GustavoMataJuly30th TopicsontheIgusa-Todorovfunctions

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Igusa-Todorov functions φ-dimension andψ-dimension Finitistic dimension conjecture

Topics on the Igusa-Todorov functions

Gustavo Mata

July 30th

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One of the most important conjectures in the Representation Theory of Artin Algebras is the finitistic dimension conjecture.

It states that the supremum of the projective dimensions for the f.

g. modules with finite projective dimension over an Artin algebra is finite.

In an attempt to prove the finitistic dimension conjecture, Igusa and Todorov defined two functions from the finitely generated modules over an Artin algebra to the natural numbers, which generalizes the notion of projective dimension.

Nowadays, these functions are known as the Igusa-Todorov functions.

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We fix the following notation A is an Artin algebra.

J⊂Ais the Jacobson ideal ofA.

ModAand modAare the right A-modules and the finitely generated rightA-modules, respectively.

S(A),P(A) andI(A) are the simple, projective and injective A-modules, respectively.

indAare the indescomposableA-modules.

Given M ∈modAwe denote by:

pd(M)its projective dimension.

id(M)its injective dimension.

Ω(M)its syzygy.

findim(A) is the finitistic dimension ofA.

gldim(A)is the global dimension of A.

repdim(A) is the representation dimension ofA.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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Ω(M)its syzygy.

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The following version of Fitting Lemma is used to define the Igusa-Todorov functions:

Lemma (Fitting Lemma)

Let R be a noetherian ring. Consider M ∈ModR and

f ∈End_R(M).Then,∀X ⊂M, such that X ∈modR there is a non-negative integer

η_f(X) = min{k ∈N:f|_fm(X):f^m(X)→^∼⁼ f^m+1(X),∀m≥k}.

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f ∈End_R(M).Then,∀X ⊂M, such that X ∈modR there is a non-negative integer

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f ∈End_R(M).Then, ∀X ⊂M, such that X ∈modR there is a non-negative integer

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Definition

Let K₀(A)be the abelian group generated by all symbols [M], with M ∈modA, modulo the relations

1 [M]−[M₁]−[M₂] if M∼=M₁⊕M₂,

2 [P]for each projective module P.

Let ¯Ω :K₀(A)→K₀(A) be the group endomorphism induced by Ω.

That is

Ω([M¯ ]) = [Ω(M)].

IfM ∈modA, then haddMi denotes the subgroup of K₀(A) generated by the classes of indecomposable (non projective) summands ofM.

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Definition

1 [M]−[M₁]−[M₂] if M∼=M₁⊕M₂,

That is

Ω([M¯ ]) = [Ω(M)].

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Definition

1 [M]−[M₁]−[M₂] if M∼=M₁⊕M₂,

That is

Ω([M¯ ]) = [Ω(M)].

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remark Observe that

for a finitely generated subgroup G ⊂K₀(A), the map Ω¯_|G :G →Ω(G¯ ) is an isomorphism if and only if rk(G) =rk( ¯Ω(G))

In general Ω(haddMi)¯ 6=haddΩ(M)i.

Given a finitely generated subgroupG of K₀(A), we defineηΩ¯(G) as in the Fitting Lemma.

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remark Observe that

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remark Observe that

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remark Observe that

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Definition (Igusa, Todorov)

The Igusa-Todorov functionφ_A of M ∈modA is defined as φ(M) =ηΩ¯(haddMi).

The Igusa-Todorov functionψ_A of M ∈modA is defined as ψ(M) =φ(M) +sup{pd(N) :N/Ω^φ(M⁾(M) andpd(N)<∞}.

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The Igusa-Todorov functionφ_A of M ∈modA is defined as φ(M) =ηΩ¯(haddMi).

The Igusa-Todorov functionψ_A of M ∈modA is defined as ψ(M) =φ(M) +sup{pd(N) :N/Ω^φ(M⁾(M) andpd(N)<∞}.

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Proposition (Igusa, Todorov) Given M, N ∈modA.

Ifpd(M)<∞, then φ(M) =ψ(M) =pd(M).

If M ∈indA andpd(M) =∞, then φ(M) =ψ(M) = 0.

φ(M)≤φ(M ⊕N) andψ(M)≤ψ(M⊕N).

φ(M^k) =φ(M) andψ(M^k) =ψ(M)if k ∈Z⁺.

Proposition (Huard, Lanzilotta, Mendoza) If M ∈modA, then

φ(M)≤φ(Ω(M)) + 1and ψ(M)≤ψ(Ω(M)) + 1.

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φ(M)≤φ(Ω(M)) + 1and ψ(M)≤ψ(Ω(M)) + 1.

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φ(M)≤φ(Ω(M)) + 1and ψ(M)≤ψ(Ω(M)) + 1.

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φ(M)≤φ(Ω(M)) + 1and ψ(M)≤ψ(Ω(M)) + 1.

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φ(M)≤φ(Ω(M)) + 1and ψ(M)≤ψ(Ω(M)) + 1.

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φ(M)≤φ(Ω(M)) + 1and ψ(M)≤ψ(Ω(M)) + 1.

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Consider the radical square zero algebraA= ^k_J^Q2, with Q as follows 1 ^//2 ^//3 ^//. . . ^//n^yy

IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

Let us computeφ(S1⊕Sn).

hadd(S₁⊕S₂)i ∼=Z⊕Z,

Ω(hadd(S¯ ₁⊕S₂)i)∼=hadd(S₂⊕S_n)i ∼=Z⊕Z, ...

Ω¯ⁿ⁻²(hadd(S₁⊕S₂)i)∼=hadd(S_n−1⊕S_n)i ∼=Z⊕Z Ω¯ⁿ⁻¹(hadd(S₁⊕S2)i)∼=hadd(S_n⊕Sn)i ∼=Z

Ω¯^n−1+k(hadd(S₁⊕S2)i)∼=hadd(S_n⊕Sn)i ∼=Z,∀ k ≥0.

Henceφ(S₁⊕S_n) =n−1.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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IfS(A) ={S₁, . . . ,S_n}, we have:

Ω(S_i) =

S_i+1 ifi = 1, . . . ,n−1, Sn ifi =n.

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Definition

Let A be an Artin algebra, andC a full subcategory ofmodA. We define

φdim(C) =sup{φ(M) :M ∈ObC}, and ψdim(C) =sup{ψ(M) :M ∈ObC}.

In particular, we denote by φdim(A) =φdim(modA), ψdim(A) =ψdim(modA)

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Definition

φdim(C) =sup{φ(M) :M ∈ObC},and ψdim(C) =sup{ψ(M) :M ∈ObC}.

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Definition

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Definition

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Definition

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The following inequalities hold for an Artin algebraA findim(A)≤φdim(A)≤ψdim(A)≤gldim(A)

ψdim(A)≤2.φdim(A).

Observe that an algebra with finiteφ-dimension verifies the finitistic dimension conjecture. This motivates the following question. Question: does every algebra have finiteφ-dimension?

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Observe that an algebra with finiteφ-dimension verifies the finitistic dimension conjecture. This motivates the following question. Question: does every algebra have finite φ-dimension?

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LetAbe an Artin algebra. We say that Ais of Ωⁿ-finite

representation typeif Ωⁿ(modA) is of finite representation type.

Examples of algebras of Ω¹-finite representation type are:

Special biserial algebras,

Radical square zero algebras (A= ^k_J^Q2), and Truncated path algebras (A= ^k_J^Q_k for k ≥2).

Monomial algebras are of Ω²-finite representation type, but in general not of Ω¹-finite representation type.

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ConsiderA= ^kQ_J2. If M ∈modA, then Ω(M) is a semisimple A-module.

Theorem (Lanzilotta, Marcos, Mata)

If A= ^k_J^Q2, thenφdim(A)≤φ(⊕_S∈S(A)S) + 1≤ |Q₀|.

The previous result can be generalizated to truncated path algebras as follows.

Theorem (Barrios, Mata, Rama)

If A= ^k_J^Qk, thenφdim(A)≤f_k(n) where

f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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f_k(m) =











0 if m= 0,

2 ^m−1_k

+ 1 if m≡1 (mod k), 2 ^m−2_k

+ 2 if m≡2 (mod k), 2_m−2

k

+ 1 otherwise.

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IfAis a monomial algebra, then Ω²(M) is a direct sum of right ideals (Huisgen-Zimmermann). Hence:

Theorem (Lanzilotta, Mata)

If A=kQ/I is a monomial algebra, then φdim(A)≤dim_kA− |Q₀|+ 2.