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Computing the Hochschild CohomologyGroups of Some Families of IncidenceAlgebrasMaría Andrea Gatica a & Andrea Alejandra Rey ba Instituto de Matemática, Universidad Nacional del Sur , BahíaBlanca, Argentinab Dto. de Matemática, Facultad de Ciencias Exactas y Naturales ,Universidad de Buenos Aires , Buenos Aires, ArgentinaPublished online: 01 Feb 2007.

To cite this article: María Andrea Gatica & Andrea Alejandra Rey (2006) Computing the HochschildCohomology Groups of Some Families of Incidence Algebras, Communications in Algebra, 34:6,2039-2056, DOI: 10.1080/00927870600549543

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Communications in Algebra®, 34: 2039–2056, 2006Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870600549543

COMPUTING THE HOCHSCHILD COHOMOLOGY GROUPSOF SOME FAMILIES OF INCIDENCE ALGEBRAS

María Andrea GaticaInstituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina

Andrea Alejandra ReyDto. de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad deBuenos Aires, Buenos Aires, Argentina

The purpose of this article is to present some computations of Hochschild cohomologygroups of particular classes of incidence algebras using one-point extensions and one-point coextensions.

Key Words: Hochschild cohomology; Incidence algebras.

2000 Mathematics Subject Classification: 16G20.

Let k be an algebraically closed field and A be an associative finite dimensionalk-algebra with identity. The Hoschschild cohomology groups Hi�A�X� of Awith coefficients in a finitely generated A-bimodule X were originally defined byHochschild (1945). When X = A, we write Hi�A� instead of Hi�A� A�, the ithHochschild cohomology group of the k-algebra A.

Since the computation of these groups is rather complicated by definition, oneoften tries to find alternative methods in order to calculate them. An example ofthis fact is the result due to Happel (1989), which shows the existence of a longexact sequence of k-vector spaces connecting the Hochschild cohomology groups ofa one-point extension (resp. coextension) with the Hochschild cohomology groupsof a particular quotient algebra.

Recall that an incidence algebra A is, for some n, a subalgebra of the algebraMn�k� of square matrices over k with elements �xij� ∈ Mn�k� satisfying xij = 0 ifvi �≤ vj , for some partial order ≤ defined in the poset (partially order set) �v1� � � � � vn�.

The Hochschild cohomology groups of incidence algebras have been studied inCibils (1988, 1989), Dräxler (1994), Gatica and Redondo (2001, 2003), Gerstenhaberand Schack (1983), Happel (1989), and Igusa and Zacharia (1990). It is well knownthat the Hochschild cohomology groups of an incidence algebra vanish if theassociated poset does not contain crowns (Dräxler, 1994; Igusa and Zacharia, 1990).

Received August 15, 2004; Revised July 15, 2005. Communicated by C. Cibils.Address correspondence to Andrea Alejandra Rey, Dto. de Matemática, Facultad de Ciencias

Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, C.P. 1428, BuenosAires, Argentina; Fax: +54-11-4576-3399; E-mail: aarey@bigua.dm.uba.ar

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Assume that A is an incidence algebra such that its poset P contains at least acrown. The aim of this article is to show how some computations of the Hochschildcohomology groups of incidence algebras associated to posets P, which have theproperty that each element x ∈ P, neither minimal nor maximal, is such that x↑ �=�y ∈ P � y ≥ x� has at least two minimal elements and x↓ �= �z ∈ P � x ≥ z� has atleast two maximal elements, can be done.

The consideration of these specific families is motivated by an algorithm givenby Igusa and Zacharia (1990) which allows us to reduce the study of the Hochschildcohomology groups of incidence algebras associated to posets P to the study of theHochschild cohomology groups of incidence algebras associated to subposets P ofP which satify the above condition.

The article is organized in two sections. In the first one, we briefly recall thedefinitions and results that will be needed throughout this article. The second sectionis devoted to computing the Hochschild cohomology groups of three classes ofincidence algebras associated to posets of the given type above.

1. PRELIMINARIES

1.1. Notation

Throughout this article, k denotes a fixed algebraically closed field. By analgebra we mean a finite dimensional k-algebra which we shall also assume basicand indecomposable. So an algebra A can be written as a bound quiver algebraA � kQ/I where Q is a finite connected quiver and I is an admissible ideal of thepath algebra kQ. The pair �Q� I� is called a presentation of A (for more details seeAuslander et al., 1995).

Given an algebra A, we denote by modA the category of finitely generatedleft A-modules. If A = kQ/I , then modA is equivalent to the category of all boundrepresentations of �Q� I�. We may thus identify a module M with the correspondingrepresentation �M� f� = �M�x�� f��x∈Q0��∈Q1

.For a given quiver Q, we denote by Q0 the set of vertices of Q and Q1 the

set of arrows between vertices. For each arrow �, s��� and e��� will be the startingand the ending vertices of �, respectively. For each x ∈ Q0, we denote by Sx thecorresponding simple A-module associated to x, and Px, Ix will denote the projectivecover and injective envelope of Sx, respectively. Then it is easy to see that there is avector spaces isomorphism between HomA�Px�M� and M�x�.

We say that an algebra B is a convex subcategory of A = kQ/I if there is a pathclosed full subquiver Q′ of Q such that B = kQ′/I ∩ kQ′. This means that any pathin Q with source and end in Q′ lies entirely in Q′.

The following known fact will be necessary in the sequel.

Lemma 1.1 (Michelena and Platzeck, 2000). Let B be a full convex subcategory ofA, and let X� Y in modB. Then

ExtiA�X� Y� � ExtiB�X� Y� for all i ≥ 0.

An algebra A = kQ/I is called triangular if Q has no oriented cycles. Thepresent work deals exclusively with triangular algebras.

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1.2. Incidence Algebras

Let �P = �v1� v2� � � � � vn��≤� be a finite partially ordered set (poset). Then theincidence algebra I�P� associated to the poset P is the following subalgebra of thesquare matrices over k, Mn�k�:

I�P� = ��xij� ∈ Mn�k� satisfying xij = 0 if vi �≤ vj��

The ordinary quiver associated to an incidence algebra I�P� is given as follows:The set of vertices Q0 is P, and there is an arrow vi → vj in Q1 whenever vi > vj andthere is no vs ∈ P such that vi > vs > vj . We say that two paths are parallel if theyhave the same starting and ending points. Then I�P� = kQ/I , where I is the idealgenerated by differences of parallel paths.

Remark 1.2. It is easy to describe, for an incidence algebra A = kQ/I , therepresentations of the indecomposable projective modules Px for each x ∈ Q0. Infact, Px�y� = k if y ≤ x and it is zero otherwise, and Px��� = idk if s��� ≤ x and it iszero otherwise. The dual facts hold for the representations of the indecomposableinjective modules Ix.

The following result will be useful for our computations. First, it is necessaryto recall that for a given A-module M , suppM = �x ∈ Q0 �M�x� �= 0�.

Proposition 1.3 (Gatica and Redondo, 2001). Let A be an incidence algebra andM1, M2 be two A-modules satisfying the following conditions:

S1) for any x ∈ suppMi, Mi�x� = k, i = 1� 2;S2) if � ∈ Q1 and s���� t��� ∈ suppMi then Mi��� = idk, i = 1� 2.

If ∅ �= suppM1 ⊆ suppM2 and suppM1 is connected, then HomA�M1�M2� � k.

1.3. Hochschild Cohomology

We briefly recall the construction of the Hochschild cohomology groupsHi�A� of an algebra A. Consider the A-bimodule A and the complex C• = �Ci� di�defined by: Ci = 0, di = 0 for i < 0, C0 = A, Ci = Homk�A

⊗i� A� for i > 0, where A⊗i

denotes the i-fold tensor product A⊗k · · · ⊗k A, d0 � A → Homk�A�A� is the mapd0�x��a� = ax − xa and for i > 0, di � Ci → Ci+1 is defined by

�dif��a1 ⊗ · · · ⊗ ai+1� = a1f�a2 ⊗ · · · ⊗ ai+1�

+i∑

j=1

�−1�jf�a1 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ ai+1�

+ �−1�i+1f�a1 ⊗ · · · ⊗ ai�ai+1

for f ∈ Ci and a1� � � � � ai+1 ∈ A. Then Hi�A� = Hi�C•� = Ker di/Im di−1 is the ithHochschild cohomology group of A with coefficients in A, see Hochschild (1945),Cartan and Eilenberg (1956), and Weibel (1994).

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It follows that H0�A� coincides with the center of A. Thus, if A is a basicindecomposable finite dimensional k-algebra whose quiver has no oriented cycles,then H0�A� � k.

Since the Hochschild cohomology groups of a given algebra are generally hardto compute directly from the definition, we try to use other methods. For example,we can use an inductive method to compute the Hochschild cohomology groupswhen we are dealing with triangular algebras. In fact, if A = kQA/I is a triangularalgebra, then the associated quiver has sinks and sources, and this allows us todescribe A as a one point extension (coextension) algebra in the following way.

If x is a source in QA, then the full convex subcategory Ax of A consisting ofall objects except x has QAx

as the quiver which is obtained from QA by deletingx and all arrows starting at x. Any presentation �QA� I� of A yields (by restriction)an induced presentation �QAx

� I ′� of Ax. The A-module M = radPx has a canonicalAx-module structure, and A is isomorphic to the one point extension algebra

AxM =(k 0

M Ax

)

where the operations are the usual addition of matrices and the multiplication isinduced by the Ax-module structure of M .

Dually, if x is a sink in QA, then the one point coextension MAx of A is bydefinition the finite dimensional k-algebra

MAx =(k M∗

0 Ax

)

where M∗ the k-dual of M . Observe that in this case M = Ix/soc Ix.The next theorem, due to Happel (1989), is useful for the computation of the

Hochschild cohomology groups of the algebras considered in this article. We willgive the result for one point extensions and leave it to the reader to derive thecorresponding statements for one point coextensions.

Theorem 1.4 (Happel, 1989). Let A = AxM be a one point extension of Ax by anAx-module M . Then there exists a long exact sequence of k-vector spaces connecting theHochschild cohomology of A and Ax:

0 → H0�A� → H0�Ax� → EndAx�M�/k → H1�A� → H1�Ax� → Ext1Ax

�M�M� → · · ·· · · → Hi�A� → Hi�Ax� → ExtiAx

�M�M� → Hi+1�A� → · · ·Observe that this result is specially useful when most of the terms in the

sequence vanish. In particular, we get the following consequences.

Corollary 1.5. Let A = AxM . If Hi�A� = 0 for all i > 0 and EndAx�M� = k, then

H0�A� = H0�Ax� and Hi�Ax� = ExtiAx�M�M��∀i > 0.

Corollary 1.6. Let A be an incidence algebra. If � is the unique arrow of QA startingat a source x, then Hi�A� = Hi�Ax�, ∀i ≥ 0.

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Proof. This follows immediately from Theorem 1.3 and Proposition 1.4 sinceM = radPs��� = Pe���. �

Now we recall other results about Hochschild cohomology groups of algebrasthat will be very useful for our computations.

Theorem 1.7 (Cibils, 1988; Happel, 1989). Let A = kQ be an hereditary algebra.Then

H0�A� = k� dimkH1�A� = 1− n+ ∑

�∈Q1

���� and Hi�A� = 0 ∀i ≥ 2�

where n is the number of vertices in Q and ���� = dimke����kQ�s���.

Theorem 1.8 (Gatica and Redondo, 2001). Let A = I�P� be an incidence algebra.If P is a poset with a unique maximal (minimal) element, then Hi�I�P�� = 0 for alli ≥ 1.

We denote by An−1qn+s, n ≥ 3, q ≥ 0, 0 ≤ s < n, the incidence algebra associated

to the following quiver:

That is An−1qn+s = I�Pn−1

qn+s�, where Pn−1qn+s =Pqn+s ∪ ��qn+ s + 1� p� � 1 ≤ p ≤ n− 1�

with Pqn+s = qn+ s× n− 1, where m = �0� � � � � m�, and the partial order isdefined by �l� t� < �l+ 1� t� and �l� t� < �l+ 1� t + 1� with �l� n� = �l� 0�.

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Theorem 1.9 (Gatica and Redondo, 2003). Let n ≥ 3, q ≥ 0 and 0 ≤ s < n. Then

Hi�An−1qn � =

{k if i = 0�

0 otherwise�q �= 0�

Hi�An−1qn+s� =

k if i = 0�

k if i = 2q + 1�

0 otherwise�

0 < s < n− 1�

Hi�An−1qn+�n−1�� =

k if i = 0�

kn−2 if i = 2q + 2�

0 otherwise�

2. HOCHSCHILD COHOMOLOGY COMPUTATIONS

In this section we calculate the Hochschild cohomology groups of threedifferent classes of incidence algebras whose associated posets P are in the family ofposets having the following two properties:

1) P contains crowns, that is, full subcategories of the form

where the intersection of the convex hull of �xi� yi� and the convex hull of �xi� yi−1�is �xi� (resp. the intersection of the convex hulls of �xi� yi� and �xi+1� yi� is �yi�) for0 ≤ i ≤ n− 1, with y−1 = yn−1, xn = x0, n ≥ 2.

In this case we say that the crown has length n.

2) P satisfies that for all x ∈ P, neither minimal nor maximal the poset x↑ =�y ∈ P � y ≥ x� has at least two minimal elements and the poset x↓ = �z ∈ P � x ≥ z�has at least two maximal elements (see Igusa and Zacharia, 1990).

We start now with the computations of Hochschild cohomology groups ofalgebras whose associated posets are superpositions of crowns of the same widthplus a bigger crown. In this case, the poset denoted by P>

qn+s is defined in thefollowing way:

For n ≥ 2, q ≥ 0, 0 ≤ s < n, and t� t′ ∈ � ∪ �0�

P>qn+s = Pqn+s ∪ ��qn+ s� p� � −t ≤ p < 0� ∪ ��qn+ s� p� � n− 1 < p ≤ n− 1+ t′�

∪ ��qn+ s + 1� p� � −t ≤ p ≤ n− 1+ t′�

with Pqn+s = qn+ s× n− 1 where m = �0� � � � � m�.

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The partial order in P>qn+s can be understood by looking at the following quiver

associated to the incidence algebra:

Let A>qn+s be the incidence algebra associated to the poset P>

qn+s. In the nexttheorem we compute the Hochschild cohomology groups of the algebras A>

qn+s usingprevious computations and Theorems 1.4 and 1.9. We can see that the Hochschildcohomology groups of these algebras do not depend on the numbers t andt′ chosen.

Theorem 2.1. Let A>qn+s be the algebra defined above. Suppose n ≥ 3 and t� t′ are not

simultaneously zero.

(i) If s = 0 and q > 0, then

Hi�A>qn� =

k if i = 0�

k if i = 1�

0 otherwise�

(ii) If 1 ≤ s ≤ n− 2, then

Hi�A>s � =

k if i = 0�

k2 if i = 1�

0 otherwise�

(iii) If 1 ≤ s ≤ n− 2 and q > 0, then

Hi�A>qn+s� =

k if i = 0�

k if i = 1�

k if i = 2q + 1�

0 otherwise�

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(iv) If s = n− 1, then

Hi�A>qn+�n−1�� =

k if i = 0�

k if i = 1�

kn−2 if i = 2q + 2�

0 otherwise�

Proof. Since the algebra A>qn+s is triangular, we can describe it as a one point

extension algebra as follows:

A>qn+s = A�qn+s+1�−t�M

where A�qn+s+1�−t� is the full convex subcategory of A>qn+s consisting of all objects

except �qn+ s + 1�−t� and M is the radical of the A�qn+s+1�−t� – projective moduleP�qn+s+1�−t�.

Bearing in mind that the immediate predecessors of �qn+ s + 1�−t� are�qn+ s�−t� and �qn+ s� n− 1+ t′�, we have that

M = radPqn+s+1�−t = P�qn+s�−t� ⊕ P�qn+s�n−1+t′��

Therefore, by Theorem 1.4, we obtain that

dimkH1�A>

qn+s� = 1+ dimkH1�A�qn+s+1�−t�� and

Hi�A>qn+s� = Hi�A�qn+s+1�−t�� ∀i ≥ 2

(I)

since EndA�qn+s+1�−t��M�M� = k2 and M is projective.

Let A = A�qn+s+1�−t�. We’ll find the Hochschild cohomology groups of Aeliminating the vertices belonging to the biggest crown that exceed from the blockof crowns of the same length, starting at �qn+ s + 1�−t� from left to right choosingalternatively one vertex in the stack qn+ s and the following in the stack qn+ s + 1and so on until �qn+ s� n− 1+ t′�. Then, starting at �qn+ s� n− 1+ t′� from rightto left alternating as before until �qn+ s + 1� n�.

Notice that in each step there exists a unique arrow starting or ending at one ofthese vertices. Therefore, by recurrence over these elements and using Corollary 1.6,we get that

Hi�A� = Hi�I�Pn−1qn+s��� ∀i ≥ 0 (II)

Hence the statement follows from (I), (II), and Theorem 1.9. �

Remark 2.2. ft = t′ = 0, then the result is known (see Gatica and Redondo, 2001).

Remark 2.3. If n = 2, then it is easy to see that

Hi�A>2q+s� =

k if i = 0�

k if i = 1�

0 otherwise�

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In fact, eliminating the same vertices mentioned in the proof of the previousTheorem, we obtain a subcategory of A>

2q+s whose associated poset has a uniquemaximal element. Therefore, our claim follows from Theorems 1.4 and 1.8.

The next class of incidence algebras in which we are interested is denoted byA∗

m, m ∈ � and m ≥ 2, and their posets are superpositions of crowns of differentwidth.

Let A∗m, m ≥ 2, be the incidence algebra associated to the poset P∗

m where

P∗m = ��l� t�∈ m× m such that 0≤ t≤m+ 1− l if l �= 0, or, 0≤ t≤m if l= 0�

Recall that m = �0� � � � � m�. The partial order can be understood by looking at thefollowing quiver associated to the incidence algebra:

The next three lemmas will be used to construct the projective resolutions thatwe will need for the computations of the Hochschild cohomology groups of A∗

m. Thefirst and second ones are very technical.

Lemma 2.4. Let �x� y�� �l� t� ∈ P∗m. Then �x� y� ≤ �l� t� if and only if l− x ≥ 0 and

one of the following two conditions holds:

i) 0 ≤ y − t ≤ l− x, orii) t = m+ 1− l, and �x� y� ∈ �l− 1� 0�↓ ∪ �l− 1�m+ 1− l�↓.

Proof. Suppose that l− x ≥ 0. If 0 ≤ y − t ≤ l− x, then �x� y� ≤ �x + �y − t��y − �y − t�� ≤ �x + �l− x�� t� = �l� t�. On the other hand, if the elements �x� y�� �l� t�

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verify ii), then it is clear that �x� y� ≤ �l� t� since �l− 1� 0� and �l− 1�m+ 1− l� arethe inmediate predecessors of �l�m+ 1− l�.

Conversely, if �x� y� ≤ �l� t�, then it follows directly from the definition of P∗m

that l− x ≥ 0. If l− x = 0, then it is trivial.Suppose that the elements �x� y�� �l� t� do not verify ii) with l− x > 0. If t =

m+ 1− l, then �x� y� < �l�m+ 1− l� yields a contradiction. If t �= m+ 1− l, thenwe shall prove, by induction on l− x, that i) holds. For l− x = 1 it is trivial.Assume that the result holds for 1 ≤ l− x < N and let �x� y� ≤ �x + N� t�. Then,by hypothesis, �x� y� < �x + N� t� with t �= m+ 1− �x + N�. Therefore �x� y� ≤ �x +N − 1� t� or �x� y� ≤ �x + N − 1� t + 1�. Hence, by inductive hypothesis, 0 ≤ y − t ≤N − 1 or 0 ≤ y − �t + 1� ≤ N − 1 and we get the desired result. �

Lemma 2.5. In P∗m, m = 2q + r, we have

i) �l� t�↓ ∩ �l� t + 1�↓ = �l− 1� t + 1�↓ for all l� t � 1 ≤ l ≤ m− 1, 0 ≤ t ≤ m− l− 1;ii) �l� 0�↓ ∩ �l�m− l�↓ = �l− �m− l��m− l�↓ for all l � q + r ≤ l ≤ m− 1;iii) �l� 0�↓ ∩ �l�m− l�↓ = ∅ for all l � 0 ≤ l ≤ q + r − 1.

Proof. i) If �x� y� ∈ �l� t�↓ ∩ �l� t + 1�↓, then �x� y� ≤ �l� t� and �x� y� ≤ �l� t + 1�.Hence, it follows from Lemma 2.4 that 0 ≤ y − t ≤ l− x and 0 ≤ y − �t + 1�≤ l− x

since t �= m+ 1− l and t �= m− l. In particular, 0 ≤ y − t − 1 ≤ l− x − 1 andtherefore, we get that �x� y� ≤ �l− 1� t + 1� using Lemma 2.4.

The other inclusion is clear since the immediate successors of �l− 1� t + 1� are�l� t� and �l� t + 1�.

ii) Let �x� y� ∈ �l� 0�↓ ∩ �l�m− l�↓. Then 0 ≤ y ≤ l− x and 0 ≤ y − �m− l� ≤l− x. In particular, 0 ≤ y − �m− l� ≤ l− x − �m− l� = l− �m− l�− x. Hence, byLemma 2.4, we have that �x� y� ≤ �l− �m− l��m− l�.

The other inclusion follows directly applying Lemma 2.4.

iii) Suppose that there exists an element �x� y� such that �x� y� ≤ �l� 0� and�x� y� ≤ �l�m− l�. As we have done in ii), we obtain that 0 ≤ y − �m− l� ≤ l− x −�m− l�. In particular, 0 ≤ l− x − �m− l� and therefore x ≤ 2l−m = r − 2 with rthe rest of the division of m by 2. This contradicts the fact that x ≥ 0. �

Lemma 2.6. Let M�l�t� be the radical of the A∗m-projective module P�l�t�, m = 2q + r.

a) If 2 ≤ l ≤ m and 0 ≤ t ≤ m− l, then

ExtiA∗m�M�l�t��M�l�t�� =

{k if i = 0�

0 otherwise�

b) If q + r + 1 ≤ l ≤ m and t = m+ 1− l, then

ExtiA∗m�M�l�m+1−l��M�l�m+1−l�� =

{k if i = 0�

0 otherwise�

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COHOMOLOGY OF INCIDENCE ALGEBRAS 2049

c) If 1 ≤ l ≤ q + r − 1 and t = m+ 1− l, then

ExtiA∗m�M�l�m+1−l��M�l�m+1−l�� =

{k2 if i = 0�

0 otherwise�

Proof. In order to compute the groups ExtiA∗m�M�M�, with M = M�l�t�, we consider

the following short exact sequences in modA∗m:

0 → P�l−2�t+1�

g−→ P�l−1�t� � P�l−1�t+1�

f−→ M�l�t� → 0

if 2 ≤ l ≤ m and 0 ≤ t ≤ m− l�

0 → P�l−m−2�m+1−l�

g−→ P�l−1�0� � P�l−1�m+1−l�

f−→ M�l�m+1−l� → 0

if q + r + 1 ≤ l ≤ m�

The morphisms f and g are induced respectively by the linear maps

f�x� y� = x + y� g�x� y� = �x�−x��

The exactness of the sequences constructed above follows directly from Lemma 2.5.Hence, applying the functor HomA∗

m�−�M� to these short exact sequences, we get

the desired result since the module M verifies the hypothesis of Proposition 1.3.Finally, in order to prove c), we consider 1 ≤ l ≤ q + r − 1 and t=m+ 1− l.

By Lemma 2.5, we have that M = radP�l�m+1−l� = P�l−1�0� � P�l−1�m+1−l� and thiscompletes the proof. �

In the next theorem we compute the Hochschild cohomology groups of thealgebras A∗

m using the same technique of Theorem 2.1.Let q be a natural number such that m = 2q + r, for r = 0 or r = 1.

Theorem 2.7.

Hi�A∗2q+r � =

k if i = 0�

kq+r if i = 1�

0 otherwise�

Proof. We start by recalling that for every source x in a triangular algebra A weget that A = AxMx where Ax is the full convex subcategory of A consisting of allobjects except x and Mx is the radical of the A-projective module Px.

Here the idea is to consider the algebra A∗m as one point extension of a factor

algebra and iterate this procedure until we get a subcategory of A∗m whose quiver

has a unique maximal element.In order to do this we eliminate the first q crowns of the top, i.e., all the

elements �l� t� ∈ P∗m such that q + r + 1 ≤ l ≤ 2q + r, and we do it stack by stack.

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2050 GATICA AND REY

In this way we obtain the algebras

A∗2q+r = B�2q+r�0�M�2q+r�0��

B�l�t� = B�l�t+1�M�l�t+1� if t �= m+ 1− l and

B�l�m+1−l� = B�l−1�0�M�l−1�0� otherwise.

In each step, the B�l�t�-module M�l�t� is the A∗m-module considered in condition a) or

b) of Lemma 2.6, for some l and t. Therefore, by recurrence over �l� t� and usingTheorem 1.4, we have that

Hi�A∗2q+r � = Hi�B�q+r+1�q�� for all i ≥ 0. (I)

Observe that the incidence algebra B�q+r+1�q� has the following quiver:

Afterwards, we delete the vertices of the form �l�m+ 1− l� for l = 1� � � � � q +r − 1. In this way we get algebras

B�q+r+1�q� = B�1�m�M�1�m� and B�l�m+1−l� = B�l+1�m−l�M�l+1�m−l��

Since the modules M�l�t� verify condition c) of Lemma 2.6 we have that, for all l =1� � � � � q + r − 1,

Hi�B�l�m+1−l�� = Hi�B�l+1�m−l��� i ≥ 2�

and that the following sequence

0 → k → k → k2/k → H1�B�l�m+1−l�� → H1�B�l+1�m−l�� → 0

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COHOMOLOGY OF INCIDENCE ALGEBRAS 2051

is exact. Thus, by induction on l, we obtain that

Hi�B�q+r+1�q�� =

k if i = 0�

kq+r−1+dimkH1�B�q+r−1�q+2�� if i = 1�

Hi�B�q+r−1�q+2�� otherwise�

(II)

Notice that the quiver associated to the incidence algebra B�q+r−1�q+2� is thefollowing:

In order to find the Hochschild cohomology groups of B�q+r−1�q+2� we eliminate allthe vertices �l� t� such that 2 ≤ l ≤ q + r and q + r − l ≤ t ≤ q, and we do it againstack by stack.

Now we obtain algebras

B�q+r−1�q+2� = B�q+r�0�M�q+r�0�

B�l�t� = B�l�t+1�M�l�t+1� if t �= q and

B�l�q� = B�l−1�q+r−l�M�l−1�q+r−l�

Since these modules verify the conditions a) or b) of Lemma 2.6, we obtain, byrecurrence over �l� t�, that

Hi�B�q+r−1�q+2�� = Hi�B�2�q�� ∀ i ≥ 0� (III)

Depending on the value of r, that is if r = 0 or r = 1, we have that the incidencealgebra B�2�q� has different quiver.

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2052 GATICA AND REY

In fact, if r = 0 then the quiver associated to B�2�q� is:

Otherwise, that is, if r = 1, then the quiver associated to B�2�q� is:

In order to calculate the Hochschild cohomology groups of B�2�q� we consider thisalgebra as one point extension of the algebra B�1�q� obtained eliminating the vertice�1� q� ∈ �Q�2�q��0. Then we get that:

B�2�q� = B�1�q�M�1�q�

where B�1�q� is the full convex subcategory of B�2�q� and M�1�q� = P�0�q� ⊕ P�0�q+1�.Therefore, by Theorem 1.4, we have that

Hi�B�2�q�� =

k if i = 0�

k1+dimkH1�B�1�q�� if i = 1�

Hi�B�1�q�� if i ≥ 2�

(IV)

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COHOMOLOGY OF INCIDENCE ALGEBRAS 2053

Since the incidence algebra B�1�q� does not contain crowns, we know (see Dräxler,1994; Igusa and Zacharia, 1990) that

Hi�B�1�q�� = 0 ∀ i ≥ 1� (V)

Now the statement is an immediate consequence of (I), (II), (III), (IV), and (V).�

We present now the last class of incidence algebras whose Hochschildcohomology groups are going to be calculated.

Let nt U , n ≥ 2 and t ≥ 1, be the incidence algebra obtained by superposing t

algebras of the form

where the partial order is defined by �l� 1� > �x� 0� for all 0 ≤ l, x ≤ n− 1.So, the quiver associated to n

t U is

Theorem 2.8. Let nt U be the incidence algebra defined above. Then

Hi�nt U� =

k if i = 0�

k�n−1�t+1if i = t�

0 otherwise�

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2054 GATICA AND REY

Proof. We know that H0�nt U� = k because the quiver associated to nt U has no

oriented cycles.We use induction on t to analyse its other Hochschild cohomology groups.If t = 1 then n

1U is an hereditary algebra and therefore, by Theorem 1.7, weget the desired result.

Suppose now that the theorem holds for all the algebras nkU with 1 ≤ k < t.

In order to compute the groups Hi�nt U�, i ≥ 1, we construct a sequence of algebrasby adding to n

t−1U the vertices �t� 0�� �t� 1�� � � � � �t� n− 1�, and consider each newalgebra as one point extension of the previous one. We will do this in the followingway.

Let B0 =nt−1U . Extending the algebra B0 by the point �t� 0� we obtain the

incidence algebra B1 = B0M0, where M0 = radP�t�0�, whose associated poset has aunique maximal element �t� 0�. Using the inductive hypothesis, Theorem 1.8 andProposition 1.3 we can apply Corollary 1.5 and we get that

ExtiB0�M0�M0� = Hi�B0� =

k if i = 0

k�n−1�t if i = t − 1�

0 if i �= t − 1

for i ≥ 1 (I)

Now, for all p = 2� � � � � n we consider the algebra Bp as one point extension of Bp−1

by the module Mp−1 = radP�t�p−1�. Notice that B0 is a convex subcategory of Bp andMp = M0, and therefore

ExtiBp�Mp�Mp� = ExtiB0

�M0�M0� (II)

Bearing in mind that Hi�B1� = 0� i ≥ 1, and the results (I), (II) we can deduce, byTheorem 1.4 and by recurrence on p, that

Hi�Bp� = 0 for i ≥ 1 and i �= t

and that the following sequence is exact:

0 −→ Extt−1Bp

�Mp�Mp� −→ Ht�Bp� −→ Ht�Bp−1� −→ 0

So, dimkHt�Bp�= dimkExt

p−1Bp

�Mp�Mp�+ dimkHt�Bp−1�= �p− 1�t + �p− 2��p− 1�t.

This concludes the proof of our Theorem since Bn = nt U . �

Recall that a triangular algebra is simply connected if, for every presentation�Q� I� of A, the fundamental group �1�Q� I� is trivial. If A is a schurian almosttriangular k-algebra then there exists a simple presentation for the fundamentalgroup �1�Q� I� in terms of generators and relations (Bustamante, to appear). Someresults in Assem and de la Peña (1996), Bardzell and Marcos (2002), Happel (1989),and Gerstenhaber and Schack (1983), for instance, show that there is a closerelation between the first Hochschild cohomology group H1�A� and the fundamentalgroup �1�Q� I�. Moreover, the existence of an injective morphism of Abelian groupss �Hom��1�Q� I�� k+� → H1�A� is known, for any presentation �Q� I� of A, where k+

denotes the underlying additive group of the field k (Assem and de la Peña, 1996).

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COHOMOLOGY OF INCIDENCE ALGEBRAS 2055

For the algebras considered in this article Hom��1�Q� I�� k+� � H1�A� (Pena andSaorín, 2001), and the fundamental group is independent of the presentation of thealgebra.

Remark 2.9. Using the isomorphism mentioned above and the fact that the firstHochschild cohomology group of the algebras A>

qn+s, A∗m and n

1U is nonzero we canconclude that these algebras are not simply connected.

Due to Lemma 2.1 proved by Bustamante (to appear) we can directly see thatthe fundamental groups of the algebras n

t U , t > 1, are trivial and therefore they aresimply connected algebras.

An algebra A is said to be rigid if any one-parameter deformation isisomorphic over k to the trivial one, see Gerstenhaber (1964). It is known that ifH2�A� = 0, then A is rigid. Moreover, if H3�A� = 0, the converse is also true.

Remark 2.10. From the computations we have done, we can conclude that all thealgebras A>

qn+s are rigid, except for q = 0, s = n− 1 and n ≥ 3. Also, all the algebrasA∗

2q+r andnt U are rigid, except for t = 2.

ACKNOWLEDGMENTS

The authors thank M. Julia Redondo, Andrea Solotar, and Sonia Trepode forinteresting comments and suggestions.

The first author has a fellowship from CONICET and the second author has afellowship from ANPCyT. This work has been partially financed by PICT 03-08280ANPCyT and UBACYT X062.

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