Cyclic homology of truncated quiver algebras and notes on ...iyama/Yamagata65/Itagaki.pdf · Cyclic homology is developed as a non-commutative variant of the de Rham cohomology by

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Introduction. . . . . . . ..

Preceding study. .. . . . .

Main result. . . . .Example

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Application

Cyclic homology of truncated quiver algebras

and notes on the no loops conjecture

for Hochschild homology

Tomohiro Itagaki

(joint work with Katsunori Sanada)

Tokyo University of Science

November 13, 2013

Perspectives of Representation Theory of Algebras Conference

honoring Kunio Yamagata on the occasion of his 65th birthday

Tomohiro Itagaki (joint work with Katsunori Sanada) Tokyo University of Science

Cyclic homology of truncated quiver algebras and notes on the no loops conjecture for Hochschild homology

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Application

1 Introduction

Definitions and the notation

Aim

2 Hochschild homology and cyclic homology of truncated quiver algebras

Hochschild homology of truncated quiver algebras

Cyclic homology of truncated quiver algebras over a field of

characteristic zero

3 Main result

Main result

The outline of the proof

4 Example

The cyclic homology of truncated cycle algebras

5 Application

The 2-truncated cycles version of the no loops conjecture

The m-truncated cycles version of the no loops conjecture



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Application

Section 1

Introduction



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Application


Notation

K: a commutative ring

N: the set of all natural numbers containing 0

∆: a finite quiver

∆n: the set of all paths of length n(n ≥ 0) in ∆

R∆: the arrow ideal of K∆

Gn: a cyclic group of order n generated by tn

A⊗n := A⊗K A⊗K · · · ⊗K A for a K-algebra A

(the n-fold tensor product of A)

Ae := A⊗K Aop: the enveloping algebra of A



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Application


Hochschild homology groups

Definition

For K-algebra A, the Hochschild complex is the following complex:

C(A) : · · · b−→ A⊗n+1 b−→ A⊗n b−→ · · · b−→ A⊗2 b−→ A −→ 0,

where the K-homomorphisms b : A⊗n+1 → A⊗n is given by

b(x0 ⊗ · · · ⊗ xn) =

n−1∑i=0

(−1)i(x0 ⊗ · · · ⊗ xixi+1 ⊗ · · · ⊗ xn)

+ (−1)n(xnx0 ⊗ x1 ⊗ · · · ⊗ xn−1).

HHn(A) := Hn(C(A)): the n-th Hochschild homology group of A

(n ≥ 0).



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Application


Hochschild homology groups

If a unital K-algebra A is projective as a module over K, then

there is an isomorphism

HHn(A) ∼= TorAe

n (A,A).

For a unital K-algebra A, the bar resolution Cbar(A) of A is

given by

Cbar(A) : · · · b′

−→ A⊗n+1 b′

−→ A⊗n b′

−→ · · · b′

−→ A⊗2 b′

−→ A −→ 0,

where the differentials b′ are given by

b′(x0 ⊗ · · · ⊗ xn) =

n−1∑i=0

(−1)i(x0 ⊗ · · · ⊗ xixi+1 ⊗ · · · ⊗ xn).



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Application


Cyclic group action

Let n be a positive integer. The action of Gn on the module A⊗n

is given by

tn · (x1⊗x2,⊗ · · ·⊗xn) := (−1)n−1(xn⊗x1⊗x2⊗· · ·⊗xn−1),

where xi ∈ A.

Let D := 1− tn and N := 1 + tn + t2n + · · ·+ tn−1n as elements

of the group ring ZGn. Then

D : A⊗n −→ A⊗n

and N : A⊗n −→ A⊗n

are K-homomorphisms.



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Application


Cyclic homology groups

DefinitionFor a K-algebra A, the following is a first quadrant bicomplexCC(A) of A: yb

y−b′yb

CC(A)∗,2 : A⊗3 D←−−−−− A⊗3 N←−−−−− A⊗3 D←−−−−−yb

y−b′yb

CC(A)∗,1 : A⊗2 D←−−−−− A⊗2 N←−−−−− A⊗2 D←−−−−−yb

y−b′yb

CC(A)∗,0 : AD←−−−−− A

N←−−−−− AD←−−−−−

CC(A)0,∗ CC(A)1,∗ CC(A)2,∗

HCn(A) := Hn(Tot(CC(A))): the n-th cyclic homology group of

A(n ≥ 0)



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Application


Cyclic homology is developed as a non-commutative variant of thede Rham cohomology by Connes (Loday, 1992). Aside from this,cyclic homology is developed by Tsygan, Loday and Quillen. Cyclichomology is closely related to Hochschild homology, because of thedefinition itself and also the periodicity exact sequence by Connes:

· · · → HHn(A)I→ HCn(A)

S→ HCn−2(A)B→ HHn−1(A)

I→ · · · .



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Application


It is well known that cyclic homology is Morita invariant.

Moreover, the following theorem is shown by Konig, Liu, Zhou.

Theorem(Konig, Liu and Zhou, 2011)

Let R and T be two finite dimensional algebras over an

algebraically closed field which are stably equivalent of Morita

type. Then,

(1) For n > 0 odd, dimHCn(R) = dimHCn(T ).

(2) Suppose that R and T have no semisimple direct summands.

Then for any n ≥ 0 even, the following statements are

equivalent:

(i) the Auslander-Reiten conjecture holds for this stable

equivalence of Morita type;

(ii) dimHCn(R) = dimHCn(T ).



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Application


The cyclic homology of algebras is investigated for classes of

various algebras, for example,

2-nilpotent algebras (Cibils, 1990)

truncated quiver algebras over a field of characteristic zero

(Taillefer, 2001)

quadratic monomial algebras (Skoldberg, 2001)

monomial algebras over a field of characteristic zero (Han,

2006)

etc.

We want to know the influence of a ground ring of an algebra on

its cyclic homology. In particular, we are going to focus on the

module structure of cyclic homology.



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Application

Aim

Aim

Aim

(1) Find the dimension formula of the cyclic homology of

truncated quiver algebras K∆/Rm∆ over a field K of positive

characteristic.

(2) Show the m-truncated cycles version of the no loops

conjecture for a class of algebras over an algebraically closed

field.



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Application

Section 2

Hochschild homology and cyclic homology oftruncated quiver algebras



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Application


Truncated quiver algebra K∆/Rm∆

Following the notation by Skoldberg(1999), we recall

characteristics of truncated quiver algebras. By adjoining an

element ⊥, we will consider the following set:

∆ = {⊥} ∪∞∪i=0

∆i.

This set is a semigroup with the multiplication defined by

δ · γ =

{δγ if t(δ) = s(γ),

⊥ otherwise,δ, γ ∈

∞∪i=0

∆i,

and

⊥ · γ = γ · ⊥ = ⊥, γ ∈ ∆.



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Application


K∆ is a semigroup algebra, and the path algebra K∆ is

isomorphic to K∆/(⊥).K∆ is ∆-graded, that is, K∆ =

⊕γ∈∆(K∆)γ , where

(K∆)γ = Kγ for γ ∈∪∞

i=0 ∆i and (K∆)⊥ = 0.

K∆ is N-graded, that is, K∆ =⊕∞

i=0 K∆i.

Rm∆ is ∆-graded and N-graded.

Thus the truncated quiver algebra K∆/Rm∆ is a ∆-graded and an

N-graded algebra.



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Application


Theorem(Skoldberg, 1999)

The following is a projective ∆-graded resolution of a truncated

quiver algebra A = K∆/Rm∆ as a left Ae-module:

PA : · · ·di+1−→ Pi

di−→ · · · d2−→ P1d1−→ P0

ε−→ A −→ 0.

Here each terms are ∆-graded modules defined by

Pi = A⊗K∆0 KΓ(i) ⊗K∆0 A for i ≥ 0,

P0 = A⊗K∆0 K∆0 ⊗K∆0 A ∼= A⊗K∆0 A,

where Γ(i) is given by

Γ(i) =

{∆cm if i = 2c (c ≥ 0),

∆cm+1 if i = 2c + 1 (c ≥ 0),



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Application


Theorem(Skoldberg, 1999)

and the differentials are defined by

d2c(α⊗ α1 · · ·αcm ⊗ β)

=

m−1∑j=0

αα1 · · ·αj ⊗ α1+j · · ·α(c−1)m+1+j

⊗ α(c−1)m+2+j · · ·αcmβ,

d2c+1(α⊗ α1 · · ·αcm+1 ⊗ β)

= αα1 ⊗ α2 · · ·αcm+1 ⊗ β − α⊗ α1 · · ·αcm ⊗ αcm+1β,

where αi ∈ ∆1 for i ≥ 1 and α, β ∈ A and ε is the multiplication.



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Application


Cycle and Basic cycle

A path is called a cycle if its source coincides with its target. A

cycle γ is called a basic cycle if there does not exist a cycle δ such

that γ = δj for some positive integer j(≥ 2).

∆cn: the set of all cycles of length n.

∆bn: the set of all basic cycles of length n.



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Application


Cycle and Basic cycle

Let a path α1 · · ·αn−1αn be a cycle, where αi is an arrow in ∆.

Then the action of Gn on ∆cn is given by

tn · (α1 · · ·αn−1αn) := αnα1 · · ·αn−1.

Similarly, Gn acts on ∆bn.

an: the cardinal number of the set of all Gn-orbits on ∆cn.

bn: the cardinal number of the set of all Gn-orbits on ∆bn.



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Application


Hcohschild homology of truncated quiver algebras

L denotes the complex A⊗K∆e0KΓ(∗), which is isomorphic to

A⊗Ae PA by the following isomorphism φ:

φ : A⊗Ae PA∼→ A⊗Ae Ae ⊗K∆e

0KΓ(∗) ∼→ A⊗K∆e

0KΓ(∗).

Since the complex L is decomposed into the subcomplexes Lγ

L ∼=∞⊕

q=0

⊕γ∈∆c

q/Gq

Lγ ,

the p-th Hochschild homology group HHp(A) is N-graded, so we

have

HHp(A) =∞⊕

q=0

HHp,q(A) for p ≥ 0.



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Application



Theorem (Skoldberg, 1999)

Let K be a commutative ring, A = K∆/Rm∆ and q = cm + e for

0 ≤ e ≤ m− 1. Then the degree q part of the p-th Hochschildhomology HHp,q(A) is the following as a K-module.

Kaq if 1 ≤ e ≤ m− 1 and 2c ≤ p ≤ 2c + 1,

⊕r|q

(K

gcd(m,r)−1 ⊕Ker

(·

m

gcd(m, r): K −→ K

))br

if e = 0 and 0 < 2c− 1 = p,

⊕r|q

(K

gcd(m,r)−1 ⊕ Coker

(·

m

gcd(m, r): K −→ K

))br

if e = 0 and 0 < 2c = p,

K#∆0 if p = q = 0,

0 otherwise.



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Application

Cyclic homology of truncated quiver algebras over a field of characteristic zero

Cyclic homology of truncated quiver algebras

Theorem (Taillefer, 2001)

Let K be a field of charactaristic zero. The cyclic homology group

of a truncated quiver algebra A is given by

dimKHC2c(A) = #∆0 +

m−1∑e=1

acm+e,

dimKHC2c+1(A) =∑r > 0

s.t. r|(c + 1)m

(gcd(m, r)− 1)br.



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Application

Section 3

Main result



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Application

Main result

Main result

Theorem (Itagaki, Sanada)

Let K be a field of characteristic ζ and A = K∆/Rm∆ . Then the

dimension formula of the cyclic homology of A is given by, forc ≥ 0,

dimKHC2c(A) = #∆0 +

m−1∑e=1

acm+e

+

c−1∑c′=0

m−1∑e=1

∑r > 0

s.t. rζ|c′m + e

br +

c∑c′=1

∑r > 0

s.t. r|c′m,

gcd(m, r)ζ|m

br

+

c∑c′=1

∑r > 0

s.t. rζ| gcd(m, r)c′

(gcd(m, r)− 1)br,



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Application

Main result

Main result

Theorem(Itagaki, Sanada)

dimKHC2c+1(A) =∑

r > 0

s.t. r|(c + 1)m

(gcd(m, r)− 1)br

+

c∑c′=0

m−1∑e=1

∑r > 0

s.t. rζ|c′m + e

br +

c+1∑c′=1

∑r > 0

s.t. r|c′m,

gcd(m, r)ζ|m

br

+

c∑c′=1

∑r > 0

s.t. rζ| gcd(m, r)c′

(gcd(m, r)− 1)br.



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Application


Cibils’ projective resolution

Lemma (Cibils, 1989)

Let I be an admissible ideal of K∆. E denotes the subalgebra of

A = K∆/I generated by ∆0. The following is a projective

resolution of A as left Ae-module:

QA : · · · → Qndn→ Qn−1 → · · · → Q1

d1→ Q0d0→ A→ 0

where Qn = A⊗E (radA)⊗n

E ⊗E A and differntials are given by

dn(x0⊗ · · · ⊗ xn+1) =

n∑i=0

(−1)i(x0 ⊗ · · · ⊗ xixi+1 ⊗ · · · ⊗ xn+1)

for x0, xn+1 ∈ A, x1, . . . , xn ∈ radA.



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Application


Cibils’ mixed complex

Let K be a field. For a K-algebra A with a decomposition

A = E⊕ r, where E is a separable subalgebra and r is a two-sided

ideal, Cibils(1990) gives the following mixed complex

(A⊗Ee r⊗n

E , b, B):

b : A⊗Ee r⊗n+1

E → A⊗Ee r⊗n

E is the Hochschild boundary and

B : A⊗Ee r⊗n

E → A⊗Ee r⊗n+1

E is given by the formula

B(x0 ⊗Ee x1 ⊗ · · · ⊗ xn)

=n∑

i=0

(−1)in(1⊗Ee xi ⊗ · · · ⊗ (x0)r ⊗ · · · ⊗ xi−1),

where x0 = (x0)E + (x0)r ∈ E ⊕ r.

The cyclic homology groups of the mixed complex and the cyclic

homology groups of A are isomorphic.Tomohiro Itagaki (joint work with Katsunori Sanada) Tokyo University of Science


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Application


The cyclic homology groups of Cibils’ mixed complex is isomorphicto the total homology groups of the following first quadrantbicomplex: yb

yb

yb

A⊗Ee r⊗2E B←−−−− A⊗Ee r

B←−−−− A ←−−−−yb

yb

yA⊗Ee r

B←−−−− A ←−−−− 0 ←−−−−yb

y yA ←−−−− 0 ←−−−− 0 ←−−−−

Denote the above bicomplex by CE(A). For a truncated quiver

algebra A = K∆/Rm∆ , there exists the decomposition

K∆/Rm∆ = K∆0 ⊕ radA.



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Application


We consider the spectral sequence associated with the filtration

FTot(CE(A)) given by

(F pTot(CE(A)))n =⊕i≤p

CE(A)i,n−i.

The cyclic homology groups HCn(A) is given by

HCn(A) ∼= Hn(Tot(CE(A))) ∼=⊕

p+q=n

E2p,q.

E1-page of the spectral sequence is drawn by the following

illustration:

HH2(A)B←− HH1(A)

B←− HH0(A)0←−

HH1(A)B←− HH0(A)

0←− 00←−

HH0(A)0←− 0

0←− 00←−



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Application


B : HHn(A)→ HHn+1(A)

Ames, Cagliero and Tirao (2009) give chain maps ι : PA → QA,and π : QA → PA which induce isomorphismHHn(A) ∼= HHn(A) for n ≥ 0. By means of these chain maps,we consider B : HHn(A)→ HHn+1(A) as follows:

Skoldberg: A⊗K∆e0KΓ(n) A⊗K∆e

0KΓ(n+1)

−→ ∼

←− ∼

A⊗Ae Pn A⊗Ae Pn+1

−→ idA ⊗ ι

←− idA ⊗ π

A⊗Ae Qn A⊗Ae Qn+1

−→ ∼

←− ∼

Cibils: A⊗K∆e0radA

⊗nK∆0

B−→ A⊗K∆e0radA

⊗n+1K∆0



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Application

Section 4

Example



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Application


Let K be a field of characteristic ζ. We consider the cyclic

homoology of a truncated quiver algebra A = K∆/Rm∆ , where ∆

is the following quiver:

0s− 1

s− 2

s− 3

2

1

αs−1 α0

αs−2α1

αs−3

The truncated quiver algebra A is called a truncated cycle algebra.

Note that ar and br is given by

ar =

{1 if s|r,0 otherwise,

br =

{1 if s = r,

0 otherwise.



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Application


For x ∈ R, we denote the largest integer i satisfying i ≤ x by [x].The dimension formula of the cyclic homology of A is given by

dimKHC2c(A)

= s +

[(c + 1)m− 1

s

]−

[cms

]+

c−1∑c′=0

([(c′ + 1)m− 1

sζ

]−

[c′m

sζ

])

+

([m

gcd(m, s)ζ

]−

[m− 1

gcd(m, s)ζ

]) c∑c′=1

([c′m

s

]−

[c′m− 1

s

])+ (gcd(m, s)− 1)

[gcd(m, s)c

sζ

]



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Application


and

dimKHC2c+1(A)

= (gcd(m, s)− 1)

([(c + 1)m

s

]−

[(c + 1)m− 1

s

]+

[gcd(m, s)c

sζ

])

+

([m

gcd(m, s)ζ

]−

[m− 1

gcd(m, s)ζ

]) c+1∑c′=1

([c′m

s

]−

[c′m− 1

s

])

+

c∑c′=0

([(c′ + 1)m− 1

sζ

]−

[c′m

sζ

]).



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Application


When ζ = 0, the dimension of the cyclic homology of A is as

follows:

dimKHC2c(A) = s +

[(c + 1)m− 1

s

]−

[cm

s

],

dimKHC2c+1(A)

= (gcd(m, s)− 1)

([(c + 1)m

s

]−

[(c + 1)m− 1

s

]).



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Application


A change of dimK HC2c(A) by ζ

We fix s = 7, m = 26. The graphs of dimK HC2c(A) of the

three cases that ζ = 0, ζ = 13 and ζ = 113 are as follows:



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Application

Section 5

Application of the chain map and truncatedquiver algebras



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Application


The no loops conjecture is as follows:

The no loops conjecture

If A is a finite dimensional algebra of finite global dimension then

Ext1A(S, S) = 0 for all simple A-modules S, or equivalently,

A has no loops in its quiver.

It is shown that this conjecture holds for a large class of the finite

dimensional algebra including all those which the ground field is an

algebraically closed field by Igusa (1990), and this is derived from

an earlier result of Lenzing (1969). Bergh, Han and Madsen show

the 2-truncated cycles version of this conjecture.



A has no 2-truncated cycles in its quiver



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Application


Definition (m-truncated cycle)

Let ∆ be a quiver with a finite number of vertices and I(⊂ R2∆)

an ideal in the path algebra K∆. A finite sequence α1, . . . , αu of

arrows in K∆/I which satisfies the equations

t(αi) = s(αi+1)(i = 1, . . . , u− 1) and t(αu) = s(α1) is an

m-truncated cycle for an integer m ≥ 2 if

αi · · ·αi+m−1 = 0 and αi · · ·αi+m−2 = 0 in K∆/I

for all i, where the indices are modulo u.

Definition (Hochschild homology dimension)

The Hochschild homology dimension of an algebra A, denoted by

HHdimA, is defined by

HHdimA = sup{n ≥ 0|HHn(A) = 0}.



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Application


Theorem (Bergh, Han, Madsen, 2012)

Let K be a commutative ring, ∆ a quiver with a finite number of

vertices and I an ideal contained in R2∆. Suppose K∆/I contains

a 2-truncated cycle x1, . . . , xu. Then for every n ≥ 1 with

un ≡ u (mod 2), the element

(x1 ⊗ · · · ⊗ xu)⊗n

represents a nonzero element in HHun−1(K∆/I). In particular,

HHdimK∆/I =∞.

By the above theorem, they show the 2-truncated cycles version of

the no loops conjecture.



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Application


Corollary (Bergh, Han, Madsen, 2012)

Let K be a field, ∆ a finite quiver and I an admissible ideal in

K∆. If the algebra K∆/I has finite global dimension, then it

contains no 2-truncated cycles.

Moreover, they propose the m-truncated cycles version of the no

loops conjecture.



A has no m-truncated cycles in its quiver

They show that the above conjecture holds for monoial algebaras.

We show that the conjecture holds for other algebras by the

similar way as the proof of the 2-truncated cycles version.



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Application


Theorem (Itagaki, Sanada)

Let K be a field, ∆ a finite quiver and I an ideal contained in Rm∆ .

Suppose that K∆/I contains an m-truncated cycle α1, . . . , αu.

Then for every n ≥ 1 with un ≡ 0 (mod m), the element

α(c−1)m+2 · · ·αcm ⊗ α1 ⊗ α2 · · ·αm

⊗ αm+1 ⊗ αm+2 · · ·α2m ⊗ α2m+1 ⊗ · · ·⊗ α(c−2)m+2 · · ·α(c−1)m ⊗ α(c−1)m+1

represents a nonzero element in HH2c−1(K∆/I), where

c = un/m. In particular, the Hochschild homology dimension

HHdim (K∆/I) =∞.



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Application


For an algebra A = K∆/I with I ⊂ Rm∆ , by the surjective map

A→ A′ = K∆/Rm∆ and the chain map π which induce the map

π : Hn(A⊗A′e QA′)→ Hn(A′ ⊗A′e PA′), we have the

composition map as follows:

Hn(C(A))→Hn(C(A′))∼→Hn(A

′ ⊗K∆e0(radA′)⊗

∗K∆0 )

∼→Hn(A′ ⊗A′e QA′)

π→Hn(A⊗A′e PA′)∼→Hn(L) = HHn(A

′).

We already find the basis of Hn(L) in order to compute the

dimension formula of the cyclic homology of truncated quiver

algebras. Therefore, it is enough to find an element of Hn(C(A))

whose image of the above composition map coincides with a

member of the basis of Hn(L).Tomohiro Itagaki (joint work with Katsunori Sanada) Tokyo University of Science


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Application


Corollary (Itagaki, Sanada)

Let K be an algebraically closed field, ∆ a finite quiver and I an

admissible ideal in K∆ with I ⊂ Rm∆ . If the algebra K∆/I has

finite global dimension, then it contains no m-truncated cycles.



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Application


Example

Let A be an algebra given by the quiver:

α1

β2

β1

α2

with relations:

(αiαi+1)3 = (βiβi+1)

2 = 0, α1β1β2α2 = (α1α2)2,

where the indices are modulo 2 and i is a positive integer modulo

2. Then, A has the 4-truncated cycle

β1, β2.

Therefore the global dimension of A is infinite.



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Application


[1] G. Ames, L. Cagliero, P. Tirao, Comparison morphisms and the

Hochschild cohomology ring of truncated quiver algebras, J.

Algebra 322(5)(2009), 1466–1497.

[2] P.A. Bergh, Y. Han, D. Madsen, Hochschild homology and

truncated cycles, Proc. Amer. Math. Soc. (2012), no. 4,

1133–1139.

[3] C. Cibils, Cohomology of incidence algebras and simplicial

complexes, J. Pure Appl. Algebra 56(3) (1989), 221–232.

[4] C. Cibils, Cyclic and Hochschild homology of 2-nilpotent

algebras, K-theory 4 (1990), 131–141.

[5] Y. Han, Hochschild (co)homology dimension, J. Lond. Math.

Soc. (2) 73 (2006), 657–668.

[6] K. Igusa, Notes on the no loops conjecture, J. Pure Appl.

Algebra 69 (1990), 161–176.



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Application


[7] K. Igusa, S. Liu, C. Paquette, A proof of the strong no loop

conjecture, e-print arxiv http://arxiv.org/abs/1103.5361.

[8] T. Itagaki, K. Sanada, The dimension formula of the cyclic

homology of truncated quiver algebras over a field of positive

characteristic, preprint.

[9] T. Itagaki, K. Sanada, Notes on the Hochschild homology

dimension and truncated cycles, in preparation.

[10] S. Konig, Y-M. Liu, G. Zhou, Transfer maps in Hochschild

(co)homology and applications to stable and derived invariants

and to the Auslander-Reiten conjecture, Trans. Amer. Math.

Soc. 364(2012) 195–232.

[11] H. Lenzing, Nilpotence Elemente in Ringen von endlicher

globaler Dimension, Math. Z. 108 (1969) 313–324.

[12] J-L. Loday, Cyclic homology, Springer-Verlag, Berlin (1992).



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Application


[13] E. Skoldberg, Hochschild homology of truncated and

quadratic monomial algebras, J. Lond. Math. Soc. (2) 59

(1999), 76–86.

[14] E. Skoldberg, Cyclic homology of quadratic monomial

algebras, J. Pure Appl. Algebra 156 (2001), 345–356.

[15] R. Taillefer, Cyclic homology of Hopf algebras, K-Theory 24

(2001), 69–85.



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Cyclic homology of truncated quiver algebras and notes on ...iyama/Yamagata65/Itagaki.pdf · Cyclic homology is developed as a non-commutative variant of the de Rham cohomology by