generalized igusaâtodorov function and finitistic dimensions

Arch. Math. 100 (2013), 309–322c© 2013 Springer Basel

0003-889X/13/040309-14

published online March 8, 2013DOI 10.1007/s00013-013-0497-0 Archiv der Mathematik

Generalized Igusa–Todorov function and finitistic dimensions

Dengming Xu

Abstract. We introduce a new function from the bounded derived cate-gory of a finite dimensional algebra over a field to the set of all naturalnumbers, which is a generalized version of the Igusa–Todorov function.Then we extend the results corresponding to the Igusa–Todorov function.As an application, we give a new proof of the finiteness of the finitisticdimension of special biserial algebras.

Mathematics Subject Classification (2010). 16E10, 16E05.

Keywords. Generalized Igusa–Todorov function, Projective dimension,Finitistic dimension, Specical biserial algebra.

1. Introduction. The finitistic dimension conjecture is a well known conjec-ture in the representation theory of Artin algebras. It says that the supremumof the projective dimension of those finitely generated (left) modules havingfinite projective dimension is finite (see [4]). A positive answer to this conjec-ture implies the solutions to other conjectures (see [2,6,20]). So far, it has onlybeen shown that the finitistic dimension conjecture holds for special classes ofalgebras (see [1,8–10,13])

In [13], to study the finitistic dimension conjecture, Igusa and Todorovintroduced a function (known as the Igusa–Todorov function) on the cate-gory of finitely generated modules over an Artin algebra. Using this function,they showed that the finitistic dimension of Artin algebras with representa-tion dimension at most three is finite, and they also gave a new proof of thefiniteness of the finitistic dimension of an Artin algebra with radical cubedzero. Recently, the Igusa–Todorov function has been widely used to study thefinitistic dimension conjecture (see [11,12,15,17–19]).

This work is supported by the scientific research foundation of the Civil Aviation Universityof China (No. 2010QD09X).

310 D. Xu Arch. Math.

The strategy of the Igusa–Todorov function is to bound the projectivedimension of a module involved in an exact sequence by those of other twomodules. Inspired by this idea, the paper aims to bound the projective dimen-sion of a module or a complex by those of complexes. Precisely, suppose thatA is a finite dimensional algebra over a field. We introduce a new function ψfrom the bounded derived category of A-mod to the set of all natural numbersand get the following theorem.

Theorem. Let A be a finite dimensional algebra over a field. Suppose thatX• −→ Y • −→ Z• −→ X•[1] is a distinguished triangle in Db(A-mod) suchthat Hj(X•) = Hj(Y •) = 0 for any j ≥ 1. If pdZ• < ∞, then pdZ• ≤ψ(X• ⊕ Y •) + 1.

For details about the definitions involved in this theorem, we refer toSection 2.2.

This paper is organized as follows. In Section 2, we first recall some basicdefinitions and notations needed in the paper, and then we introduce the gener-alized Igusa–Todorov function and give some basic properties of this function.In Section 3, we give the proof of the main theorem and illustrate some appli-cations of it. In Section 4, we give a new proof of the finiteness of the finitsticdimension of special biserial algebras.

2. Preliminaries. In this section, we recall some basic notations and factsneeded in the paper and give the definition of the generalized Igusa–Todorovfunction.

2.1. Basic notations and facts. Firstly, we recall some basic notations andfacts.

Let A be a finite dimensional algebra over a field. The Jacobson radicalof A is denoted by rad(A), the category of all finitely generated left A-mod-ules is denoted by A-mod, and the full subcategory of A-mod consisting offinitely generated projective A-modules is denoted by A-proj. In the following,unless otherwise specified, all algebras are assumed to be finite dimensionalalgebras over a fixed field, and all modules considered are finitely generatedleft modules. Let X be an A-module. The full subcategory of A-mod whoseobjects are direct summands of finite sums of copies of M is denoted byadd (M); the Jacobson radical of X and the module X/rad (X) are denotedby radA(X) and topA(X), respectively. The global dimension of A, which isdenoted by gl.dim(A), is defined to be the supremum of the projective dimen-sion of all finitely generated left A-modules, the finitistic dimension of A,which is denoted by fin.dim(A), is defined to be the supremum of the projec-tive dimension of all finitely generated left A-modules with finite projectivedimension. In the whole paper, all the maps are composed from left to right.

Let C be an additive category. A complex P • over C is a sequence of mor-

phisms diP • between objects P i in C: · · · −→ P i−1

di−1P •−→ P i di

P •−→ P i+1 −→ · · ·,such that di−1

P • diP • = 0. The width l(P •) of a complex P • is defined by l(P •) :=

1 + sup {i | P i �= 0} − inf {j | P j �= 0}. For a complex P •, [1] denotes theshift functor, that is, P •[1]j = P j+1 for any j ∈ Z and dP •[1] = −dP • , and

Vol. 100 (2013) Generalized Igusa–Todorov 311

τ≤i(P •) denotes a new complex defined by τ≤i(P •)j is P j for all j ≤ i andzero otherwise. The category of complexes over C with chain maps is denotedby C (C). The homotopy category of complexes over C is denoted by K (C).The full subcategory of K (C) consisting of bounded above (resp. bounded)complexes over C is denoted by K −(C) (resp. K b(C)); the full subcategory ofK −(C) consisting of complexes over C with only finitely many nonzero coho-mologies is denoted by K −,b(C); the derived category of bounded complexesover A-mod is denoted by Db(A-mod).

Now we recall the definition of the projective dimension of a complex (see

[3]). Let M and N be in A-mod. A homomorphism Mf−→ N of A-modules

is called a radical map if hfg is not an isomorphism for any Xh−→ M and

Ng−→ X with X an indecomposable A-module. A complex P • over A-mod

is called a radical complex if all of its differential maps are radical maps. Itis easy to check that if two radical complexes are isomorphic in K (A) if andonly if they are isomorphic in C (A). Thanks to this observation, for each com-plex P • ∈ K −(A-proj), we can define the projective dimension pdP • of P •

as follows.

pdP • := sup {n |Q−n �= 0, Q• is a radical complex in K −(A− proj) andQ• P • in K (A− proj)},

and put pdP • = 0 if P • is zero in K −(A-proj). It should be noted accord-ing to the definition that the projective of P • can be negative. Let C be anabelian category and X• a complex over C. The i-th cohomology group isdenoted by Hi(X•). A morphism f : X• −→ Y • of C (C) is quasi-isomorphicif the induced morphisms Hi(f) : Hi(X•) −→ Hi(Y •) are isomorphisms forall i, and we say that X• is quasi-isomorphic to Y • in this case. Let X• be abounded above complex over A-mod. It is known that there exists a radicalcomplex P •

X• in K −(A-proj) such that P •X• is quasi-isomorphic to X• (see

[16]). Thanks to this observation, we can define the projective dimension pdX•

of X• by pdP •X• , that is, pdX• := pdP •

X• . For this reason, we only considerthe projective dimension of complexes in K −(A-proj).

2.2. Definition of the generalized Igusa–Todorov function. In this subsection,we give the definition of the generalized Igusa–Todorov function and somebasic properties of it. First, the following known observation is needed.

Lemma 2.1. [14, pp.113–114, FITTING’S LEMMA]. (1) Let M be a moduleover a Noetherian ring R, and let f : M → M be an endomorphism of M .Then, for any finitely generated submodule X of M , there is an integer ηf (X)so that f sends fm(X) isomorphically onto fm+1(X) for all m ≥ ηf (X).

(2) If Y is a submodule of X, then ηf (Y ) ≤ ηf (X).(3) If R is an Artin algebra and X = M , then there is a direct sum decom-

position X = Y ⊕ Z so that Z = ker fm and Y = imfm for all m ≥ ηf (X).

Let A be a finite dimensional algebra over a field. For convenience, we write

RK ≤0(A-proj) := {P • ∈ K −,b(A-proj) |P • is a radical complex with

P j = 0 for each j ≥ 1}.


Let K0 be the abelian group generated by all symbols [P •], where P • ∈RK ≤0(A-proj), modulo the relations

(1) [P •] = [P •1 ] + [P •

2 ] if P • P •1 ⊕ P •

2 and(2) [P •] = 0 if P • is a stalk complex concentrated in degree zero.

Then K0 is the free abelian group generated by the isomorphism classes ofall indecomposable objects in RK ≤0(A-proj) except stalk complexes con-centrated in degree zero. For any complex P • ∈ RK ≤0(A-proj), we defineL(P •) = (τ≤−1(P •))[−1]. Obverse that L commutes with direct sums, andL(P •) = 0 if P • is a stalk complex concentrated in degree zero. Then Linduces a homomorphism from K0 to K0, which is also denoted by L. Forany P • ∈ RK ≤0(A-proj), we denote by 〈addP •〉 the subgroup of K0 gener-ated by the isomorphism classes of all indecomposable direct summands of P •.Note that Db(A-mod ) is a Krull–Schmidt category (see [5]). We know that〈addP •〉 is a finitely generated abelian group. Then by the Fitting,s lemma,we can define

φ(P •) := ηL(〈addP •〉),and

ψ(P •) := φ(P •)+sup {pdS• | pdS•<∞, S• is a direct summand of Lφ(P •)(P •)}.

Further, for any X• ∈ Db(A-mod) with Hj(X•) = 0 for all j ≥ 1, there existsa unique P •

X• ∈ RK ≤0(A-proj) such that P •X• is quasi-isomorphic to X•.

Thanks to this fact, we define

ψ(X•) := ψ(P •X•).

IfX• is a stalk complex concentrated in degree zero, then ψ(X•) is equal to theone defined in [13]. For this reason, we call ψ the generalized Igusa–Todorovfunction.

The following two lemmas are similar to some lemmas in [13], which givecertain basic properties of the functions φ and ψ. For completeness, we includetheir proofs here.

Lemma 2.2. Let P • and Q• be in RK ≤0(A-proj). Then we have the following.(1) If pdP • is finite, then φ(P •) = pdP •.(2) If P • is indecomposable and pdP • is infinite, then φ(P •) = 0.(3) φ(P •) ≤ φ(P • ⊕Q•).(4) φ(kP •) = φ(P •) if k ≥ 1.

Proof. By definition. �

Lemma 2.3. Let P • and Q• be in RK ≤0(A-proj). Then we have the following.(1) If pdP • is finite, then ψ(P •) = φ(P •) = pdP •.(2) ψ(kP •) = ψ(P •) if k ≥ 1.(3) ψ(P •) ≤ ψ(P • ⊕Q•).(4) Let T • be a direct summand of Ln(P •) with n ≤ φ(P •). If pdT • < ∞,

then pdT • + n ≤ ψ(P •).


Proof. (1) and (2) follow from the definition of ψ.(4) Since T • is a direct summand of Ln(P •), Lφ(P •)−n(T •) is a direct

summand of Lφ(P •)(P •). Then we have pdT • + n ≤ φ(P •) − n +pd(Lφ(P •)−n(T •))+n = φ(P •)+pd(Lφ(P •)−n(T •)) ≤ ψ(P •) by the definitionof the function ψ.

(3) We have

ψ(P •) = φ(P •) + sup {pdS•|pdS• < ∞, S• is a direct summand of Lφ(P •)(P •)}≤ φ(P •) + sup {pdS•|pdS•<∞, S• is a direct summand of Lφ(P •)(P • ⊕Q•)}≤ ψ(P • ⊕Q•),

where the last inequality follows from (4) and Lemma 2.2 (3). �

3. Proofs and applications. In this section,we give the proof of the main resultsand illustrate its applications.

3.1. Proof of the main result. To prove the theorem, we need some basicresults related to the projective dimension of complexes.

Lemma 3.1. [3, Theorem 2.4.P] Let n be a naturel number and P • in K −(A-proj). Then the following are equivalent.

(1) pd(P •) ≤ n.(2) Hom K −(A-mod )(P •, N [j]) = 0 for any A-module N and j ≥ n+ 1.(3) Hj(P •) = 0 for j ≤ −(n+ 1) and Coker(d−(n+1)) is projective.

Proof. For completeness, we include a proof here. Suppose pd(P •) = n. Thenthere exists a radical complexQ• in K −(A-proj) such thatQ• P • in K −(A-proj). Then by definition, Qj = 0 for j ≤ −(n+ 1). It is clear that

Hom K −(A-mod )(P •, N [j]) Hom K −(A-mod )(Q•, N [j]) = 0

for any A-module N and j ≥ n+ 1. Thus (1) implies (2).Now we prove that (2) implies (3). Let N = D(AA). By assumption,

Hom K −(A-mod )(P •,D(AA)[j]) = 0 for j ≥ n+ 1.

Recall that Hom A(X,D(YA)) Hom Aop(YA,D(AX)). Then we haveHj (D(P •)) = 0 for j ≥ n + 1. Thus Hj(P •) = 0 for j ≤ −(n + 1). As aresult, we get an exact sequence

· · · −→ P−(n+2) −→ P−(n+1) −→ P−(n) δ−→ Coker(d−(n+1)) −→ 0.

Let N be an A-module. Then

Ext1A(Coker(d−(n+1)), N) Hom K −(A-mod )(P •, N [n+ 1]) = 0

by assumption. This implies that Coker(d−(n+1)) is projective.


Next we show that (3) implies (1). Firstly, we have an exact sequence ofcomplexes

˜P •

��

· · · �� P −(n+1)

=

��

�� im(d−(n+1)) ��

��

0

��P •

��

· · · �� P −(n+1)

��

�� P −n

��

�� P −n+1

=

��

�� · · ·

P • 0 �� P −n/im(d−(n+1) �� P −n+1 �� · · · .

Since Coker(d−(n+1)) is projective, im(d−(n+1)) is projective. Then ˜P • is zero inK −(A-proj). Thus P • P • in K −(A-proj). By definition, pdP • = pdP • ≤n. This implies (1). �

Thanks to this lemma, we can compare the projective dimension of com-plexes involved in a distinguished triangle.

Lemma 3.2. Let P •, Q•, and R• be in K −(A-proj) and P • → Q• → R• →P •[1] a distinguished triangle. Then we have the following.

(1) If pdP • < pdQ•, then pdQ• = pdR•.(2) pdR• ≤ 1 + max{pdP •,pdQ•}.(3) pdQ• ≤ max{pdP •,pdR•}.

Proof. Let N be an A-module and C = K −(A-mod ). There is a long exactsequence

· · · −→ Hom C(P •, N [n− 1]) −→ Hom C(R•, N [n]) −→ Hom C(Q•, N [n])−→ Hom C(P •, N [n]) −→ · · · .

Using Lemma 3.1, we get this lemma. �Let f : X• → Y • be a morphism of complexes. We define the cone of

f , denoted by Cone(f), to be the complex (Cone(f))n = Y n ⊕ Xn+1 with

differential dnCone(f) =

(

dnY • 0

fn+1 −dn+1X•

)

.

Lemma 3.3. Let P • f−→ Q• be a morphism of radical complexes in K −,b(A-proj) with Cone(f)j = 0 for j ≥ 1. Suppose that pdCone(f) = m with 2 ≤m < ∞. Then we have the following.

(1) f−t is an isomorphism for any t ≥ m+ 1.(2) Suppose φ(P • ⊕Q•) ≤ m− 2. Then there is an integer 0 ≤ k ≤ m− 2,

a projective module Q′, and a distinguished triangle

τ≤−k(P •) −→ τ≤−k(P •) ⊕Q′[k] −→ ˜Z• −→ τ≤−k(P •)[1]

in K −,b(A-proj) such that pdτ≤−k(Cone(f)) = pd˜Z•.


Proof. By definition, Cone(f) is the following complex:

· · · �� Q−m−1 ⊕ P −mδ−m−1 �� Q−m ⊕ P −m+1 ��

(

h1h2

)

�� Q−m+1 ⊕ P −m+2 �� · · ·

L

��

where δ−m−1 =(

d−m−1Q• 0

f−m −d−mP •

)

. Since pdCone(f) = m, by Lemma 3.1, wehave an exact sequence

Cone(f) = · · · �� Q−m−1 ⊕ P−m δ−m−1�� Q−m ⊕ P−m+1 �� L �� 0

with L projective. Now δ−m−1 ·(

h1

h2

)

= 0 implies that d−m−1Q• h1 = 0 and

f−mh1 = d−mP • h2. Thus we have a homomorphism of complexes

˜P •

˜f��

· · · �� P−m−1

f−m−1

��

�� P−md−m

P • ��

f−m

��

P−m+1

h2

��

�� 0

˜Q• · · · �� Q−m−1

d−m−1Q•

�� Q−mh1

�� L �� 0.

Since L is projective, Cone( ˜f) = ˜Cone(f) is zero in K −,b(A-proj). This impliesthat ˜f is an isomorphism in K −,b(A-proj). Let f be the inverse of ˜f . Then˜f f ∼ 1

˜P • and f ˜f ∼ 1˜Q• . Thus we have homomorphisms

˜P •

˜f��

· · · P−m−1

l−m−1

��

��

��

��

�f−m−1

��

�� P−m

l−m

��

��

��

��

�

d−mP • ��

f−m

��

P−m+1

l−m+1

��

��

��

��

�h2

��

�� 0

0

��

��

��

��

˜Q•

f

��

· · · �� Q−m−1

f−m−1

��

�� Q−m

f−m

��

�� L

f−m+1

��

�� 0

˜P • · · · �� P−m−1

d−m−1P •

�� P−m

d−mP •

�� P−m+1 �� 0

of complexes such that 1−h2f−m+1 = l−m+1d−m

P • and 1−f−tf−t = d−t

P • l−t+1+l−td−t−1

P • for any t ≥ m. Since d−tP • and d−t−1

P • are radical maps, we knowf−t

f−t = 1−d−tP • l−t+1 − l−td−t−1

P • is an isomorphism for any t ≥ m. Similarly,h2f−m+1 is an isomorphism, and f−tf−t is an isomorphism for any t ≥ m+1.

Thus h2 and f−m are split monomorphisms, and f−t is an isomorphism forany t ≥ m+1. This also shows that ˜P • is a summand of ˜Q• in C −(A-proj) andτ≤−m(Q•) τ≤−m(P •) ⊕Q′[m] as complexes for some projective module Q′.

(2) By (1), f−j is an isomorphism for any j ≥ m+1. Let k be the minimalinteger with k ≤ m such that there is a morphism ϕ : τ≤−k(Q•) → τ≤−k(P •)⊕Q′[k] which is an isomorphism of complexes. Note that k ≤ φ(P • ⊕ Q•). By


assumption, φ(P • ⊕ Q•) ≤ m − 2, then k ≤ m − 2. Since Cone(f)j = 0 forj ≥ 1, we can choose k ≥ 0. Then we get a distinguished triangle

(∗) τ≤−k(P •)τ≤−k(f)◦ϕ−→ τ≤−k(P •) ⊕ Q′[k] −→ Cone(τ≤−k(f) ◦ ϕ) −→ τ≤−k(P•)[1].

Since ϕ is an isomorphism of complexes, Cone(τ≤−k(f) ◦ϕ) Cone(τ≤−k(f))in K −(A-proj). Meanwhile, we have a distinguished triangle

P−k+1[k] → τ≤−k(Cone(f)) → Cone(τ≤−k(f)) −→ P−k+1[k + 1].

If pdτ≤−k(Cone(f))≤k, then, by Lemma 3.1, we know thatHi(τ≤−k(Cone(f)))= 0 for i ≤ −(k + 1) and that coker δ−(k+1) is projective. This implies thatcoker δ−(k+2) is projective. Again by Lemma 3.1, we have pdCone(f) ≤ k +1 ≤ m − 1. This contradicts the assumption that pdCone(f) = m. Thuspdτ≤−k(Cone(f)) > k. By Lemma 3.2, we have pdτ≤−k(Cone(f)) = pdCone(τ≤−k(f)). Then

pdτ≤−k(Cone(f)) = pdCone(τ≤−k(f)) = pdCone(τ≤−k(f) ◦ ϕ).

Write ˜Z• = Cone(τ≤−k(f) ◦ ϕ), the triangle (∗) is as desired. �To prove the main theorem, we need the following proposition.

Proposition 3.4. Let P • f−→ Q• be a morphism of radical complexes inK −,b(A-proj) with Cone(f)j = 0 for j ≥ 1. Suppose pdCone(f) = m < ∞.Then pdCone(f) ≤ ψ(P • ⊕Q•) + 1.

Proof. (1) If m = 0 or 1, there is nothing to show.(2) Suppose m ≥ 2.

Case 1. If m − 1 ≤ φ(P • ⊕ Q•), then pdCone(f) = m ≤ φ(P • ⊕ Q•) + 1 ≤ψ(P • ⊕Q•) + 1.Case 2. Suppose φ(P • ⊕ Q•) ≤ m − 2. Keep the notations from the proof ofLemma 3.3. By (∗), there is an integer 0 ≤ k ≤ m − 2 and a distinguishedtriangle

τ≤−k(P •)τ≤−k(f)◦ϕ−→ τ≤−k(P •) ⊕˜Q′ −→ ˜Z• −→ (τ≤−k(P •))[1]

such that pdτ≤−k(Cone(f)) = pd˜Z•. Denote by ϕ the composition

τ≤−k(f) ◦ ϕ ◦ (1τ≤−k(P •), 0) : τ≤−k(P •) −→ τ≤−k(P •).

Since f−j is an isomorphism for any j ≥ m + 1 by Lemma 3.3 (1), ϕ−j isan isomorphism for any j ≥ m + 1. By the choice of k, we know that foreach integer j with k + 1 ≤ j ≤ m, ϕ−j is an isomorphism. Then, by Lemma2.1, there is an integer t so that ϕ−k sends (ϕ−k)l(P−k) isomorphically onto(ϕ−k)l+1(P−k) for l ≥ t. Then we have a commutative diagram of complexes

(ϕ)t(τ≤−k(P •))

i1

��

θ �� (ϕ)2t(τ≤−k(P •))

i2

��τ≤−k(P •)

(ϕ)t

�� (ϕ)t(τ≤−k(P •)),


where θ = (ϕ)t|(ϕ)t(τ≤−k(P •)) and i1, i2 are the inclusion maps. By the choiceof t, we deduce that θ ◦ i2 is an isomorphism of complexes, so (ϕ)t is a splitepimorphism. Consequently, we have τ≤−k(P •) (ϕ)t(τ≤−k(P •))⊕ker(ϕ)t ascomplexes. Note that since ϕ−j is an isomorphism for any j ≥ m+ 1, we havepd ker(ϕ)t < ∞. Now we have a distinguished triangle

ker(ϕ)t −→ ker(ϕ)t ⊕˜Q′ −→ Z• −→ ker(ϕ)t[1]

which is quasi-isomorphic to the triangle stated at the beginning of theproof, where Z• ˜Z• in K −,b(A-proj). This implies pdZ• ≤ 1 +max{pd ker(ϕ)t, pd(Q′[k])} by Lemma 3.2 (2). To complete the proof of thisproposition, we shall prove the following claim first.Claim: ker(ϕ)t �= 0 in K −(A-proj). Hence, pd ker(ϕ)t ≥ k.

Proof of the claim. Since P • is a radical complex, τ≤−k(P •) is a radical com-plex. Then, as a direct summand of τ≤−k(P •), ker(ϕ)t is a radical complex.If ker(ϕ)t = 0 in K −(A-proj), then ker(ϕ)t = 0 is zero in C −(A-proj). Thisimplies that (ϕ)t is an isomorphism. Then ϕ is an isomorphism. As a result,f−j is an isomorphism for each j ≥ k. In particular, f−j is an isomorphism foreach j ≥ m − 2. This implies that Cone(τ≤−m+2(f)) is zero in K −(A-proj).Then, by Lemma 3.1, we have pdCone(f) ≤ m−1, a contradiction. This provesthe claim.

Now we continue to prove this proposition. By the claim, we havepd ker(ϕ)t ≥ pd(Q′[k]). Then pdZ• ≤ 1 + pd ker(ϕ)t. Let W • be the com-plex defined by

W i ={

0 if i ≤ −k;Cone(f)i if i > −k.

Then we get a triangle

W • −→ Cone(f) −→ τ≤−k(Cone(f)) −→ W •[1].

Since k ≤ m − 2, it follows from Lemma 3.2 (1) that pdCone(f) =pdτ≤−k(Cone(f)). Then, by Lemma 3.3 (2), we have pdCone(f) = pdτ≤−k

(Cone(f)) = pd˜Z• = pdZ• ≤ 1 + pd(ker(ϕ)t). Since ker(ϕ)t is a directsummand of τ≤−k(P •), ker(ϕ)t[−k] is a direct summand of τ≤−k(P •)[−k] =Lk(P •), which is again a direct summand of Lk(P • ⊕ Q•). Note that k ≤φ(P • ⊕Q•). Then, by Lemma 2.3 (4), we get pd(ker((ϕ)t)[−k])+k ≤ ψ(P • ⊕Q•). Consequently, pdCone(f) ≤ 1+pd(ker(ϕ)t) ≤ 1+pd(ker(ϕ)t[−k])+ k ≤ψ(P • ⊕Q•) + 1. �

Now we can prove the main result, which is restated as follows.

Theorem 3.5. Let X• −→ Y • −→ Z• −→ X•[1] be a distinguished trian-gle in Db(A-mod ) such that Hj(X•) = Hj(Y •) = 0 for all j ≥ 1. SupposepdZ• < ∞. Then pdZ• ≤ ψ(X• ⊕ Y •) + 1.


Proof. Recall that K −,b(A-proj) is equivalent to Db(A-mod ) as triangu-lated categories. Then we can form the following isomorphism of triangles inDb(A-mod ).

PX•

��

f �� PY •

��

�� Cone(f) ��

��

PX• [1]

��X• �� Y • �� Z• �� X•[1],

where PX• and PY • are radical complexes in K −,b(A-proj). Since Hj(X•) =Hj(Y •) = 0 for any j ≥ 1, the upper triangle satisfies the conditions of Prop-osition 3.4. In fact, P j = Qj = 0 for j ≥ 1. Since Z• is quasi-isomorphic toCone(f), we have pdZ• = pdCone(f) ≤ ψ(PX• ⊕PY •) + 1 = ψ(X• ⊕Y •) + 1,where the equations follow from the definition and the inequality follows fromProposition 3.4. This proves the theorem. �

Recall that if X is an A-module, then ψ(X) := ψ(P •X) defined at the begin-

ning of this section is equal to the one defined in [13], where P •X is the minimal

projective resolution of X. Meanwhile, it is known that any exact sequence inA-mod can be embedded into a distinguished triangle in Db(A-mod ). Notethat if A is an Artin algebra, then ψ(X) can be defined for any finitely gener-ated A-module X. As a consequence of Theorem 3.5, we get the following wellknown result.

Corollary 3.6. [13] Let A be an Artin algebra and 0 −→ X −→ Y −→ Z −→ 0an exact sequence in A-mod. Suppose pdA(Z) < ∞. Then pdA(Z) ≤ ψ(X ⊕Y ) + 1. �

3.2. Applications of the main result. In this subsection, we use the main resultto bound the finitistic dimension of some subcategories.

A subcategory C of Db(A-mod ) is said to be add (M•)-hereditary in Db

(A-mod ) if there exists an object M ∈ Db(A) such that(1) Hj(M•) = 0 for j ≥ 1.(2) For any X ∈ C, there exists a natural number n and a distinguished

triangle

M•1 −→ M•

0 −→ X•[n] −→ M•1 [1]

with M•0 ,M

•1 ∈ add (M•).

For convenience, we write

fin.dimA(C) := sup {pd(W ) | pd(W ) < ∞ and W ∈ C}.Proposition 3.7. Let A be a finite dimensional algebra and C a subcategory ofDb(A-mod ). Suppose that there exists an object M• ∈ Db(A) such that C isadd (M•)-hereditary in Db(A-mod ). Then fin.dim (C) < ∞.

Proof. Let X• ∈ C with pd(X•) < ∞. By assumption, there exists a naturalnumber n and a distinguished triangle

M•1 −→ M•

0 −→ X•[n] −→ M•1 [1]


with M•0 ,M

•1 ∈ add (M•). By Theorem 3.5, we have pd(X•[n]) ≤ ψ(M•

0 ⊕M•

1 ) + 1 ≤ ψ(M•) + 1. Thus pd(X•) = pd(X•[n]) − n ≤ ψ(M•) + 1 − n ≤ψ(M•) + 1. Consequently, fin.dim(C) < ∞. �

Let A be an Artin algebra. A generator–cogenerator is an A-moduleM suchthat any indecomposable projective or injective A-module is a direct summandof M . The representation dimension rep.dim (A) of A is defined as follows:

rep.dim(A) = inf{gl.dimEndA(M) | M is a generator − cogerator}.

Immediately from the previous proposition, the following corollary follows.

Corollary 3.8. Let A be a finite dimensional algebra. Suppose that thereexists an object M• ∈ Db(A) such that A-mod is add (M•)-hereditary inDb(A-mod ). Then fin.dim (A) < ∞. In particular, if rep.dim(A) ≤ 3, thenfin.dim (A) < ∞.

4. The finitistic dimension conjecture and quotient algebras. In [8], theauthors showed that the representation dimension of a special biserial alge-bra is at most three and then proved that the finitistic dimension of a specialbiserial algebra is finite by a result in [13]. In this section, we first constructexact sequences containing syzygy modules of one module over both a givenalgebra and its quotient algebra. Then by using the characteristics of the sec-ond syzygies of modules over a monomial algebra proved in [21], we give a newproof of the finiteness of the finitistic dimension of special biserial algebras,which avoids calculating the representation dimension of these algebras.

Firstly, the following basic homological fact is needed.

Lemma 4.1. Let A be an Artin algebra and 0 → X → Y → Z → 0 an exactsequence in A-mod. Then there are exact sequences

(1) 0 −→ ΩA(Z) −→ X ⊕Q −→ Y −→ 0,(2) 0 −→ ΩA(Y ) −→ ΩA(Z) ⊕ P −→ X −→ 0,(3) 0 −→ ΩA(X) −→ ΩA(Y ) ⊕R −→ ΩA(Z) −→ 0,

where P , Q, and R are projective A-modules.

Let A be an Artin algebra. Let

Ω2A(A- mod ) := {Ω2

A(X) | X ∈ A- mod }.

The algebra A is said to be 2-syzygy finite if there exist finitely many noniso-morphic indecomposable modules in Ω2

A(A-mod ).The following theorem provides a way to bound the finitistic dimension of

certain algebras through some properties of their quotient algebras.

Proposition 4.2. Suppose that I is a nilpotent ideal of an Artin algebra A suchthat Irad (A) = 0. Let A = A/I. If A is 2-syzygy finite, then fin.dim (A) < ∞.


Proof. Let Y be an A-module. Then ΩA(Y ) is an A-module since Irad (A) = 0.Write X = ΩA(Y ). Then, by the snake lemma, we can form the following com-mutative diagram in A-mod.

IPX=−−−−→ IPX

⏐

⏐

�

⏐

⏐

�

0 −−−−→ ΩA(X) −−−−→ PXf−−−−→ X −−−−→ 0

⏐

⏐

�

⏐

⏐

�

⏐

⏐

�

=

0 −−−−→ ΩA(X) −−−−→ PX/IPX −−−−→ X −−−−→ 0⏐

⏐

�

⏐

⏐

�

0 0 .

Since I is nilpotent, I ⊆ rad (A). Then I2 ⊆ Irad (A) = 0. Thus I2 = 0.Consequently, IPX is an A-module. As a result, we have an exact sequence

0 −→ IPX −→ ΩA(X) −→ ΩA(X) −→ 0 (4.1)

in A-mod.By (4.1) and Lemma 4.1 (1), we have an exact sequence

0 −→ Ω2A(X) −→ IPX ⊕QX −→ ΩA(X) −→ 0 (4.2)

in A-mod, where QX is a projective A-module. By assumption, Ω2A(A-mod) is

of finite type in A-mod. Then there is an A-module M such that add A(M) =Ω2

A(A-mod). If pdA(Y ) < ∞, then pdA(X) < ∞. By Corollary 3.6 and (4.2),

we have

pdA(ΩA(X)) ≤ ψ(Ω2A(X) ⊕ IP ⊕Q) + 1 ≤ ψ(M ⊕ I ⊕A) + 1.

As a result,

pdA(Y ) ≤ pdA(Ω2A(Y )) + 2 = pdA(ΩA(X)) + 2 ≤ ψ(M ⊕ I ⊕A) + 3.

Consequently, fin.dim(A) < ∞. �Immediately from Theorem 4.2, the following result follows.

Corollary 4.3. Let A be a finite dimension algebra given by a quiver with rela-tions. Suppose that I is a nilpotent ideal of A such that Irad (A) = 0 and A/Iis a monomial algebra. Then fin.dim (A) < ∞.

Proof. Suppose that A/I is a monomial algebra. Write A = A/I. Then, by[21, Theorem I], we know that Ω2

A(A-mod) is of finite type. Then the corollary

follows from Proposition 4.2. �The following result is known.


Lemma 4.4. [7, Corollary 9.2.5] Let A be a connect basic Artin algebra andP an indecomposable summand of the regular module. Suppose P is injective.Then we have the following.

(1) soc (P )rad (A) = 0. In particular, soc (P ) is an ideal of A.(2) If A is not a simple algebra, then soc (P ) is a nilpotent ideal of A.

Before proving the main result of this section, we recall the definition ofspecial biserial algebras. Let A = kQ/I be a finite dimensional algebra givenby a quiver with relations. Then A is called a special biserial algebra if thefollowing hold.

(1) Any vertex of Q is the starting point of at most two arrows and alsothe end point of at most two arrows;

(2) Let β be an arrow in Q1. Then there is at most one arrow α with αβ /∈ Iand at most one arrow γ with βγ /∈ I.

Theorem 4.5. [8] Let A be a special biserial algebra. Then fin.dim (A) < ∞.

Proof. Let A = P ⊕Q with P a direct sum of indecomposable projective-injec-tive A-modules and Q a direct sum of indecomposable projective noninjectiveA-modules. Let I = soc (P ). Then we have by Lemma 4.4 that I is an idealof A. Write A = A/I. It is known that A is a monomial algebra. Then thetheorem follows from Lemma 4.4 and Corollary 4.3. �

Acknowledgements. The author thanks the referee for helpful comments andrecommendations which helped to improve the readability and quality of thepaper.

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Dengming Xu

Sino-European Institute of Aviation Engineering,Civil Aviation University of China,Tianjin 300300,People’s Republic of Chinae-mail: [email protected]

Received: 5 September 2012

generalized igusaâtodorov function and finitistic dimensions

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