数理解析研究所講究録
第 1784 巻 2012 年 42-56

42Hochschild cohomology of quiver algebras defined by twocycles and a quantum-like relation( 2 サイクルを持つ擬量子多元環のホッホシルトコホモロジー )東京理科大学理学研究科数学専攻博士課程 3 年 小原 大樹 (Daiki Obara)
Department of Mathematics
Tokyo University of Science
Abstract
This paper is based on my talk given at the Symposium on Cohomology Theoryof Finite Groups and Related Topics held at Kyoto University, Japan, 29 August
over a fieldto 2 September 2011. In this paper, we consider quiver algebras
defined by two cycles and a quantum-like relation depending on a non-zero elementin . We determine the ring structure of the Hochschild cohomology ring of
to satisfy themodulo nilpotence and give a necessary and sufficient condition for
finiteness conditions given in [4].
$k$$A_{q}$$k$$q$$A_{q}$$A_{q}$

Introduction
Let be an indecomposable finite dimensional algebra over a field . We denote by
the enveloping algebraof , so that left -modules correspond to A-bimodules.
The Hochschild cohomology ring is given by HH $*(A)=Ext_{A^{e}}^{*}(A, A)=\oplus_{n\geq 0}Ext_{A^{e}}^{n}(A, A)$with Yoneda product. It is well-known that HH $*(A)$ is a graded commutative ring, thatis, for homogeneous elements $\eta\in HH^{m}(A)$ andHH $n(A)$ , we have.
Let
denote the ideal of HH $*(A)$ which is generated by all homogeneous nilpotent
elements. Thenis contained in every maximal ideal of HH $*(A)$ , so that the maximal
ideals of HH $*(A)$ are in 1-1 correspondence with those in the Hochschild cohomology.
ring modulo nilpotence HH’
Let be a non-zero element in andintegers with $t\geq 1$ . We consider thequiver algebra $A_{q}=kQ/I_{q}$ defined by the two cycles with $s+t-1$ vertices and $s+t$
arrows as follows:
$A$

$k$

$A\otimes_{k}A^{\sigma p}$$A$$A^{e}$$A^{e}$$\theta\in$$\eta\theta=(-1)^{mn}\theta\eta$$\mathcal{N}$$\mathcal{N}$

. ..

$q$

$(A)/\mathcal{N}$

$k$

$s,$

$t$

$s,$

$Q$

$b(2)arrow^{\beta_{2}}b(3)$$a(3)a(2)\underline{\alpha 2}$.

$\backslash ^{\alpha_{1}}$$\prime^{3}\alpha$and the ideal$I_{q}$$\nearrow^{\beta_{1}}$$\backslash ^{\beta_{3}}$$\nearrow_{\alpha_{s}}^{1}\backslash _{\beta_{t}}$$.\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$.

$..arrow\alpha_{s-1}a(s)$$\cdot$$b(t)\overline{\beta_{t-1}}$

of $kQ$ generated by$X^{sa},$ $X^{s}Y^{t}-qY^{t}X^{s},$ $Y^{tb}$for $b\geq 2$ where we set
and $Y:=\beta_{1}+\beta_{2}+\cdots+\beta_{t}$ . We denotethe trivial path at the vertex $a(i)$ and at the vertex $b(j)$ by
and byrespectively.
We regard the numbers in the subscripts ofmodulo and in the subscripts ofmodulo . In this paper, we describe the ring structure of HH.
$X:=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}$$a,$$e_{b(j)}$$e_{a(i)}$$i$$e_{b(j)}$$t$$e_{a(i)}$

$s$

$j$

$*(A_{q})/\mathcal{N}$43
In [19], Snashall and Solberg used the Hochschild cohomology ring modulo nilpotence
HH
to define a support variety for any finitely generated module over . This
led us to consider the structure of HH. In [19], Snashall and Solberg conjectured
that HHis always finitely generated as a k-algebra. But a counterexample tothis conjecture was given by Snashall [18] and Xu [23]. This example makes us consider
whether we can give necessary and sufficient conditions on a finite dimensional algebraforto be finitely generated as a k-algebra.
On the other hand, in the theory of support varieties, it is interesting to know whenthe variety of a module is trivial. In [4], Erdmann, Holloway, Snashall, Solberg andTaillefer gave the necessary and sufficient conditions on a module for it to have trivial
variety under some finiteness conditions on A. In the paper, we show thatsatisfies
the finiteness conditions given in [4] if and only if is a root of unity.
The content of the paper is organized as follows. In Section 1 we deal with thedefinition of the support variety given in [19] and precedent results about the Hochschildcohomology ring modulo nilpotence. In Section 2, we describe the finiteness conditions
given in [4] and introduce precedent results about these conditions. In Section 3, we
determine the Hochschild cohomology ring ofmodulo nilpotence and show thatsatisfies the finiteness conditions if and only if is a root of unity.
$A$$*(A)/\mathcal{N}$$*(A)/\mathcal{N}$$*(A)/\mathcal{N}$$A$$HH^{*}(A)/\mathcal{N}$$A_{q}$$q$$A_{q}$$A_{q}$$q$1

Support variety

In [19], Snasall and Solberg defined the support variety of a finitely generated A-module$M$ over a noetherian commutative graded $suba!gebraH$ of HH $*(A)$ with $H^{0}=HH^{0}(A)$ .
In this paper, we consider the case $H=HH^{*}(A)$ .

Definition 1.1 ([19]). The support variety of $M$ is given by$V(M)=$ { $m\in$ MaxSpec HH $*(A)/\mathcal{N}|$ AnnExt $A*(M,$ $M)\subseteq m’$ }where AnnExt $A*(M, M)$ is the annihilator of $Ext_{A}^{*}(M, M),$ $m’$ is the pre-image of $m$ for
the natural epimorphism and the HH $*(A)$ -action on $Ext_{A}^{*}(A, A)$ is given by the gradedalgebra homomorphism $HH^{*}(A)^{-\otimes M}arrow Ext_{A}^{*}(M, M)$ .
Since is indecomposable, we have that $HH^{0}(A)$ is a local ring. Thus
has a unique maximal graded ideal $m_{gr}=$ rad HH $*(A)$ , HH
. We say that the$M$variety ofis trivial if $V(M)=\{m_{gr}\}$ .
$A$$HH^{*}(A)/\mathcal{N}$$\geq 1(A)\rangle/\mathcal{N}$$\langle$

In [18], Snashall gave the following question.

Question ([18]). Whether we can give necessary and sufficient conditions on a finite
dimensional algebra for the Hochschild cohomology ring modulo nilpotence to be finitely
generated as a k-algebra.

With respect to sufficient condition, it is shown that HH
is finitely generated
as a k-algebra for various classes of algebras by many authors as follows:$*(A)/\mathcal{N}$(1) In [6], [22], Evens and Venkov showed that HH
block of a group ring of a finite group.

$*(A)/\mathcal{N}$is finitely generated for any(2) In [7], Friedlander and Suslin showed that HH
is finitely generated for any
block of a finite dimensional cocommutative Hopf algebra.
$*(A)/\mathcal{N}$44
is finitely generated(3) In [11], Green, Snashall and Solberg showed that HH
for finite dimensional self-injective algebras of finite representation type over analgebraically closed field.
$*(A)/\mathcal{N}$(4) In [12], Green, Snashall and Solberg showed that HH
for finite dimensional monomial algebras.
(5) In [13], Happel showed that HH
algebras of finite global dimension.

$*(A)/\mathcal{N}$$*(A)/\mathcal{N}$is finitely generated

is finitely generated for finite dimensional

is finitely generated for the(6) In [17], Schroll and Snashall showed that HHwith $q=-1$ defined by the quiver
principal block of the Heche algebra$*(A)/\mathcal{N}$$H_{q}(S_{5})$$Q:\epsilon C1\vec{\frac{\alpha}{\overline{\alpha}}}2\mathfrak{O}\overline{\epsilon}$

and the ideal$I$of $kQ$ generated by$\epsilon^{2}-\alpha\overline{\alpha},\overline{\epsilon}^{2}-\overline{\alpha}\alpha$$\alpha\overline{\epsilon},\overline{\alpha}\epsilon,\overline{\epsilon\alpha},$(7) In [20], Snashall and Taillefer showed that HHclass of special biserial algebras.

.

$*(A)/\mathcal{N}$is finitely generated for a(8) In [14], Koenig and Nagase produced many examples of finite dimensional algebras

with a stratifying ideal for which HH$*(A)/\mathcal{N}$is finitely generated as a k-algebra.

(9) In [18] and [23], Snashall and Xu gave the example of a finite dimensional algebra
for which HH
is not a finitely generated k-algebra.
$*(A)/\mathcal{N}$

Example 1.2. ([18, Example 4.1]) Let $A=kQ/I$ where$Q$

is the quiverゆ$1arrow^{c}2$$t_{\tilde{b}})$and $I=\{a^{2},$$b^{2}$, ab–ba, $ac\}$ . Then Snashall showed the following in [18, Theorem

4.5].

(a) HH$*(A)/\mathcal{N}\cong\{\begin{array}{ll}k\oplus k[a, b]b if char k=2,k\oplus k[a^{2}, b^{2}]b^{2} if char k\neq 2.\end{array}$(b) HH$*(A)/\mathcal{N}$is not finitely generated as a -algebra.
$k$

Xu showed this in the case char $k=2$ in [23].

2Finiteness conditions

In [4], Erdmann, Holloway, Snashall, Solberg and Taillefer gave the following two conditions (Fgl) and (Fg2) for an algebra and a graded subalgebra $H$ of HH $*(A)$ .
$A$(Fgl) $H$ is a commutative Noetherian algebra with $H^{0}=HH^{0}(A)$ .

45(Fg2) $Ext_{A}^{*}(A/radA, A/radA)$ is a finitely generated H-module.
In [4], under the finiteness conditions above, some geometric properties of the support
variety and some representation theoretic properties are related. In particular, the following theorem hold.

Theorem 2.1 ([4, Theorem 2.5]). Suppose that(a)is Gorenstein, that is,and as a rright A-module.

$A$

$A$

satisfies the finiteness conditions.
has finite injective dimension both as a left A-module$A$(b) The following are equivalent for an A-module $M$ .
(i) The variety of $M$ is trivial.
(ii) The projective dimension of $M$ is finite.

(iii) The injective dimension of $M$ is finite.

There are some papers which deal with the finiteness conditions as follows.
(1) In [2], Bergh and Oppermann show that a codimension

quantum complete intersection satisfies the finiteness conditions if and only if all the commutators
are roots of unity.
$n$$q_{ij}$Definition 2.2. A codimension

quantum complete intersection is defined by$n$$k\langle x_{1},$$\ldots,$$x_{n}\}/I$

where I genemted by$x_{i}^{a_{i}},$$x_{j}x_{i}-q_{ij}x_{i}x_{j}$for $1\leq i<j\leq n,$$a_{i}\geq 2,$ $q_{ij}\in k$.

(2) In [5], Erdmann and Solberg gave the necessary and sufficient conditions on a
Koszul algebra for it to satisfy the finiteness conditions.

Theorem 2.3 ([5, Theorem 1.3]). Let
be a finite dimensional Koszul algebm
over an algebmically closed field, and let $E(A)=Ext_{A}^{*}(A/radA, A/radA)$ .
satisfies the finiteness conditions if and only if $Z_{gr}(E(A))$ is Noethenan and $E(A)$is a finitely genemted $Z_{gr}(E(A))$ -module.
$A$

$A$

(3) In [9], Furuya and Snashall provided examples of $(D, A)$ -stacked monomial algebras
which are not self-injective but satisfy the finiteness conditions.

Example 2.4. ([9, Example 3.2]) Let$Q$

be the quiver$\delta_{4}\uparrow 1arrow 3arrow 2\gamma\alpha\downarrow$

and

$I$

$\beta$the ideal of $kQ$ generated by$\alpha\beta\gamma\delta\alpha\beta,$.

$\gamma\delta\alpha\beta\gamma\delta$

Then, $A=kQ/\mathcal{I}$ is not self-injective but satisfies the finiteness conditions.
(4) In [17], Schroll and Snashall show that the finiteness conditions hold for the principal block of the Heche algebrawith $q=-1$ .
$H_{q}(S_{5})$46
Hochschild cohomology ring of quiver algebras definedby two cycles and a quantum-like relation

3

In this section, we consider the quiver algebras $A_{q}=kQ/I_{q}$ defined by the quiver
follows:$Q$

as$b(2)arrow^{\beta_{2}}b(3)$$a(3)arrow^{\alpha_{2}}a(2)$$\gamma^{\alpha_{3}}$$\backslash ^{\beta_{3}}$$\backslash _{1}^{\alpha_{1}}\nearrow^{\beta_{1}}$$............$.
$arrow\alpha_{s-1}a(s)\nearrow^{\alpha_{s}}$and the ideal$I_{q}$$\backslash _{b(t)}\beta_{t}\overline{\beta_{t-1}}.\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$of $kQ$ generated by$X^{sa},$ $X^{s}Y^{t}-qY^{t}X^{s},$ $Y$め

for $b\geq 2$ where we set $X:=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}$ and $Y:=\beta_{1}+\beta_{2}+\cdots+\beta_{t}$ , andis non-zero element in . Paths are written from right to left. We will determine themodulo nilpotence HH
Hochschild cohomology ring ofand show thatsatisfies the finiteness conditions if and only if is a root of unity.
First, we note that the following elements inform a k-basis of .
$a,$

$q$

$k$

$*(A_{q})/\mathcal{N}$$A_{q}$$A_{q}$$q$$A_{q}$$A_{q}$$X^{sl+l’}e_{a(i)}$for $2\leq i\leq s,$ $0\leq l\leq a-1,0\leq l’\leq s-1$ ,$Y^{tl+l’}e_{b(j)}$for $1\leq j\leq t,$ $0\leq l\leq b-1,0\leq l’\leq t-1$ ,$X^{si+l}Y^{tj+l’}$for $0\leq i\leq a-1,0\leq j\leq b-1,1\leq l\leq s-1,0\leq l^{l}\leq t-1$ ,$X^{si}Y^{tj+l’}$for $1\leq i\leq a-1,0\leq j\leq b-1,0\leq l’\leq t-1$ ,$X^{si}Y^{tj}X^{l}$for $0\leq i\leq a-1,1\leq j\leq b-1,1\leq l\leq s-1$ ,$Y^{l’}X^{si}Y^{tj}$for $1\leq i\leq a-1,0\leq j\leq b-1,1\leq l’\leq t-1$ ,$Y^{l’}X^{si}Y^{tj}X^{l}$for $0\leq i\leq a-1,0\leq j\leq b-1,1\leq l\leq s-1,1\leq l’\leq t-1$ ,$X^{si+l}Y^{tj}X^{l’}$for $0\leq i\leq a-1,1\leq j\leq b-1,1\leq l,$ $l’\leq s-1$ ,$Y^{l}X^{si}Y^{tj+l’}$for $1\leq i\leq a-1,0\leq j\leq b-1,1\leq l,$ $l’\leq t-1$ .

So we have $\dim_{k}A_{q}=ab(s+t-1)^{2}$ .

3.1

Projective resolution of

For $n\geq 0$ , we define left

$A_{q}^{e}$$A_{q}$-modules, equivalently$A_{q}$-bimodules,$P_{2n}= \coprod_{l=0}^{2n}A_{q}e_{1}\otimes e_{1}A_{q}\oplus\prod_{i=2}^{s}A_{q}e_{a(i)}\otimes e_{a(i)}A_{q}\oplus\coprod_{j=2}^{t}A_{q}e_{b(j)}\otimes e_{b(j)}A_{q}$$P_{2n+1}= \coprod_{l=1}^{2n}A_{q}e_{1}\otimes e_{1}A_{q}\oplus\prod_{i=1}^{s}A_{q}e_{a(i+1)}\otimes e_{a(i)}A_{q}\oplus\coprod_{j=1}^{t}A_{q}e_{b(j+1)}\otimes e_{b(j)}A_{q}$where$U_{l=1}^{0}A_{q}e_{1}\otimes e_{1}A_{q}=0$

are labeled$\epsilon_{l}^{2n}$for. The generators$0\leq l\leq 2n,$$\epsilon_{a(i)}^{2n}$for$e_{1}\otimes e_{1},$$2\leq i\leq s$$e_{a(i)}\otimes e_{a(i)}$, and$\epsilon_{b(j)}^{2n}$

for

and

of
respectively.

$e_{b(j)}\otimes e_{b(j)}$$2\leq j\leq t$,$P_{2n}$47
Similarly, we denote the generators
of $P_{2n+1}$ by
and$1\leql\leq
2n,$$1\leqi\leq
s$$1\leq
j\leq
t$
for
, and
for
respectively. In [15],forwe give the minimal projective bimodule resolution of
as follows.
$e_{1}\otimes e_{1},$$\epsilon_{l}^{2n+1}$$e_{a(i+1)}\otimes e_{a(i)}$$\epsilon_{a(i)}^{2n+1}$$e_{b(j+1)}\otimes e_{b(j)}$$\epsilon_{b(j)}^{2n+1}$$A_{q}$

Theorem 3.1 ([15, Theorem 1.1]). The following sequence
resolution of the left:
-module$A_{q}^{e}$$\mathbb{P}$is a minimal pmjective$A_{q}$$\mathbb{P}:\cdotsarrow P_{2n+1}arrow d_{2n+1}P_{2n}arrow P_{2n-1}d_{2n}arrow\ldotsarrow P_{2}arrow^{d_{2}}P_{1}arrow^{d_{1}}P_{0}arrow^{\pi}A_{q}arrow 0$

where $\pi:P_{0}arrow A_{q}$ is the multiplication map, and we define left
and $d_{2n+2}$ by$A^{e}$.

-homomorphisms$d_{2n+1}$$d_{2n+1}$:$\{\begin{array}{l}\epsilon_{b(j)}^{2n+1}\mapsto\epsilon_{b(j+1)}^{2n}Y-Y\epsilon_{b(j)}^{2n} for 1\leq j\leq t,\epsilon_{l}^{2n+1}\mapsto[Case]\epsilon_{\text{ゆ}(i)}^{2n+1}\mapsto\epsilon_{a(i+1)}^{2n}X-X\epsilon_{a(i)}^{2n} for 1\leq i\leq s,\end{array}$$d_{2n+2}$:$\{\begin{array}{l}\epsilon_{0}^{2n+2}\mapsto\sum_{l=0}^{b-1}\sum_{l=0}^{t-1}Y^{tl+l’}\epsilon_{b(t-l)}^{2n+1}Y^{t(b-1-l)+t-l’-1},\epsilon_{b(j)}^{2n+2}\mapsto\sum_{l=0}^{b-1}\sum_{l’=0}^{t-1}Y^{tl+l’}\epsilon_{b(j-1-l’)}^{2n+1}Y^{t(b-1-l)+t-l’-1} for 2\leq j\leq t,\epsilon_{1}^{2n+2}\mapsto qY^{t}\epsilon_{1}^{2n+1}-\epsilon_{1}^{2n+1}Y^{t}-\sum_{j=1}^{t}X^{s}Y^{t-j}\epsilon_{b(j)}^{2n+1}Y^{j-1}+q^{bn+1}\sum_{j=1}^{t}Y^{t-j}\epsilon_{b(j)}^{2n+1}Y^{j-1}X^{s},\epsilon_{l}^{2n+2}\mapsto\epsilon_{2n+1}^{2n+2}\mapsto[Case]q^{an+1}\sum_{i=1}^{s}Y^{t}X^{s-i}\epsilon_{a(i)}^{2n+1}X^{i-1}-\sum_{i=1}^{s}X^{s-i}\epsilon_{a(i)}^{2n+1}X^{i-1}Y^{t}-X^{s}\epsilon_{2n}^{2n+1}+q\epsilon_{2n}^{2n+1}X^{s},\epsilon_{a(i)}^{2n+2}\mapsto\sum_{l=0}^{a-1}\sum_{l=0}^{s-1}X^{sl+l’}\epsilon_{a(i-1-l’)}^{2n+1}X^{s(a-1-l)+s-l’-1} for 2\leq i\leq s,\epsilon_{2n}^{2n}\ddagger_{2}^{2}\mapsto\sum_{l=0}^{a-1}\sum_{l=0}^{s-1}X^{sl+l’}\epsilon_{a(s-l’)}^{2n+1}X^{s(a-1-l)+s-l’-1},\end{array}$for$n\geq 0$, where in the case $n=0,$$\epsilon_{l}^{1},$, and$\epsilon_{l}^{2},$, vanish, and the image of$\epsilon_{1}^{2}$by$d_{2}$

is$- \sum_{j=1}^{t}X^{s}Y^{t-j}\epsilon_{b(j)}^{1}Y^{j-1}+q\sum_{j=1}^{t}Y^{t-j}\epsilon_{b(j)}^{1}Y^{j-1}X^{s}$.

$+q \sum_{i=1}^{s}Y^{t}X^{s-i}\epsilon_{a(i)}^{1}X^{i-1}-\sum_{i=1}^{s}X^{s-i}\epsilon_{a(i)}^{1}X^{i-1}Y^{t}$

3.2

Hochschild cohomology group of$A_{q}$

Next, we give a basis of the n-th Hochschild cohomology group HH $n(A_{q})$for $n\geq 0$ , using the minimal projective -resolution given in Theorem 3.1. Now we$:=Ext_{A_{q}^{e}}^{n}(A_{q}, A_{q})$$A_{q}^{e}$

48
consider the case $t\geq 2$ . In the case $s=1$ or $t=1$ , we can give a basis of $HH^{n}(A_{q})$ by
the same method.
Now, we consider the vector space structure of $HH^{n}(A_{q})$ for all $n\geq 0$ . By thedefinition of , we have isomorphisms$s,$$P_{n}$$u_{2n}:$u_{2n+1}:

where$P_{2n+1}^{*}$,

Hom_{A_{q}^{e}}(P_{2n},A_{q})arrow-IIe_{1}A_{q}e_{1}\oplus\prod_{i=2}^{s}e_{a(i)}A_{q}e_{a(i)}\oplus\prod_{j=2}^{t}e_{b(j)}A_{q}e_{b(j)}$Hom_{A_{q}^{e}}(P_{2n+1},A_{q})arrow\sim\prod_{l=1}^{2n}e_{1}A_{q}e_{1}\oplus\prod_{i=1}^{s}e_{a(i+1)}A_{q}e_{a(i)}\oplus\prod_{j=1}^{t}e_{b(j+1)}A_{q}e_{b(j)}$$\prod_{l=1}^{0}e_{1}A_{q}e_{1}=0$. We denote the k-modules

respectively. We see that the dimension of$P_{n}^{*}$

andis given by${\rm Im} u_{2n}$${\rm Im} u_{2n+1}$by$P_{2n}^{*}$,

and$\dim_{k}P_{n}^{*}=nab+ab(s+t-1)$ .

for $0\leq l\leq 2n,$for $2\leq i\leq s$ofare labeled
andThe elements
andfor $2\leq j\leq t$ respectively. Similarly, we denote the elements
andfor $1\leq i\leq s$ andfor $1\leq j\leq t$
byof
for $1\leq l\leq 2n,$
. Using the maps
respectively. These labels correspond to that of generators of$n\geq0$$d_{2n+1},$$d_{2n+2}$, we obtain the following diagram:$u_{2n+1},$for$e_{1},$$e_{a(i)}$$e_{l}^{2n}$$P_{2n}^{*}$$e_{b(j)}$$e_{a(i)}^{2n}$$e_{1},$$e_{b(j)}^{2n}$$e_{l}^{2n+1}$$P_{2n+1}^{*}$$e_{b(j)}$$e_{a(i)}$$e_{b(j)}^{2n+1}$$e_{a(i)}^{2n+1}$$P_{n}$$u_{2n},$$0$$arrow^{d_{2}^{*}}$$arrow^{d_{1}^{*}}$$0arrow$$P_{1}^{*}$$P_{0}^{*}$

where we putcomplex$\sim\downarrow u2$$\sim\downarrow u_{1}$$\sim\downarrow u0$$\overline{d}_{n}=Hom_{A_{q}^{e}}(d_{n}, A_{q}),$$d_{n}^{*}=u_{n}\overline{d}_{n}u_{n-1}^{-1}$$arrow^{d_{3}^{*}}$$P_{2}^{*}$

See [16] for the homomorphism$d_{n}^{*}$... ,for $n\geq 1$ . Hence we have the$\mathbb{P}^{*}:0arrow P_{0}^{*}arrow^{d_{1}^{*}}P_{1}^{*}arrow\cdotsarrow P_{n-1}^{*}arrow^{d_{n}^{*}}P_{n}^{*}arrow\cdots$$\bullet$...

$Hom_{A_{q}^{e}}(P_{0}, A_{q})arrow^{d_{1}\overline}Hom_{A_{q}^{e}}(P_{1}, A_{q})arrow^{\overline d_{2}^{\backslash }}Hom_{A_{q}^{e}}(P_{2}, A_{q})arrow^{\overline d_{3}}$—.

. Now, we denote some elements of$P_{n}^{*}$

as follows:

For $n=0$ :$T_{l,l’}:=X^{sl} Y^{tl’}e_{0}^{0}+\sum_{j=2}^{t}Y^{j-1}X^{sl}Y^{t(l’-1)+t-j+1}e_{b(j)}^{0}$$+ \sum_{i=2}^{s}X^{s(l-1)+i-1}Y^{tl’}X^{s-i+1}e_{a(i)}^{0}$$\bullet$

For

$n$

for $1\leq l\leq a$ and $1\leq l’\leq b$ ,odd, $n\geq 1$ :$U_{0,l,l}^{n}$, $:=YX^{sl}Y^{tl’}e_{b(1)}^{n}$ for $0\leq l\leq a-1$ and $0\leq l^{l}\leq b-1$ ,$U_{m,l,l}^{n}$$U_{n,l,l}^{n}$, $:=X^{sl}Y^{tl’}e_{m}^{n}$ for $1\leq m\leq n-1,0\leq l\leq a-1$ and $0\leq l’\leq b-1$ ,, $:=X^{sl+1}Y^{tl’}e_{a(1)}^{n}$ for $0\leq l\leq a-1$ and $0\leq l^{l}\leq b-1$ ,49$\bullet$

For

$n$

even, $n\geq 2$ :$W_{0,l,l’}^{n} \cdot=X^{sl}Y^{tl’}e_{0}^{n}+\sum_{j=2}^{t}Y^{j-1}X^{sl}Y^{t(l’-1)+t-j+1}e_{b(j)}^{n}$for $0\leq l\leq a-1$ and $0\leq l’\leq b$ ,$W_{0,l,b-1}^{n*}:=bX^{sl} Y^{t(b-1)}e_{0}^{n}+b\sum_{j=2}^{t}Y^{j-1}X^{sl}Y^{t(b-2)+t-j+1}e_{b(j)}^{n}$for $0\leq l\leq a-1$ ,$+(q^{b(n/2-1)+1}-1)X^{s(l+1)}e_{1}^{n}$$W_{m,l,l’}^{n}\cdot=X^{sl}Y^{tl’}e_{m}^{n}$for $1\leq m\leq n-1,0\leq l\leq a-1$ and $0\leq l’\leq b-1$ ,$W_{n,l,l’}^{n}:=X^{sl}Y^{tl’}e_{n}^{n}+ \sum_{i=2}^{s}X^{s(l-1)+i-1}Y^{tl’}X^{s-i+1}e_{a(i)}^{n}$for $0\leq l\leq a$ and $0\leq l’\leq b-1$ ,$W_{n,a-1,l’}^{n*}:=aX^{s(a-1)} Y^{tl’}e_{n}^{n}+a\sum_{i=2}^{s}X^{s(a-2)+i-1}Y^{tl’}X^{s-i+1}e_{a(i)}^{n}$for $0\leq l’\leq b-1$ .

$+(q^{a(n/2-1)+1}-1)Y^{t(l’+1)}e_{n-1}^{n}$In the following results we use the complex
to compute a basis of the Hochschildcohomology group $HH^{n}(A_{q})=Kerd_{n+1}^{*}/{\rm Im} d_{n}^{*}$ of our algebra
for $n\geq 0$ . First, weconsider the case where is an r-th root of unity for integer $r\geq 1$ . Now, we set 7 is theremainder when we divide by for any integer . Then we have.
$\mathbb{P}^{*}$$A_{q}$

$q$

$z$

$r$

$0\leq\overline{z}\leq r-1$

$z$

Theorem 3.2 ([15, Proposition 3.3] and [16, Theorem 2.1]). If is an r-th mot of unity
for $r\geq 1$ and $t\geq 2$ , then the following elements form a basis of $HH^{n}(A_{q})$ .
$q$

$s,$

(1) Basis of $HH^{0}(A_{q})$ :

(a)

(b)
(c)
(d)

$1_{A}=e_{0}^{0}+ \sum_{j=2}^{t}e_{b(j)}^{0}+\sum_{i=2}^{s}e_{a(i)}^{0}$$T_{l,l’}$$T_{l,b}$$T_{a,l’}$,for $1\leq l\leq a-1,1\leq l’\leq b-1$ if
for $1\leq l\leq a-1$ ,for $1\leq l’\leq b-1$ ,$\overline{l}=\overline{l’}=0$,(2) Basis of $HH^{2n}(A_{q})$ for $n\geq 1$ :

(a)
(b)

(c)
(d)

$W_{0,0,l}^{2n}$$W_{0,l,l}^{2n}$, for $0\leq l’\leq b-1if\overline{l’}=\overline{bn}$ ,, for $1\leq l\leq a-1,1\leq l’\leq b-1if\overline{l}=0,$$W_{0,l,b}^{2n}$for $1\leq l\leq a-1$ if$W_{0,a-1,l}^{2n}$$\overline{l}=0,$, for $0\leq l^{l}\leq b-1$ if$\overline{b(n-1)}=0$$\overline{a}=1,$$\overline{l’}=\overline{bn}$,, char $k|b$ ,$\overline{bn-l’}\neq 0$,(e)$\{W_{l,l+l,0}^{2n}for0\leq l\leq a-2if^{\overline{\frac{l}{l}}}=0^{\overline{\frac{b(n-1)+1}{b(n-1)+1}}}\neq 0,chark|bW_{0,l,b-1}^{2n*}for0\leq l\leq a-1if=0,\neq 0,chark(b$(f)$W_{1,l+1,1}^{2n}$(g)for $1\leq l\leq a-2$ if$\overline{l}=0,$$\overline{b(n-1)}\neq 0$,$\{\begin{array}{l}W_{2l-1,l,l}^{2n}, for 1\leq l\leq a-1,1\leq l^{l}\leq b-1 if char k(a, char k|b,W_{2l-1,l,l}^{2n}, for 1\leq l\leq a-1,0\leq l’\leq b-1 if char k(a, char k|b,W_{2l-1,l,l}^{2n}, for 0\leq l\leq a-1,1\leq l’\leq b-1 if char k|a, char k\{bW_{2l-1,l,l}^{2n}, for 0\leq l\leq a-1,0\leq l’\leq b-1 if char k|a, char k|b,\end{array}$for$1\leq l^{\prime l}\leq n$

if$\overline{l}=\overline{a(l’’-1)+1},$$\overline{l’}=\overline{b(n-l^{\prime l})+1}$

,

50

(h)

$\{\begin{array}{l}W_{2l,l,l}^{2n}, for 0\leq l\leq a-2,0\leq l’\leq b-2 if char k(a, char k(b,W_{2l,l,l}^{2n}, for 0\leq l\leq a-2,0\leq l’\leq b-1 if char k(a, char k|b,W_{2l,l,l}^{2n}, for 0\leq l\leq a-1,0\leq l’\leq b-2 if char k|a, char k(bW_{2l,l,l}^{2n}, for 0\leq l\leq a-1,0\leq l^{l}\leq b-1 if char k|a, char k|b,\end{array}$for$1\leq l^{Jl}\leq n-1$

if$\overline{l’}=\overline{b(n-l^{\prime l})}$$\overline{l}=\overline{al^{l\prime}},$(i) $W_{2n-1,1,l+1}^{2n}$ for $1\leq l’\leq b-2$ if(j)$\overline{l’}=0,$,$\overline{a(n-1)}\neq 0$$\{W_{2n,a-1,l’}^{2n*},for0\leq l’\leq b-lif\overline{l’}=0,\overline{a(}0,chark\{aW_{2n-1,0,l+1}^{2n}for0\leq l’\leq b-2if\overline{l^{l}}=0^{\frac{n-l)+1\neq}{a(n-1)+l}}\neq 0,chark|a$,for $0\leq l\leq a-1$ if, for $1\leq l\leq a-1,1\leq l’\leq b-1$ if(1), for $1\leq l’\leq b-1$ if(m)(n)
for $0\leq l\leq a-1$ if(o) Additionally in the case of $q=-1$ :.
for $0\leq l\leq a-1if\overline{an}=0,$,ii.
for $1\leq l\leq a-1$ if
,
iii.
if
for
iv.
for $1\leq l’’\leq n-1$ if, for $1\leq l’\leq b-1$ if.
, for $0\leq l’\leq b-1$ if
vi.

(k)

,

$\overline{l}=\overline{an}$$W_{2n,l,0}^{2n}$$\overline{l’}=0,$$W_{2n,l,l}^{2n}$$\overline{b}=1,$$W_{2n,l,b-1}^{2n}$$i$$\overline{b}=0,$$\overline{b}=0,$$1\leq l’’\leq n$$W_{2l-1,0,0}^{2n}$$\overline{l}=1$,$\overline{l}=0$$\overline{a}=\overline{b}=0$$\overline{a}=\overline{b}=0$$W_{2l,a-1,b-1}^{2n}$$v$$\overline{an-l}\neq 0$$W_{0,l,b}^{2n}$$W_{1,l,0}^{2n}$,, char $k|a$ ,$\overline{a(n-1)}=0$$\overline{l’}=0,$$W_{2n,a,l}^{2n}$$\overline{l}=\overline{an}$$\overline{l’}=0$$\overline{a}=0,$$W_{2n-1,0,l}^{2n}$$\overline{a}=0,$$W_{2n,a,l}^{2n}$$\overline{bn}=0,$,,$\overline{l’}=1$,(3) Basis of $HH^{2n+1}(A_{q})$ for $n\geq 0$ :for $0\leq l\leq a-1,1\leq l’\leq b-1$ if, for $1\leq l’\leq b-1$ if,$b(n-1)+1\neq 0$ ,
for

(a)

$U_{0,l,l}^{2n+1}$(b)$U_{0,a-1,l}^{2n+1}$(c)$U_{0,l,b-1}^{2n+1}$(d)$\{\begin{array}{l}U_{0,0,0}^{2n+1}if\overline{bn}=0, char k(b,U_{0,l,0}^{2n+1} for 0\leq l\leq a-1if\overline{l}=0,\overline{bn}=0, char k|b,\end{array}$(e)$U_{1,l+1,0}^{2n+1}$(f)$\overline{a}=1,$$\overline{l}=0$,$\overline{bn-l’}\neq 0$$0\leq l\leq a-1if\overline{l}=0,$

for $0\leq l\leq a-2$ if$\overline{l}=0,$$\overline{bn}\neq 0$,$\{\begin{array}{l}U_{2l,l,l}^{2n+1}, for 0\leq l\leq a-2,1\leq l’\leq b-1 if char k(a, char k(b,U_{2l,l,l}^{2n+1}, for 0\leq l\leq a-1,1\leq l^{l}\leq b-1 if char k|a, char k\{b,U_{2l,l,l}^{2n+1}, for 0\leq l\leq a-2,0\leq l^{l}\leq b-1 if char k(a, char k|b,U_{2l,l,l}^{2n+1}, for 0\leq l\leq a-1,0\leq l^{l}\leq b-1 if char k|a, char k|b,\end{array}$

for
(g)

$\overline{l’}=\overline{bm},$$1\leq l’’\leq n$

if$\overline{l}=\overline{al’’},$$\overline{l’}=\overline{b(n-l’’)+1}$,$\{\begin{array}{l}U_{2l+1,l,l}^{2n+1}, for 1\leq l\leq a-1,0\leq l’\leq b-2 if char k \dagger a, char k(b,U_{2l+1,l,l}^{2n+1}, for 1\leq l\leq a-1,0\leq l’\leq b-1 if char k(a, char k|b,U_{2l+1,l,l}^{2n+1}, for 0\leq l\leq a-1,0\leq l’\leq b-2 if char k|a, char k\{b,U_{2l+1,l,l}^{2n+1}, for 0\leq l\leq a.-1,0\leq l’\leq b-1 if char k|a, char k|b,\end{array}$for$0\leq l^{lJ}\leq n-1if\overline{l}=\overline{al^{l\prime}+1},$$\overline{l’}=b(n-l’’)$

,

51
(h)

,
for
, for $1\leq l\leq a-1,0\leq l’\leq b-1if\overline{l’}=0,$, for,
for $1\leq l\leq a-1$ if$0\leq l’\leq b-2if\overline{l’}=0,$$U_{2n,0,l+1}^{2n+1}$(i)$U_{2n+1,l,l}^{2n+1}$(j)$U_{2n+1,a-1,l}^{2n+1}$(k)$U_{2n+1,l,b-1}^{2n+1}$(1)$\overline{an}\neq 0$$\overline{l}=\overline{an}$$0\leq l’\leq b-1if\overline{l’}=0,$ $\overline{a(n-1)+1}\neq 0$$\overline{b}=1,$,,$\overline{an-l}\neq 0$$\{\begin{array}{l}U_{2n+1,0,0}^{2n+1}if\overline{an}=0, char k(a,U_{2n+1,0,l}^{2n+1}, for 0\leq l’\leq b-1if\overline{l^{l}}=0,\overline{an}=0, char k|a,\end{array}$(m) Additionally in the case of $q=-1$ :
.

$i$

ii.

for $0\leq l\leq a-1$ if
for $1\leq l\leq a-1$ if
for
if
for
if
, for $1\leq l’\leq b-1$ if, for $0\leq l’\leq b-1$ if$\overline{a(n-1)}=0,$$U_{0,l,0}^{2n+1}$$U_{1,l,b-1}^{2n+1}$iii.

$U_{2l-1,0,b-1}^{2n+1}$iv.

$U_{2l,a-1,0}^{2n+1}$$\overline{b}=0,$$1\leq l’’\leq n$$1\leq l’’\leq n$$\overline{l}=0$$\overline{a}=\overline{b}=0$$\overline{a}=\overline{b}=0$.

$U_{2n,a-1,l}^{2n+1}$$\overline{a}=0,$vi.

$U_{2n+1.0,l}^{2n+1}$$\overline{a}=0,$$v$$\overline{b}=0,$$\overline{l}=1$

,

,

,

,
$\overline{l’}=0$,$\overline{b(n-1)}=0,$$\overline{l’}=1$.

In the case $q=1,$ is a first root of unity. Thenfor any integer . Hence if $q=1$
then a basis of $HH^{n}(A_{q})$ is formed by the elements of (1), (2)(a),(b),(c),(g),(h),(k),(l),(m)and (3) (a), (d), (f), (g), (i), (1).
$\overline{z}=0$$q$

Next, in the case where$q$

is not a root of unity, we give a basis of HH $n(A_{q})$ for $n\geq 0$ .

Theorem 3.3 ([16, Theorem 2.2]). If is not a root of unity andfollowing elements form a basis of $HH^{n}(A_{q})$ .
$q$(1) Basis of $HH^{0}(A_{q})$ :(a) 1$A_{q}=e_{0}^{0}+ \sum_{j=2}^{t}e_{b(j)}^{0}+\sum_{i=2}^{s}e_{a(i)}^{0}$(b)$T_{l,b}$(c)$T_{a,l’}$,for $1\leq l\leq a-1$ ,for $1\leq l\leq b-1$ ,(2) Basis of $HH^{2n+1}(A_{q})$ for $n\geq 0$ :(a)

$U_{0,l+1,b-1}^{1}$(b)$U_{0,0,b-1}^{2n+1}$(c)$U_{0,0,0}^{1}$

for $0\leq l\leq a-2$ if $n=0$ ,,if $n=0$ ,for $0\leq l\leq b-3$ if $n=0$ ,(d) $U_{1,a-1,l+1}^{1}$(e)$U_{2n+1,a-1,0}^{2n+1}$(f)$U_{1,0,0}^{1}$,

if $n=0$ ,(3) Basis of $HH^{2n+2}(A_{q})$ for $n\geq 0$ :if $n=0$ ,(a)$W_{1,1,1}^{2}$(b)$W_{0,0,b-1}^{2n+2*}$(c)$W_{2n+2,a-1,0}^{2n+2*}$

,

.

$z$

$s,$ $t\geq

2$, then the

52
Hochschild cohomology ring of

3.3$A_{q}$

modulo nilpotence.
In this section, we determin the Hochschild cohomology ring of
Now we recall the Yoneda product in HH $*(A)$ (see [8]). For homogeneous elements$\eta\in HH^{m}(A)$ andHH $n(A)$ represented by $\eta:P_{m}arrow A$ and $\theta:P_{n}arrow A$ respectively,
the Yoneda product $\eta\theta\in HH^{m+n}(A)$ is given as follows: There exists a commutative
diagram of A-bimodules$A_{q}$$\theta\in$$\cdotarrow P_{m}arrowarrow P_{m+n_{d_{m}}^{arrow}}^{d_{m+n}}\ldotsarrow P_{1}arrow P_{0^{arrow}}^{\backslash 0^{\theta}}Aarrow P_{n+1}arrow P_{n}d_{n+2}d_{n+1}d_{2}d_{1\pi}\downarrow\sigma_{1}\downarrow\sigmaarrow 0$..

,

where are liftings of . Here we see that such liftings always exist but are not unique.
is independent of the choice of
HH $m+n(A)$ . We note that
Then we have$\sigma_{i}(0\leq
i\leq
m)$
3.1] for the liftings
Proposition
[16,
. See
representationand liftingsof the basis of HH $n(A_{q})(n\geq 0)$ . In the case where is a root of unity, by the liftingsgiven in [16, Proposition 3.1], we see that $HH^{n+2r}(A_{q})$ is generated by the elements in$HH^{n}(A_{q})$ and that in $HH^{2r}(A_{q})$ for $n\geq 2r$ .
By corresponding Yoneda product of the$n(A_{q})$$*(A_{q})$given in Theorem 3.2, we now have the
HHbasis elements of HH.
generators of HH $*(A_{q})$ . In this paper. we consider the case where $t\geq 2$ andIn the other cases, we have similar results to the following theorem and corollary. See[16] for the other cases.
$\theta$$\sigma_{i}$$\eta\theta=\eta\sigma_{m}\in$$\eta\theta$$\theta$$\eta,$$q$$:=\oplus_{n\geq 0}$$s,$

Theorem 3.4 ([16, Theorem 3.2]). In the case wherean algebm by the following genemtors:$\overline{a},$$\overline{b}\neq 0$$\overline{a},$, HH $*(A_{q})$ is genemted as(1) The genemtors of $HH^{*}(A_{q})$ in degree :$0$1$A_{q},$

for

.
.
.
.

$T_{r,r},$ $T_{l_{1},r},$$1\leq l_{1},$$T_{r,l_{1}’},$$T_{l_{2},b},$$T_{a,l_{2}’}$$l_{2}\leq a-1,1\leq l_{1}’,$ $l_{2}^{l}\leq b-1$

if$\overline{l_{1}}=\overline{l_{1}’}=0$.

(2) The genemtors of $HH^{*}(A_{q})$ in degree 1:.

.
.

for $0\leq l\leq a-1,0\leq l^{l}\leq b-1$ if
, for
,
, for
,
if
,
for $1\leq l’\leq a-1$ if,
for $1\leq l\leq a-1$ if, for $1\leq l’\leq b-1$ if, char,, char.
for $1\leq l\leq a-1$ if$U_{0,0,l}^{1},,$$U_{1,l,0}^{1}$$U_{0,a-1,l}^{1}$$1\leq l’\leq b-1if\overline{a}=1,\overline{l’}\neq 0$$U_{1,a-1,l}^{1}$$0\leq l^{l}\leq b-1$$\{$

$U_{0,l,b-1}^{1}$$U_{1,0,l}^{1}$$U_{0,l,0}^{1}$$\overline{l}=\overline{l’}=0$,$\overline{a}\neq 1,\overline{l’}=0$$\overline{l}\neq 0,$ $\overline{b}=1$$U_{1,l,b-1}^{1}$$\{$$\overline{l}=0,\overline{b}\neq 1$$k|a$$\overline{l^{l}}=0$$\overline{l}=0$$k|b$(3) The genemtors of $HH^{*}(A_{q})$ in degree 2:$W_{2,\overline{a},l_{2}’}^{2}$$for1\leq l_{1},l_{2}\leq a-1,1\leq l_{1},l_{2}’\leq b-1W_{0,0,l_{1}’}^{2},W_{2,l_{1},0}^{2},W_{0,l_{2},\overline{b}}^{2},$if$\overline{l_{1}}=\overline{a},$$\{\begin{array}{l}W_{0,l,b-1}^{2*} for 0\leq l\leq a-1if\overline{l}=0, char k(b,W_{1,l+1,0}^{2} for 0\leq l\leq a-2 if \overline{l}=0, char k|b,\end{array}$$\overline{b}\neq 0$$\overline{l_{1}’}=\overline{b},$$\overline{l_{2}}=\overline{l_{2}’}=0$

,

53

.
..

$\{\begin{array}{l}W_{2,a-1,l}^{2*}, for 0\leq l’\leq b-1 if \overline{l’}=0, char k(a,W_{1,0,l+1}^{2} for 0\leq l’\leq b-2 if \overline{l’}=0, char k|a,\end{array}$$W_{0,a-1,l}^{2}$, for $0\leq l^{l}\leq b-1$ if$W_{2,l,b-1}^{2}$

for $0\leq l\leq a-1$ if$\overline{a}=1,$$\overline{b}=1,$$\overline{l’}\neq\overline{b}$$\overline{l}\neq\overline{a}$,.

(4) The genemtors of $HH^{*}(A_{q})$ in degree $2n$ for $2\leq n\leq r$ :.

$\{\begin{array}{l}W_{0,0,\overline{bn}}^{2n} if \min\{\overline{bn’}|1\leq n’\leq n-1\}>\overline{bn},W_{0,l,\overline{bn}}^{2n} for 1\leq l\leq a-1 if \overline{l}=0,\overline{bn}\neq 0, \min\{\overline{bn’}|1\leq n’\leq n-1\}>\overline{bn},W^{2n}0,a-1,\overline{l_{1}’}+\overline{l_{2}’}-r\end{array}$$if= \geq r+1,\overline{l_{1}’}=\min\{\overline{bn’}|1\leq n’\leq n-1\}\overline{\frac{a}{l_{2}’}}\frac{1,\overline{l_{1}^{l}}+\overline{l_{2}’}}{b(n-n’)}\neq wheren^{l}isintegersuchthat1\leq n^{l}\leq n-1$

and

.
.
.
.
.
.

for any $1\leq n’’\leq n-1$ ,, char $k(b$ ,for $0\leq l\leq a-1$ if$b(n-1)+1\neq 0$ , char,for $0\leq l\leq a-2$ if,
for $1\leq l\leq a-2$ if“$\overline{bn’}\leq\overline{bn}$$W_{0,l,b-1}^{2n*}$$\overline{l}=0,\overline{b(n-1)+1}\neq 0$$W_{1,l+1,0}^{2n}$$\overline{l}=0,$$\{$$W_{1,l+1,1}^{2n}$$\overline{l}=0,$$k|b$$\overline{b(n-1)}\neq 0$$\{\begin{array}{l}W_{2n-1,0,l}^{2n}, for1\leq l’\leq b-1if=0,\overline{l’}=l,chark|aW_{1,l,0}^{2n}for1\leq l\leq a-1if\overline{l}=1=0,chark|b\frac{1,\overline{b(n-1)+}}{a(n-1)+1},,W_{2}^{2}if’,==0, char k|a, char k|b\frac{\iota^{n}-1,0,0}{a(l’-1)+1}\frac{l’’\leq n}{b(n-l’’)+1}\leq,\end{array}$$W_{2n-1,1,l+1}^{2n}$for$1\leq l’\leq b-2if\overline{l’}=0,$ $\overline{a(n-1)}\neq 0$,$chark(a,$$\{$$chark|aW_{2n,a-1,l’}^{2n*},for0\leq
l’\leq b-1if\overline{l’}=0,\overline{a(}0,$$W_{2n-1,0,l+1}^{2n}for0\leq
l’\leq b-2if\overline{l’}=0^{\frac{n-1)+1\neq}{a(n-1)+1}}\neq
0,$

for

$1\leq l’\leq b-1if\overline{l’}=0,\overline{an}\neq 0,$ $\min\{\overline{an’}|1\leq n’\leq n-1\}>\overline{an}$,,$W^{2n}$$\{\begin{array}{l}W_{2n,\overline{an},0}^{2n}if \min\{\overline{an’}|1\leq n’\leq n-1\}>\overline{an},W_{2n,\overline{an},l’}^{2n}\end{array}$$2n,\overline{l_{1}}+\overline{l_{2}}-r,b-1$$if_{\frac{\overline{b}}{l_{2}}\frac{\overline{l_{1}}+\overline{l_{2}}}{a(n-n)}}r+1, \overline{l_{1}}=\min_{integersuchthat1\leq n\leq n-1},’$and$\overline{an’}\leq\overline{an’’}$.
.

for any$1\leq n’’\leq n-1$.

(5) The genemtors of $HH^{*}(A_{q})$ in degree $2n+1$ for $1\leq n\leq r-1$ :.
.

$U_{0,l}^{2n}, \frac{+1}{bn}$$U_{0,l,b-1}^{2n+1}$

for $0\leq l\leq a-1$ if
for $0\leq l\leq a-1$ if
for $0\leq l\leq a-2$ if$U_{1,l+1,0}^{2n+1}$$\overline{l}=0,$$\min\{\overline{bn’}|1\leq n’\leq n-1\}\geq\overline{bn}$$\overline{l}=0,$$\overline{b(n-1)+1}\neq 0$$\overline{l}=0,$$\overline{bn}\neq 0$,, char $k|b$ ,,$\{\begin{array}{l}U_{2,l,0}^{2n+1} for 0\leq l\leq a-2if\overline{l}=\overline{a}, \overline{b(n-1)+1}=0,U_{2l^{\frac{1}{al}},0}^{2n+} for 2\leq l’’\leq n, 0\leq\overline{al’’}\leq[Case]if\overline{b(n-l’’)+1}= O, min \{\overline{al’’’}|1\leq l’’’\leq l’’-1\}>\overline{al’’},\end{array}$if char

$k|b$

,

54

.
.
.
.
.
.

$\{U_{2},,for0\leq l’\leq b-2if^{\frac{)|l’’+1\leq n’}{a(n-1)+1}}=0,\overline{l,}=\overline{b}U_{2+,\overline{b(n-l’’)}}for0\leq l’’\leq n-l,0\leq\overline{b(n-l’’)}\leq\{if^{\frac{2n+1l’1,0}{2n+1al’,+,1n-1,0l}}=0,\min\{\overline{b(n’-l’’}\leq n-l\}>\overline{b(n-l’)}$if char

$k|a$

,

for
, for
, for

$0\leq l^{l}\leq b-2$

$U_{2n,0,l+1}^{2n+1}$$U_{2n+1,\overline{an},l}^{2n+1}$

if$\overline{l’}=0,$$\overline{an}\neq 0$,$0\leq l’\leq b-1if\overline{l’}=0,$ $\min\{\overline{an’}|1\leq n’\leq n-1\}\geq\overline{an}$$U_{2n+1,a-1,l}^{2n+1}$$0\leq l’\leq b-1if\overline{l’}=0,$,$a(n-1)+1\neq 0$ , char $k|a$ .

(6) The genemtors of HH $*(A_{q})$ in degree $2r+2n+1$ for $0\leq n\leq r-2$ :for $n+1\leq l’’\leq r,$$U_{2l^{\prime^{\frac{2n}{al’}}},,0}^{2r++1}$

if, char $k|b$ ,$\min\{\overline{al’’’}|1\leq l’’’\leq l^{\prime l}-1\}>\overline{al’’},\overline{b(n-l’’)+1}=0$$U_{2l’+1,0^{\frac{1}{b(n-l’’)}}}^{2r+2n+}$if$0\leq\overline{al^{\prime l}}\leq\{\begin{array}{l}a-2 if char k(a,a-1 if ch\text{下}J k|a,\end{array}$$\overline{al^{\prime l}+1}=0,$for$1\leq l’’\leq r-1,0\leq\overline{b(n-l’’)}\leq\{\begin{array}{l}b-2 if char k(b,b-1 if char k|b,\end{array}$, char $k|a$ .

$\min\{\overline{b(n’-l’’)}|l’’+1\leq n’\leq r+n-1\}>\overline{b(n-l’’)}$

are not nilpotent andandIt follows from the Theorem 3.4 thatthe other generators are nilpotent. Thus we have the following corollary.
$1_{A_{q}},$$W_{0,0,0}^{2r}$$W_{2r,0,0}^{2r}$, then the quotient of the Hochschild cohomology
Corollary 3.5. If $t\geq 2$ andring ofmodulo nilpotence is isomorphic to the polynomial ring of two variables in all
chamcteris $tic$ :.
HH$\overline{a},\overline{b}\neq 0$$s,$$A_{q}$$*(A_{q})/\mathcal{N}\cong k[W_{0,0,0}^{2r}, W_{2r,0,0}^{2r}]$

Finaly, we consider the ring structure of HH $*(A_{q})$ in the case where is not a rootofof unity. It follows from the liftings given in [16] that all basis elements except$n\geq0$. Thus we have the following results.
HH $n(A_{q})$ are nilpotent elements for$q$$1_{A_{q}}$Theorem 3.6. If
k-algebm.
Corollary 3.7. If

$q$

$q$

is not a root of unity then $HH^{*}(A_{q})$ is not a finitely genemted
is not a root of unity then

$HH^{*}(A_{q})/\mathcal{N}\cong k$.

is not self-injective, monomial or Koszul. Moreover
In general, our algebrais new example of a class of algebras
does not have a stratifying ideal. Therefore
for which the Hochschild cohomology ring modulo nilpotence is finitely generated as aisk-algebra. For example, in the case where $s=2,$ $t=1$ and $a=b=2$ , our algebra
stratifying
not
have
a
Moreoverdoes
ideal.
orKoszul.
not self-injective, monomial$A_{q}$$A_{q}$$A_{q}$$A_{q}$$A_{q}$

3.4

Finiteness conditions for$A_{q}$

Finally, we show thatsatisfies the finiteness conditions in the case where is a rootof unity.
. In
Now we consider the case where is an r-th root of unity, $t\geq 2$ andsatisfies the finiteness conditions by the same method.
the other case, we see thatis given by
The Yoneda algebra or $Ext$ algebra of$A_{q}$

$q$

$s,$

$q$

$\overline{a},$$\overline{b}\neq 0$$A_{q}$$A_{q}$$E(A_{q})=\oplus_{n\geq 0}Ext_{A_{q}}^{n}(A_{q}/\mathfrak{r}, A_{q}/\mathfrak{r})$55with the Yoneda product. We use the notationn-th graded component of $E(A_{q})$ . Then it is easy to see that

$E(A_{q})^{n}=Ext_{A_{q}}^{n}(A_{q}/\mathfrak{r}, A_{q}/\mathfrak{r})$$LI_{j=2}^{t}ke_{b(j)}^{n}\oplus U_{i=2}^{s}ke_{a(i)}^{n}$Let$\varphi$.

: HH $*(A_{q})arrow E(A_{q})$ be a homomorphism of graded rings given by. Then it is easy to see that the image of is precisely the graded ring $k[x, y]$andin degree $2r$ .

$\varphi(\eta)=$$\eta\otimes_{A_{q}}A_{q}/\mathfrak{r}$wherefor the$E(A_{q})^{n}\simeq\coprod_{l=0}^{n}ke_{l}^{n}\oplus$$\varphi$$x:=e_{0}^{2r}+ \sum_{j=2}^{t}e_{b(j)}^{2r}$$y:=e_{2r}^{2r}+ \sum_{i=2}^{s}e_{a(i)}^{2r}$Proposition 3.8. $E(A_{q})$ is a finitely genemted left $k[x, y]$ -module with genemtors:$e_{l}^{2n},$$e_{b(j)}^{2n},$$e_{a(i)}^{2n}$for $0\leq l\leq 2n,$$2\leq j\leq t$

and $2\leq i\leq s$
in degree $2n$ for $0\leq n\leq r-1$ ,$e_{l}^{2n+1},$$e_{b(j)}^{2n+1}$,$e_{a(i)}^{2n+1}$

for $1\leq l\leq 2n,$$1\leq j\leq t$

and $1\leq i\leq s$
in degree $2n+1$ for $0\leq n\leq r-1$ ,$e_{l}^{2r}$for $1\leq l\leq 2r-1$ in degree
for $2n+1\leq l\leq 2r$
for $2n+1\leq l\leq 2r-1$

$2r$

,

$e_{l}^{2r+2n+1}$

in degree $2r+2n+1$ for $0\leq n\leq r-1$ ,$e_{l}^{2r+2n+2}$

in degree $2r+2n+2$ for $0\leq n\leq r-2$ .

Now we consider the conditions (Fgl) and (Fg2). The element $W_{0,0,0}^{2r}\in HH^{2r}(A_{q})$is a pre-image of and the element $W_{2r,0,0}^{2r}\in HH^{2r}(A_{q})$ is a pre-image of . Let $H$ be
the graded subalgebra of $HH^{*}(A_{q})$ generated by $HH^{0}(A_{q}),$, so that $H$
andis a pre-image of $k[x, y]$ in HH. Then we have the following immediate consequenceof Proposition 3.8.
$x$

$y$

$W_{2r,0,0}^{2r}$$W_{0,0,0}^{2r}$$*(A_{q})$Theorem 3.9. The conditions (Fgl) and (Fg2) hold for the algebm
the subptng $H$ of HH.

$A_{q}$

with respect to$*(A_{q})$By [2], Theorem 3.6 and 3.9, we have the necessary and sufficient conditions forto satisfy the finiteness conditions.

Theorem 3.10.

$A_{q}$

satisfies the finiteness conditions if and only if$q$$A_{q}$

is a root of unity.

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