ι_{−1}} B ⟶^{π_{−1}} C, where π_{−1} is an X –epic and ι_{−1} is the kernel of π_{−1}, there is a commutative diagram

\begin{matrix}⋮ & & ⋮ & & ⋮ \\ ↓ & & ↓ & & ↓ \\ X_{1} & ⟶^{ι_{1}} & X_{1} ⊕ Y_{1} & ⟶^{π_{1}} & Y_{1} \\ x_{1}↓ & & z_{1}↓ & & y_{1}↓ \\ X_{0} & ⟶^{ι_{0}} & X_{0} ⊕ Y_{0} & ⟶^{π_{0}} & Y_{0} \\ x_{0}↓ & & z_{0}↓ & & y_{0}↓ \\ A & ⟶^{ι_{−1}} & B & ⟶^{π_{−1}} & C \\ ↓ & & ↓ & & ↓ \\ 0 & & 0 & & 0 \end{matrix} (2.3)

with ι_{j} (resp. π_{j}) j = 0, 1, 2, … the canonical injection (resp. canonical projection) and where the columns are right X –resolutions of A, B and C.

Proof. Let x_{0} : X_{0} ⟶ A (resp. y_{0} : Y_{0} ⟶ C) be a right X –approximation of A (resp. C) with kernel g_{1} : K_{1} ⟶ X_{0} (resp. ℓ_{1} : L_{1} ⟶ Y_{0}). Since π_{−1} is an X –epic, there is a morphism s_{0} : Y_{0} ⟶ B with π_{−1} ∘ s_{0} = y_{0}. Let z_{0} : X_{0} ⊕ Y_{0} ⟶ B denote the morphism (ι_{−1} ∘ x_{0}, s_{0}). The morphism z_{0} is a right X –approximation, since it is an X –epimorphism by Lemma 2.9 and since X_{0} ⊕ Y_{0} is an object of X. Let h_{1} : H_{1} ⟶X_{0} ⊕ Y_{0} be the kernel of z_{0}.

The fully commutative diagram below is obtained in the same way as the diagram (2.2):

K_{1} ⟶^{m_{1}} H_{1}  ⟶^{n_{1}} L_{1}
g_{1}↓  ^{h_{1}}↓  ^{ℓ_{1}}↓
X_{0} ⟶^{ι_{0}} X_{0} ⊕ Y_{0} ⟶^{π_{0}} Y_{0}
x_{0}↓  ^{z_{0}}↓  ^{y_{0}}↓
A ⟶^{ι_{−1}} B ⟶^{π_{−1}} C

Hence, by Lemma 2.10, the morphism n_{1} is an X –epic with kernel m_{1}.

We consider the sequence K_{1} ⟶^{m_{1}} H_{1} ⟶^{n_{1}} L_{1} and continue inductively, repeating the above procedure. This leads to the construction of the promised right X –resolutions, according to Lemma 2.5.

We close this section with some observations concerning relative Ext–functors. The introduction of these functors will be justified in Section 4.

Let f : A ⟶B be a morphism of C and

\begin{matrix}X : ⋯ ⟶X_{i+1} ⟶^{x_{i+1}} X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{0} ⟶^{x_{0}} A ⟶0, \\ Y : ⋯ ⟶Y_{i+1} ⟶^{y_{i+1}} Y_{i} ⟶^{y_{i}} Y_{i−1} ⟶⋯ ⟶Y_{0} ⟶^{y_{0}} B ⟶0 \end{matrix}

be right X –resolutions of A and B respectively.

For any object A of C,we denote by X_{d} the deleted right X –resolution:

⋯ ⟶X_{i+1} ⟶^{x_{i+1}} X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{0} ⟶0.

One shows as in the classical “Comparison Theorem”, that f induces a chain transformation from X to Y and that any two such chain transformations are chain homotopic.

Using deleted right X –resolutions, one defines for any object D of C the contravariant functors:

Ext_{X}^{n} (−, D) : C^{op} ⟶AB, ∀n ≥ 0

as the n–th right derived functors of C(−, D). We call these functors the X –relative Ext–functors.

Obviously, if rt. dim_{X} A ≤ n, then Ext_{X}^{j} (A, D) = 0, ∀j ≥ n + 1 and any object D of C.

By Lemma 2.11, we know that given a sequence A ⟶^{ι_{−1}} B ⟶^{π_{−1}} C, with π_{−1} an X –epic and ι_{−1} the kernel of π_{−1}, there are deleted right X –resolutions X_{d}, Z_{d} and Y_{d} of A, B and C respectively, such that there is an exact sequence of complexes of abelian groups:

0 ⟶C(Y_{d}, D) ⟶^{π_{D}} C(Z_{d}, D) ⟶^{ι_{D}} C(X_{d}, D) ⟶0

for any object D of C. Hence, there is a long–exact sequence of X –relative Ext–functors in AB:

0 ⟶Ext_{X}^{0} (C, D) ⟶Ext_{X}^{0} (B, D) ⟶Ext_{X}^{0} (A, D) ⟶^{∂} Ext_{X}^{1} (C, D) ⟶⋯

⋯ ⟶Ext_{X}^{i} (C, D) ⟶Ext_{X}^{i} (B, D) ⟶Ext_{X}^{i} (A, D) ⟶^{∂} Ext_{X}^{i+1}(C, D) ⟶⋯

(2.4)

3 Stable Grothendieck Groups

Let C be a category and X a subcategory contravariantly finite in C such that any X –epic has a kernel. In the present section, we shall introduce and study group structures related to the pair (C, X) and to the stable category C/J_{X}, cf. [5]. These group structures are similar to the classical Grothendieck groups studied in K–theory.

Let Iso(C) be a set of representatives of the isomorphism classes of the objects of C and let F(C) be the free abelian group with base Iso(C). For any object A of C the corresponding element of Iso(C) will be denoted by (A). We also consider the subset Iso(X) of Iso(C) having as elements the representatives of the isomorphism classes of the objects of X. Let F(X) be the free abelian group with base Iso(X). Obviously, F(X) ⊆ F(C).

Let F_{0}(C, X) be the subgroup of F(C) generated by the set of formal sums

Γ(C, X) = {(A) − (B) + (C) : where there is an X –epic B ⟶^{g}C

with kernel A ⟶^{f} B}

We denote by K_{0}(C, X) the quotient group F(C)/F_{0}(C, X) and by [A] the elements (A) + F_{0}(C, X) of K_{0}(C, X).

Definition 3.1. The group K_{0}(C, X) is said to be the Grothendieck group of the pair (C, X).

Remark 3.2. Let A be an object of C with rt. dim_{X} A ≤ n and let

0 ⟶X_{n} ⟶^{x_{n}} X_{n−1} ⟶⋯ ⟶X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{1} ⟶^{x_{1}} X_{0} ⟶^{x_{0}} A ⟶0

be a right X –resolution of A, then [A] = ∑_{i=0}^{n}(−1)^{i}[X_{i}]. So, if C is right X –regular, then the set {[X] : (X) ∈ Iso(X)} forms a set of generators of K_{0}(C, X).

Remark 3.3. Given two objects A and B of C, the canonical projection A ⊕ B ⟶B is an X –epic with kernel A. Hence, [A ⊕ B] = [A] + [B].

Let F_{0}(X , X) be the subgroup of F(C) generated by the set of formal sums

Γ(X , X) = {(A) − (B) + (C) : where there is an X –epic B ⟶^{g}C

with kernel A ⟶^{f} B and with B, C objects of X }

Lemma 3.4. Let g : B ⟶C be an X –epic with kernel f : A ⟶B. If B and C are objects of X, then B ≅ A ⊕ C and A is an object of X.

Proof. Since g is an X –epic, there is a morphism h : C ⟶B with g ∘h = id_{C}. Since f is a kernel of g, there is a morphism k : B ⟶ A satisfying f ∘ k =

id_{B} − h ∘ g. Now A ⇆_{f}^{k} B ⇄_{h}^{g} C is a byproduct diagram and B ≅ A ⊕ C.

Because X is closed under direct summands A is an object of X.

Corollary 3.5. The group F_{0}(X , X) is a subgroup of F(X) and of F_{0}(C, X).

Proof. By 3.4, any element (A)−(B)+(C) of Γ(X , X) has the property that (A) is also in Iso(X). Hence, Γ(X , X) ⊆ F(X).

For any two objects (A), (B) of Iso(C) it holds (A) − (A ⊕ B) + (B) ∈ F_{0}(C, X). So, again by 3.4, Γ(X , X) ⊆ F_{0}(C, X).

Definition 3.6. The group K_{0}(X , ⊕) := F(X)/F_{0}(X , X) is said to be the Grothendieck group of X.

We shall denote the elements (X) + F_{0}(X , X) of K_{0}(X , ⊕) by ∥X∥.

Remark 3.7. For any (A), (B), (C) ∈ Iso(X), the formal sum (A) − (B) + (C) of F(X) belongs to Γ(X , X) if and only if B ≃ A ⊕ C, because of Lemma 3.4. Hence, ∥A ⊕ B∥ = ∥A∥ + ∥B∥.

Let ι_{0} : F(X) ⟶ F(C), ι_{1} : F(X , X) ⟶ F(X) and ι_{2} : F(X , X) ⟶ F_{0}(C, X) be the canonical inclusions.

The short exact sequences

0 ⟶F_{0}(C, X) ⟶^{s}F(C) ⟶^{p}K_{0}(C, X) ⟶0 (3.1)

and

0 ⟶F_{0}(X , X) ⟶^{ι_{1}} F(X) ⟶^{w} K_{0}(X , ⊕) ⟶0 (3.2)

yield in AB the commutative diagram

\begin{aligned}0 & ⟶ & F_{0}(X , X) & ⟶^{ι_{1}} & F(X) & ⟶^{w} & K_{0}(X , ⊕) & ⟶ & 0 \\ & & ι_{2}↓ & & ι_{0}↓ & & c_{X} ↓ \\ 0 & ⟶ & F_{0}(C, X) & ⟶^{s} & F(C) & ⟶^{p} & K_{0}(C, X) & ⟶ & 0 \end{aligned} (3.3)

where the homomorphism c_{X} is canonically induced by ι_{0}.

Definition 3.8. The group homomorphism c_{X} : K_{0}(X , ⊕) ⟶ K_{0}(C, X) is said to be the Cartan map of the pair (C, X).

In the following we are interested in Coker c_{X}. As we shall see Coker c_{X} appears as “Grothendieck group” of the left triangulated structure of the stable category C/J_{X}. We start by introducing “Grothendieck groups” for any left triangulated category. Firstly, we recall some facts about left triangulated categories developed in [5].

Let Ω be an endofunctor of a category D and let LTR(D, Ω) be the category with objects the diagrams of the form ΩC ⟶^{f} A ⟶^{g} B ⟶^{h} C and with

set of morphisms from ΩC ⟶^{f} A ⟶^{g} B ⟶^{h} C to ΩC′ ⟶^{f′} A′ ⟶^{g′} B′ ⟶^{h′} C′, the

triples (α, β, γ) of morphisms of D from (A, B, C) to (A′, B′, C′), which make the diagram below commutative:

ΩC ⟶^{f} A ⟶^{g} B ⟶^{h} C
Ωγ↓  ^{α}↓  ^{β}↓  ^{γ}↓
ΩC′ ⟶^{f′} A′ ⟶^{g′} B′ ⟶^{h′} C′

The objects of LTR(D, Ω) are said to be the left triangles of (D, Ω) .

Whenever it is convenient the objects ΩC ⟶^{f} A ⟶^{g}B ⟶^{h} C of LTR(D, Ω) will be denoted by (A, B, C, f, g, h).

Definition 3.9. A full subcategory Δ of LTR(D, Ω) is said to be a left triangulation of (D, Ω) if it is closed under isomorphisms and satisfies the following four axioms:

LT1i For any object A of D, the left triangle 0 ⟶^{0} A ⟶^{1_{A}} A ⟶^{0} 0 belongs to Δ.

LT1ii For any morphism x : B ⟶C of D, there is a left triangle in Δ of the form (A, B, C, f, g, x).

LT2 For any left triangle (A, B, C, f, g, h) in Δ, the left triangle (ΩC, A, B, −Ωh, f, g) is also in Δ.

LT3 For any two left triangles (A, B, C, f, g, h), (A′, B′, C′, f′, g′, h′) in Δ and any two morphisms β : B ⟶ B′ and γ : C ⟶ C′ of D with γ ∘ h = h′ ∘ β, there is a morphism α : A ⟶A′ of D such that the triple (α, β, γ) is a morphism from the first triangle to the second.

LT4 For any two left triangles (A, B, C, f, g, h) and (E, C, D, i, ℓ, k) in Δ, there is a third left triangle (F, B, D, j, m, k ∘ h) in Δ and two morphisms α : A ⟶F and β : F ⟶E of D, such that the diagram below is fully commutative and where the second column from the left, i.e. ΩE ⟶^{f∘Ωℓ} A ⟶^{α} F ⟶^{β} E, is a left triangle in Δ.

ΩE
f∘Ωℓ↓
ΩC ⟶^{f}  A  ⟶^{g} B ⟶^{h} C
Ωk↓  α↓  ∥  k↓
ΩD ⟶^{j}  F  ⟶^{m} B ⟶^{k∘h} D
∥  β↓  h↓  ∥
ΩD ⟶^{i}  E  ⟶^{ℓ} C ⟶^{k} D

Definition 3.10. The triple (D, Ω, Δ) is said to be a left triangulated category.

Definition 3.11. A covariant functor G : D ⟶ D′ is said to be a triangle equivalence from the left triangulated category (D, Ω, Δ) to the left triangulated category (D′, Ω′, Δ′) , if it is an equivalence and there is a natural isomorphism ϕ : G ∘ Ω ⟶Ω′ ∘ G, such that if (A, B, C, f, g, h) is a left triangle of Δ, then the left triangle (GA, GB, GC, Gf ∘ ϕ_{C}^{−1}, Gg, Gh) is in Δ′.

We also make the following definition for later use:

Definition 3.12. Given a left triangulated category (D, Ω, Δ), the functor Ω is said to be locally nilpotent if for any object A of D, there is an n(A) ∈ ℕ, such that Ω^{n(A)}A = 0.

Now, we are ready to construct a “Grothendieck group” for any left triangulated category. A notion which was introduced already in by A. Grothendieck for triangulated categories, see [7] and [8].

Let (D, Ω, Δ) be a left triangulated category and let Iso(D) = {(A) : A object of D} be a set of representatives of the isomorphism classes of the objects of D.

Let F(D) be the free abelian group with base Iso(D) and F_{0}(D, Ω, Δ) be the subgroup of F(D) generated by the set of formal sums

Γ(D, Ω, Δ) = {(A) − (B) + (C) : where there is a left triangle (A, B, C, f, g, h) in Δ}

We denote the quotient group F(D)/F_{0}(D, Ω, Δ) by K_{0}(D, Ω, Δ) and its elements (A) + F_{0}(D, Ω, Δ) by [A].

Definition 3.13. The group K_{0}(D, Ω, Δ) is said to be the Grothendieck group of the left triangulated category (D, Ω, Δ).

Remark 3.14. Any triangle equivalence G : D ⟶D′ between two left triangulated categories (D, Ω, Δ) and (D′, Ω′, Δ′), induces a group isomorphism between F_{0}(D, Ω, Δ) and F_{0}(D′, Ω′, Δ′). Hence, G induces an isomorphism between the corresponding Grothendieck groups.

Lemma 3.15. (i) Let (A, B, C, f, g, h) be a left triangle in Δ, then g (resp. f) is a weak kernel of h (resp. g). Moreover, if g = 0 (resp. f = 0), then h (resp. g) is a monomorphism.

(ii) For any two objects A and B of D, the left triangle (A, A⊕B, B, 0, ι_{A}, π_{B}) belongs to Δ, where ι_{A} is the canonical injection and π_{B} is the canonical projection .

Proof. (i) By LT1i, the left triangle (C, C, 0, 0, id_{C}, 0) belongs to Δ. Hence by LT2, the left triangle (0, C, C, 0, 0, id_{C}) belongs also to Δ. We consider the commutative diagram:

ΩC ⟶^{f} A ⟶^{g} B ⟶^{h} C
id_{ΩC} ↓  ^{ℓ}↓  ^{h}↓  ^{id_{C}} ↓
ΩC ⟶^{0} 0 ⟶^{0} C ⟶^{id_{C}} C

where ℓ exists by LT3. Hence, h ∘ g = 0.

Let k : X ⟶B be a morphism with h ∘ k = 0. We consider the triangle (X, X, 0, 0, id_{X}, 0) and the commutative diagram:

0 ⟶^{0} X ⟶^{id_{X}} X ⟶^{0} 0
0↓  ^{m}↓  ^{k}↓  ^{0}↓
ΩC ⟶^{f} A ⟶^{g} B ⟶^{h} C

where m exists by LT3. Hence, g ∘ m = k. Obviously, if g = 0 then k = 0. So, in this case h is a monomorphism.

The remaining part of (i) follows applying the same arguments as above on the left triangle (ΩC, A, B, −Ωh, f, g), which belongs to Δ because of LT2.

(ii) By LT1ii, there is a left triangle ΩB ⟶^{f} K ⟶^{g}A⊕B ⟶^{π_{B}} B in Δ. Since Ωπ_{B} ∘ f = 0, the morphism f equals zero and the first part of our Lemma implies that g is a monomorphism. Hence, (K, g) is a kernel of π_{B} and so there is an isomorphism σ : K ⟶A with ι_{A} ∘ σ = g. Since, (σ, id_{A⊕B}, id_{A}) is an isomorphism from (K, A ⊕ B, B, 0, g, π_{B}) to (A, A ⊕ B, B, 0, ι_{A}, π_{B}), this last left triangle belongs also to Δ.

Corollary 3.16. (i) If [A] is an element of K_{0}(D, Ω, Δ), then

[A] = (−1)^{n}[Ω^{n}A], ∀n ∈ ℕ ∪ {0}.

(ii) If [A] and [B] are elements of K_{0}(D, Ω, Δ), then [A] + [B] = [A ⊕ B].

(iii) If Ω is locally nilpotent, then K_{0}(D, Ω, Δ) is the trivial group.

Proof. (i) Given an object A of D, the left triangle ΩA ⟶^{−id_{ΩA}} ΩA ⟶^{0} 0 ⟶^{0} A belongs to Δ by LT1i and LT2 . Hence, [A] = −[ΩA] and so [A] = (−1)^{n}[Ω^{n}A].

(ii) By Lemma 3.15(ii) the left triangle (A, A ⊕ B, B, 0, ι_{A}, π_{B}) belongs to Δ for any two objects A and B of D. Hence, [A] + [B] = [A ⊕ B].

(iii) Obvious.

Now, we are ready for a closer study of the Grothendieck groups of stable categories.

Let C be an additive category and a subcategory X contravariantly finite in C , such that any X –epic has a kernel. We recall, that the stable category C_{X} = C/J_{X} has the same objects as C and that for any two objects A and B of C_{X} the set of morphisms C_{X} (A, B) equals the factorgroup C(A, B)/X (A, B), where X (A, B) is the subgroup of C(A, B) having as elements the morphisms of C which factor over some object of X. The composition in C_{X} is induced by the corresponding composition in C.

Let F be the projection functor C ⟶C_{X}. The image FA of any object A of C will be denoted by \underline{A} and the image Ff of any morphism f of C will be denoted by \underline{f}.

In [5], we constructed a functor Ω_{X} : C_{X} ⟶C_{X} and a left triangulation Δ_{X} of C_{X}, such that the triple (C_{X} , Ω_{X} , Δ_{X}) forms a left triangulated category. We recall the definition of Ω_{X} on the objects of C_{X} for later use. We refer for further details to [5]. For any object A of C we consider a right X – approximation g : X ⟶A and its kernel f : K ⟶X. We define Ω_{X} \underline{A} := \underline{K}.

Definition 3.17. The factor group

K_{0}(C_{X} , Ω_{X} , Δ_{X}) := F(C_{X} , Ω_{X} , Δ_{X})/F_{0}(C_{X} , Ω_{X} , Δ_{X})

is said to be the Grothendieck group of the stable category C_{X}.

Let π_{0} : F(C) ⟶ F(C_{X}) be the epimorphism defined by (A) ↦ (\underline{A}). Its kernel equals the canonical inclusion ι_{0} : F(X) ⟶F(C).

Lemma 3.18. The restriction of π_{0} to F_{0}(C, X) defines a group epimorphism π_{1} : F_{0}(C, X) ⟶F_{0}(C_{X} , Ω_{X} , Δ_{X})

Proof. By [5], Prop. 2.10, any X –epic g : B ⟶ C with kernel f : A ⟶ B induces a left triangle (\underline{A}, \underline{B}, \underline{C}, \underline{f}, \underline{g}, \underline{h}) in Δ_{X}. Conversely, the same Proposition implies that given a left triangle in Δ_{X}, there is an X –epimorphism ℓ : E ⟶ F with kernel m : D ⟶ E, such that the induced left triangle (\underline{D}, \underline{E}, \underline{F} , \underline{k}, \underline{ℓ}, \underline{m}) is isomorphic to the given one. Hence, the restriction of π_{0} on Γ(C, X) defines a surjective map Γ(C, X) ⟶ Γ(C_{X} , Ω_{X} , Δ_{X}), (A) − (B) + (C) ↦ (\underline{A}) − (\underline{B}) + (\underline{C}) and so π_{0} restricted on F_{0}(C, X) defines an epimorphism π_{1} : F_{0}(C, X) ⟶F_{0}(C_{X} , Ω_{X} , Δ_{X}).

The epimorphisms π_{0}, π_{1} and the short exact sequences

0 ⟶F_{0}(C_{X} , Ω_{X} , Δ_{X}) ⟶^{t}F(C_{X} , Ω_{X} , Δ_{X}) ⟶^{q}K_{0}(C_{X} , Ω_{X} , Δ_{X}) ⟶0. (3.4)

and (3.1) yield in AB the following commutative diagram:

\begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ & & & & F_{0}(C, X) & ⟶^{π_{1}} & F_{0}(C_{X} , Ω_{X} , Δ_{X}) & ⟶ & 0 \\ & & & & s↓ & & t↓ \\ 0 & ⟶ & F(X) & ⟶^{ι_{0}} & F(C) & ⟶^{π_{0}} & F(C_{X}) & ⟶ & 0 \\ & & r↓ & & p↓ & & q↓ \\ 0 & ⟶ & Ker π & ⟶^{ι} & K_{0}(C, X) & ⟶^{π} & K_{0}(C_{X} , Ω_{X} , Δ_{X}) & ⟶ & 0 \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0 \end{aligned} (3.5)

where π and r are canonically induced by π_{0} and p respectively. Since π_{0} is an epimorphism, π is also an epimorphism.

With the assumptions made at the beginning of our paper we have:

Theorem 3.19. The cokernel of the Cartan map c_{X} : K_{0}(X , ⊕) ⟶K_{0}(C, X) is isomorphic to the Grothendieck group K_{0}(C_{X} , Ω_{X} , Δ_{X}).

Proof. Because of the diagram (3.3) we obtain Im c_{X} = Im(p ∘ ι_{0}). By the diagram (3.5), Im(p ∘ ι_{0}) = Im(ι ∘ r). Since π_{1} is an epimorphism, the application of the “Snake Lemma” on the two bottom rows of diagram (3.5) implies that r is an epimorphism. Hence, Im c_{X} = Ker π.

The next Theorem gives a sufficient condition for the Cartan map c_{X} to be an isomorphism. With the assumptions made at the beginning of our paper we have:

Theorem 3.20. Let C be a category and X be a subcategory contravariantly finite in C. If C is right X –regular, then the Cartan map c_{X} : K_{0}(X , ⊕) ⟶ K_{0}(C, X) is an isomorphism.

Proof. Since C is right X –regular, the functor Ω_{X} is locally nilpotent. By Corollary 3.16(iii), the Grothendieck group K_{0}(C_{X} , Ω_{X} , Δ_{X}) is the trivial group and so c_{X} is an epimorphism.

Let

0 ⟶X_{n} ⟶^{x_{n}} X_{n−1} ⟶⋯ ⟶X_{i} ⟶^{x_{i}} X_{i−1} ⟶⋯ ⟶X_{0} ⟶^{x_{0}} A ⟶0.

be a right X –resolution of an object A of C. We claim that the assignment (A) ↦ ∑_{i=0}^{n}(−1)^{i}∥X_{i}∥ determines a well–defined homomorphism δ_{X} : F(C) ⟶K_{0}(X , ⊕). Indeed, given another right X –resolution

0 ⟶Y_{n} ⟶^{y_{n}} Y_{n−1} ⟶⋯ ⟶Y_{i} ⟶^{y_{i}} Y_{i−1} ⟶⋯ ⟶Y_{1} ⟶^{y_{1}} Y_{0} ⟶^{y_{0}} A ⟶0

of A, we obtain by Lemma 2.8 and Remark 3.7 that

∑_{i even} ∥X_{i}∥ + ∑_{i odd} ∥Y_{i}∥ = ∑_{i even} ∥Y_{i}∥ + ∑_{i odd} ∥X_{i}∥, 0 ≤ i ≤ n.

So, ∑_{i=0}^{n}(−1)^{i}∥X_{i}∥ = ∑_{i=0}^{n}(−1)^{i}∥Y_{i}∥ and δ_{X} is well–defined.

Now, Lemma 2.11 implies that δ_{X} ((A) − (B) + (C)) = 0 for any element of Γ(C, X). Hence, there is a homomorphism d_{X} : K_{0}(C, X) ⟶ K_{0}(X , ⊕) induced by δ_{X}. For any object X of X we have d_{X} ([X]) = ∥X∥ , since 0 ⟶X ⟶^{id_{X}} X ⟶0 is trivially a right X –resolution of X.

We claim that the composition d_{X} ∘ c_{X} is the identity homomorphism of K_{0}(X , ⊕). The set Γ(∥X ∥) = {∥X∥ : (X) ∈ Iso(X)} forms a set of generators of K_{0}(X , ⊕). For any element ∥X∥ of Γ(∥X ∥) we have d_{X} ∘ c_{X} (∥X∥) = d_{X} ([X]) = ∥X∥. So d_{X} ∘ c_{X} = id_{K_{0}(X ,⊕)}. This implies that c_{X} is a monomorphism and completes the proof.

4 Cartan Matrices and Euler Characteristic

Let k be a field, C be a finite dimensional k–additive Krull–Schmidt category, cf. [9] Section 2.2. and {X_{1}, …, X_{m}} be a finite family of non–isomorphic indecomposable objects of C. Let X denote the direct sum X_{1} ⊕ ⋯ ⊕ X_{m}. The full subcategory X = add X is contravariantly finite in C, cf. [1] Prop.1.9.

In the present section, we shall deal with subcategories of the above form with the property that any X –epic has a kernel and we shall define a quadratic matrix behaving analogously to the classical Cartan matrix. If this matrix is ℤ–invertible, we shall introduce a bilinear form K_{0}(C, X) × K_{0}(C, X) ⟶ℤ and shall give an interpretation of the corresponding quadratic form using relative homological terms.

Definition 4.1. For any object A of C, the m–tuple

\underline{dim}_{X} A := (dim_{k} C(X_{1}, A), dim_{k} C(X_{2}, A), …, dim_{k} C(X_{m}, A))

is said to be the X –dimension vector of A.

Lemma 4.2. The assignment [A] ↦ \underline{dim}_{X} A determines a group homomorphism \underline{dim}_{X} : K_{0}(C, X) ⟶ℤ^{m}.

Proof. We consider the group homomorphism ϕ : F(C) ⟶ ℤ^{m} defined by (A) ↦ \underline{dim}_{X} A, ∀(A) ∈ Iso(C). For any generator (A) − (B) + (C) of Γ(C, X) we have ϕ((A) − (B) + (C)) = 0, since there is an X –epic B ⟶^{g}C with kernel

A ⟶^{f} B, yielding ∀i = 1, …, m, the short exact sequence 0 ⟶C(X_{i}, A) ⟶^{f^{X_{i}}}

C(X_{i}, B) ⟶^{g^{X_{i}}} C(X_{i}, C) ⟶0 in AB.

Hence, there is a well defined group homomorphism \underline{dim}_{X} : K_{0}(C, X) ⟶ ℤ^{m} induced by ϕ.

For an object A of C we have [A] = 0 if and only if \underline{dim}_{X} A = 0. Indeed, if \underline{dim}_{X} A = 0, then the zero morphism 0 ⟶ A is a right X –approximation of A, implying that (A) ∈ F_{0}(C, X).

Definition 4.3. The m × m matrix C_{X} = (c_{ij}) with c_{ij} = dim_{k} C(X_{i}, X_{j}) is said to be the Cartan matrix of the pair (C, X).

Suppose that the Cartan matrix C_{X} is ℤ–invertible, then C_{X}^{−1} defines a bilinear form:

⟨−, −⟩_{X} : K_{0}(C, X) × K_{0}(C, X) ⟶ℤ^{m} by ⟨[A], [B]⟩_{X} = \underline{dim}_{X} AC_{X}^{−t}(\underline{dim}_{X} B)^{t}

The corresponding quadratic form q_{X} ([A]) = ⟨[A], [A]⟩_{X} is said to be the X –Euler characteristic of the pair (C, X).

Our next Proposition is the relative homological version of the corresponding one in [9], Section 2.4.

Proposition 4.4. Let A and B be two objects of C. If rt. dim_{X} A < ∞, then

⟨[A], [B]⟩_{X} = ∑_{j≥0}(−1)^{j} dim_{k} Ext_{X}^{j} (A, B) (4.1)

Proof. We proceed by induction on n = rt. dim_{X} A.

For d = 0, there is a X -resolution 0 ⟶ Q ⟶ A with Q an object of X. Hence, [A] = [Q] and \underline{dim}_{X} A = \underline{dim}_{X} Q. Moreover, Ext_{X}^{j} (A, B) = 0, ∀j ≥ 1 and Ext_{X}^{0} (A, B) = C(Q, B). Hence, the right hand side of (4.1) equals dim_{k} C(Q, B). Suppose that Q = 0, then obviously both sides of (4.1) equal zero. Suppose that Q ≠ 0, then without loss of generality we may assume that Q = X_{i} for some i = 1 … m. Since C_{X}^{−1}(\underline{dim}_{X} X_{i})^{t} = ϵ(i), where ϵ(i) denotes the column vector with 1 at the i–th position and zero anywhere else, the left hand side of (4.1) equals \underline{dim}_{X} Bϵ(i) = dim_{k} C(X_{i}, B). Hence, again both sides coincide.

Assume that our claim is true for all objects A, such that rt. dim_{X} A < d. Let A be an object of C with rt. dim_{X} A = d. There is a sequence A′ ⟶^{f} Q ⟶^{g} A, where Q is an object of X and g is a X –epic with kernel f. Hence,

[A] = [Q] − [A′] and \underline{dim}_{X} A = \underline{dim}_{X} Q − \underline{dim}_{X} A′.

We have:

⟨[A], [B]⟩_{X} =⟨[Q], [B]⟩_{X} − ⟨[A′], [B]⟩_{X} =

∑_{j≥0}(−1)^{j} dim_{k} Ext_{X}^{j} (Q, B) − ∑_{j≥0}(−1)^{j} dim_{k} Ext_{X}^{j} (A′, B). (4.2)

The sequence A′ ⟶^{f} Q ⟶^{g}A, yields the long exact sequence

⋯ ⟶Ext_{X}^{i} (A, B) ⟶Ext_{X}^{i} (Q, B) ⟶Ext_{X}^{i} (A′, B) ⟶^{∂} Ext_{X}^{i+1}(A, B) ⟶⋯

discussed at the end of Section 2. Since dim_{k} acts additively on this long exact sequence, we obtain for the last term of (4.2):

∑_{j≥0}(−1)^{j} dim_{k} Ext_{X}^{j} (Q, B) − ∑_{j≥0}(−1)^{j} dim_{k} Ext_{X}^{j} (A′, B) =

∑_{j≥0}(−1)^{j} dim_{k} Ext_{X}^{j} (A, B).

Let Γ denote the k-algebra End_{C}(X), mod Γ the category of finitely generated right Γ-modules and P_{Γ}^{f} the full subcategory of mod Γ with objects the projective finitely generated Γ-modules. Let G denote the functor C(X, −) : C ⟶mod Γ. It is well-known that G restricted on X induces an equivalence

σ : X ⟶^{∼} P_{Γ}^{f}.

The X -relative Ext-functors, which appear in formula (4.1) are related with the absolute Ext-functors of mod Γ as follows:

Proposition 4.5. For any object B of C and ∀i ≥ 0, the functors Ext_{X}^{i} (−, B) and Ext_{Γ}^{i}(G−, GB) are natural isomorphic .

Proof. We give a brief proof since all arguments are standard. Let

⋯ ⟶Y_{i+1} ⟶^{y_{i+1}} Y_{i} ⟶^{y_{i}} Y_{i−1} ⟶⋯ ⟶Y_{0} ⟶^{y_{0}} B ⟶0,

be a right X –resolution of B.

For any object X of X we obtain from the resolution above the diagram

... ⟶  C(X,Y_{1})  ⟶^{y_{1}^{X}}  C(X,Y_{0})  ⟶^{y_{0}^{X}}  C(X,B) ⟶ 0
↓  ↓  ^{ϕ_{X,B}} ↓
... ⟶ Hom_{Γ}(GX,GY_{1}) ⟶^{Gy_{1}^{GX}} Hom_{Γ}(GX,GY_{0}) ⟶^{Gy_{0}^{GX}} Hom_{Γ}(GX,GB) ⟶ 0

This is a fully commutative diagram with exact rows. Moreover, the vertical homomorphisms C(X, Y_{i}) ⟶Hom_{Γ}(GX, GY_{i}) are isomorphisms for all i ≥ 0, hence the morphism ϕ_{X,B} is also an isomorphism. We leave for the reader to verify that ϕ_{X,B} is natural on both arguments.

Let A and B be objects of C and let

⋯ ⟶Z_{i+1} ⟶^{z_{i+1}} Z_{i} ⟶^{z_{i}} Z_{i−1} ⟶⋯ ⟶Z_{0} ⟶^{z_{0}} A ⟶0, (4.3)

be a right X –resolution of A. By using this resolution we obtain the diagram

0 ⟶  C(Z_{0},B)  ⟶^{z_{1},_{B}}  C(Z_{1},B)  ⟶^{z_{2},_{B}}  C(Z_{2},B) ⟶ ...
ϕ_{Z_{0},B} ↓  ^{ϕ_{Z_{1},B}} ↓  ^{ϕ_{Z_{2},B}} ↓
0 ⟶ Hom_{Γ}(GZ_{0},GB) ⟶^{Gz_{1},_{GB}} Hom_{Γ}(GZ_{1},GB) ⟶^{Gz_{2},_{GB}} Hom_{Γ}(GZ_{2},GB) ⟶ ...

where the rows are complexes and the vertical maps are isomorphisms.

This shows that the i–th right derived functor Ext_{X}^{i} (−, B) of C(−, B) is isomorphic to the i–th right derived functor Ext_{Γ}^{i}(G−, GB) of Hom_{Γ}(G−, GB).

Remark 4.6. For a given object A of C, we describe explicitly the induced isomorphism ψ from Ext_{X}^{0} (A, B) = Ker z_{1},_{B} = {g : Z_{0} ⟶ B : g ∘ z_{1} = 0} to Ext_{Γ}^{0}(GA, GB) = Ker Gz_{1},_{GB} = Hom_{Γ}(GA, GB) for later use.

For any element g ∈ Ext_{X}^{0} (A, B) we have Gg∘ Gz_{1} = 0. Since Gz_{0},_{GB} is the cokernel of Gz_{1},_{GB}, there is a uniquely determined morphism m : GA ⟶GB satisfying m ∘ Gz_{0},_{GB} = Gg. The assignment g ↦ m defines the isomorphism ψ : Ext_{X}^{0} (A, B) ⟶Hom_{Γ}(GA, GB).

Our next Proposition relates the right X –dimension of C with the global dimension of Γ.

Proposition 4.7. If any morphism between the objects of X has a weak kernel in C, then

rt. dim_{X} C ≤ gl. dim Γ ≤ 2 + rt. dim_{X} C

Proof. Since, any right X –resolution of an object A of C, gives rise to a projective resolution of GA, the left–hand inequality is true.

For the right–hand inequality, it is enough to show that if GY_{1} ⟶^{f_{1}} GY_{0} ⟶^{f_{0}} M is a projective presentation of a Γ–module M, then Ker f_{1} is isomorphic to GL for some object L of C. Let g_{1} : Y_{1} ⟶ Y_{0} be a morphism of C with Gg_{1} = f_{1}. We consider the complex Y_{3} ⟶^{g_{3}} Y_{2} ⟶^{g_{2}} Y_{1} ⟶^{g_{1}} Y_{0}, where g_{i} = h_{i} ∘ ℓ_{i+1}, i = 1, 2 with h_{i} : L_{i} ⟶ Y_{i} a weak kernel of g_{i} and ℓ_{i+1} : Y_{i+1} ⟶ L_{i} a right X –approximation. The exactness of GY_{3} ⟶^{Gg_{3}} GY_{2} ⟶^{Gg_{2}} GY_{1} ⟶^{Gg_{1}} GY_{0} yields Ker Gg_{1} ≃ GY_{2}/ Im Gg_{3}. But Im Gg_{3} ≃ Im Gh_{2} ≃ Ker Gℓ_{2} and GY_{2}/ Ker Gℓ_{2} ≃ Im Gℓ_{2} ≃ GL_{1}.

Corollary 4.8. If any morphism between objects of X has a weak kernel in C and rt. dim_{X} C ≤ n, then the Cartan matrix C_{X} is ℤ–invertible

Proof. The Cartan matrix C_{X} coincides with the Cartan matrix of Γ, which is ℤ–invertible because gl. dim Γ ≤ n + 2.

Remark 4.9. Suppose that Λ is a finite dimensional k–algebra of finite representation type and let {X_{1}, X_{2}, …, X_{m}} be a complete set of nonisomorphic indecomposable Λ–modules. Then X = add(X_{1} ⊕ X_{2} ⊕ ⋯ ⊕ X_{m}) equals to mod Λ and rt. dim_{X} mod Λ = 0. In this case, the Lemma above specializes to the well–known fact that the Auslander-algebra of mod Λ has global dimension ≤ 2.

5 Indecomposable Objects Determined by their Dimension Vectors

In the present section we keep the same assumptions and notation as in Section 4. Our aim is to study when given an indecomposable object A of C, the Γ–module GA remains indecomposable. This enables us to develop criteria for when indecomposable objects of C are determined, up to isomorphism, by their X –dimension vectors linking this question with results developed in representation theory of finite dimensional–algebras.

Let Gen X be the full subcategory of C with objects those A of C for which there is an epimorphism Z′ ⟶ A with Z′ an object of X. Obviously, any right X -approximation of an object A of Gen X is an epimorphism.

Let D_{X} be the full subcategory of C formed by the objects A of Gen X for which there is a right X -approximation Z′′ ⟶^{f} A with Ker f also in Gen X.

We shall show that under certain assumptions, the indecomposable objects of C which are in D_{X} are determined, up to isomorphism by their X – dimension vectors.

Lemma 5.1. If any morphism between the objects of X has a cokernel in C, then the functor G restricted on D_{X} is a fully faithful functor.

Proof. Let A be an object of D_{X}. Because of the definition of D_{X}, there is a right X –presentation Z_{1} ⟶^{z_{1}} Z_{0} ⟶^{z_{0}} A ⟶0 of A, where z_{0} is an epimorphism and Ker z_{0} is an object of Gen X. By Lemma 2.5, z_{1} = f_{1} ∘ h_{0}, where f_{1} : Ker z_{0} ⟶ Z_{0} is the kernel of z_{0} and h_{0} : Z_{1} ⟶ Ker z_{0} is a right X – approximation. The hypothesis of our Lemma implies that z_{0} is the cokernel of f_{1}.

Given an object B of C we consider the relative cohomology group Ext_{X}^{0} (A, B). The assignment ℓ ↦ ℓ ∘ z_{0} determines a homomorphism ω : C(A, B) ⟶ Ext_{X}^{0} (A, B) which is monomorphism because z_{0} is an epimorphism. For any g ∈ Ext_{X}^{0} (A, B) we have g ∘ z_{1} = g ∘ f_{1} ∘ h_{0} = 0. This implies g ∘f_{1} = 0, since h_{0} is an epimorphism. Hence, there is a morphism ℓ : A ⟶B with ℓ ∘ z_{0} = g and so ω is an isomorphism.

Let ψ : Ext_{X}^{0} (A, B) ⟶ Hom_{Γ}(GA, GB) be the isomorphism described in Remark 4.6. We evaluate the composition ψ ∘ ω : C(A, B) ⟶Hom_{Γ}(GA, GB) on an element ℓ ∈ C(A, B). We have ψ ∘ ω(ℓ) = ψ(ℓ ∘ z_{0}) = m, where m : GA ⟶ GB is uniquely determined by the property m ∘ Gz_{0} = Gℓ ∘ Gz_{0}. Since Gz_{0} is an epimorphism, m = Gℓ. The proof is completed.

We recall that given a finite –dimensional k–algebra Γ and a finite–dimensional indecomposable Γ–module M, its dimension vector \underline{dim}M is defined to be the m-tuple

(dim_{k} Hom_{Γ}(P_{1}, M), dim_{k} Hom_{Γ}(P_{2}, M), …, dim_{k} Hom_{Γ}(P_{m}, M)),

where {P_{1}, P_{2}, …, P_{m}} is a complete set of non-isomorphic indecomposable projective Γ–modules of finite k–dimension.

The fact that G : D_{X} ⟶ mod Γ is fully faithful enable us to prove the next Theorem, which is based on the main result of [6].

Theorem 5.2. Suppose that any morphism between the objects of X has a cokernel in C. Let A and B be objects of D_{X} which are indecomposable as objects of C with \underline{dim}_{X} A = \underline{dim}_{X} B. If GA belongs to a postprojective component of the Auslander–Reiten quiver of Γ, then A ≃ B.

Proof. Since the restriction of G on D_{X} is a fully faithful functor, the Γ– modules GA and GB are indecomposable. Moreover, for the dimension vectors we have

\underline{dim}GA = \underline{dim}_{X} A = \underline{dim}_{X} B = \underline{dim}GB

By our assumption GA is in a postprojective component of Γ. Hence, by the main result of [6], the Γ–module GA is isomorphic to GB. Using again the fact that G is a fully faithful functor restricted on D_{X}, we obtain that A is isomorphic to B.

Suppose that C is the category mod Λ of finitely generated Λ–modules over a finite–dimensional k–algebra Λ. Let {X_{1}, X_{2}, …, X_{m}} be a set of non–isomorphic indecomposable Λ–modules, X be their direct sum and X be the full subcategory add X. Let Γ be the k-algebra End_{Λ}(X) and let G be the functor Hom_{Λ}(X, −) : mod Λ ⟶ mod Γ. In this case the following facts can be easily proven:

Remark 5.3. Suppose that A is an object of D_{X}.

(i) If f : Y ⟶A is a minimal right X –approximation of A, then Gf : GY ⟶GA is a projective cover of GA and conversely any projective cover of GA determines a minimal right X –approximation of A.

(ii) If f : Y ⟶ A is a minimal right X –approximation of A, then Ker f is an object of Gen X.

A right X –presentation Z_{1} ⟶^{z_{1}} Z_{0} ⟶^{z_{0}} A is said to be a minimal right X – presentation if z_{0} and k_{1} : Z_{1} ⟶Ker z_{0} are minimal right X –approximations, where i_{0} : Ker z_{0} ⟶Z_{0} is the canonical monomorphism and z_{1} = i_{0} ∘ k_{1}. Our next Theorem is based also on the main result of [6]. We leave for the reader the proof, which can be easily derived using the arguments of the Remark above.

Theorem 5.4. Let A and B be two indecomposable Λ–modules which belong to D_{X}. Let Z_{1} ⟶ Z_{0} ⟶ A be a minimal right X –presentation of A. If GA belongs to a postprojective component of the Auslander–Reiten quiver of End_{Λ}(X), then the following are equivalent:

(i) A ≃ B.

(ii) \underline{dim}_{X} A = \underline{dim}_{X} B.

(iii) B has a minimal X –presentation Z_{1} ⟶Z_{0} ⟶B.

Remark 5.5. A more general statement, than that of Lemma 5.1, is true for the category C and the subcategory D_{X}:

The functor G : C ⟶ mod End_{C}(X) has a left adjoint functor F if and only if any morphism between the objects of X has a cokernel in C.

Using this statement we are able to detect the subcategory D_{X} in the following sense:

An object A of C is in D_{X} if and only if the counit τ_{A} : FGA ⟶A of the adjoint pair (F, G) is an isomorphism.

References

1. M. Auslander and I. Reiten, Grothendieck groups of algebras and orders, J. Pure Appl. Algebra 39 (1986), 1–51.

2. M. Auslander and S.O. Smalø, Preprojective modules over artin algebras, J. Algebra 66 (1980), 61–122.

3. M. Auslander and O. Solberg, Relative homology and representation theory i: Relative homology and homologically finite subcategories, Comm. in Algebra 21 (1993), no. 9, 2995–3031.

4. H. Bass, Algebraic k-theory, Benjamin, Inc., New York, 1968.

5. A. Beligiannis and N. Marmaridis, Left triangulated categories arising from contravariantly finite subcategories, Comm. in Algebra 22 (1994), no. 12, 5021–5036.

6. K. Bongartz and S. O. Smalø, Modules determined by their tops and socles, Proc. Amer. Math. Soc. 96 (1986), no. 1, 34–38.

7. A. Grothendieck, Groupes des classes des cateégories abeliennes et trianguleés complexes parfait, Lecture Notes in Mathematics, vol. 589, Springer-Verlag, Berlin, 1977.

8. D. Happel, Triangulated structures in the representation theory of finitedimensional algebras, LMS Lecture Notes, vol. 119, Cambridge University Press, Cambridge, 1988.

9. C.M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, 1984.