数理解析研究所講究録 1222 巻 2001 年 24-33

24Some type of commutative artin algebras*

山口大学理学部 菊政 勲 (Isao Kikumasa)

山口大学理学部 吉村 浩 (Hiroshi Yoshimura)
Department of Mathematics, Faculty of Science
Yamaguchi University
Yamaguchi 753-8512, Japan

Introduction
The class of quasi-Frobenius (QF) algebras, i.e., selfinjective artin algebras, is oneof the most interesting classes of non-semisimple artin algebras. For example,
any artin algebra is afactor algebra of the trivial extension of by its dual,
algebra. Our aim is to consider the possibility of QF test of factor
which is aalgebras of apolynomial ring over afield by using computers and to determine theisomorphism classes of commutative QF algebras Aover afield satisfying thecondition$A$$A$$\mathrm{Q}\mathrm{F}$$k$’$\Lambda/J\underline{\simeq}k^{(\mathfrak{n})}$and$J^{3}=0$

is aproduct of $m$ copies of for apositive
where is the radical of Aandinteger .
In \S 1 we show the existence of an algorithm for deciding whether agiven factor, and demonstrate some examples by our imalgebra of apolynomial ring is
plementation. In \S 2 we consider some type of commutative QF algebras satisfying
the condition above, which will be said to be of type (1, 2, 1), and determine theirisomorphism classes.
the characteristic,andThroughout this paper, is afield withlgebras mean finite dimensional commutative algebras over and isomorall
phisms between algebras mean -algebra isomorphisms. For a algebra , wedenote by Rad(A) and Soc(A) the radical and the socle of , respectively.
$k^{(m)}$

$J$

$k$

$m$

$\mathrm{Q}\mathrm{F}$$k^{*}=k\backslash \{0\}$$k$$\mathrm{c}\mathrm{h}k$$k$$k-\dot{\mathrm{a}}$

$k$

$k$

$k$

$\Lambda$“More general results of the last section in this paper will appear elsewhere$\Lambda$

25
1

QF test of polynomial algebras by computer

The following result is most essential for investigating local QF algebras.

Lemma 1.1 ([1]) Let Abe a local -algebra. Then the following conditions are$k$equivalent:(1) Ais QF;(2) $\dim_{k}\Lambda/Rad(\Lambda)=\dim_{k}$ Soc(A).

Using the lemma above and some results related to Gr\"obner basis theory, we
have the following.

Theorem 1.2 There exists an algorithm for deciding whether, for a given idealof the polynomial ring $=k[x_{1}, \ldots, x_{n}]$ over a field , $R/I$ is $QF$.
$R$

$I$

$k$

In fact, under some conditions on the base field , the algorithm can be implemented in several computer algebra systems. The following are some examples
computed by our program worked on SINGULAR [4], acomputer algebra system.
$k$

Example 1.3 Let $R=\mathbb{Q}[x, y, z]$ . By our program, we can easily check that thealgebra $R/\langle z^{3},yz -z^{2}, y^{2}, xz-z^{2}, xy,x^{2}-z^{2}\rangle$ is alocal QF algebra, while thealgebra $R/\langle z^{3}, yz-z^{2}, y^{2}, xz-z^{2}, xy, x^{2}\rangle$ is local but not.
$\mathrm{Q}\mathrm{F}$

Example 1.4 (Tachikawa [6]) Let be the polynomial ring$n+1$ variables, ,,over afield . For apositive integer , set$R$$x_{0}$$x_{1}$$\ldots$$k[x_{0},x_{1}, \ldots,x_{n}]$$k$$x_{n}$

in$t$$I_{t}=\langle x_{[]}x_{[+1]}\cdots x_{[:+n-1]}-x_{[\dot{|}+n]}^{t}|i=0,1, \ldots, n\rangle$

where $[h]$ is the residue of modulo $n+1$ . Then, Tachikawa proved that if $t\neq n$ ,then the algebra
is isomorphic to adirect sum of agroup algebra and alocal QF algebra (and hence, it is). Now, we shall particularly consider the
algebra in case that $n=t-1$ and denote it by . The following is an execution
result for the algebraby ourprogram, where the first three lines are our input and the rest are output.
$h$

$k$

$R/I_{t}$$\mathrm{Q}\mathrm{F}$$T_{t}$$T_{3}=\mathbb{Q}[x_{0}, x_{1}, x_{2}]/\langle x_{0}x_{1}-x_{2}^{3}, x_{1}x_{2}-x_{0}^{3}, x_{2}x_{0}-x_{1}^{3}\rangle$$\mathbb{Q}$,ideal I– ($>\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{r}=0$$>$$(\mathrm{x},\mathrm{y},\mathrm{z})$$\mathrm{x}\mathrm{y}$

-z3 ,

.

$1\mathrm{p}j$$\mathrm{y}\mathrm{z}-\mathrm{x}3,$$\mathrm{z}\mathrm{x}-\mathrm{y}3$)$j$$>\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}(\mathrm{I})j$$<\mathrm{z}9-\mathrm{z}5,\mathrm{y}\mathrm{z}6-\mathrm{y}\mathrm{z}2,\mathrm{y}2\mathrm{z}5-\mathrm{y}2\mathrm{z}$,$\mathrm{y}4-\mathrm{z}4,\mathrm{x}\mathrm{z}-\mathrm{y}3,\mathrm{x}\mathrm{y}-\mathrm{z}3.\mathrm{x}3-\mathrm{y}\mathrm{z}>$k-dimension $=27$
number of component $\mathrm{s}=11$26—–1
Local QF ?
Groebner Basis of

———-

True
$\mathrm{I}=$of ( :rad(I))
of rad(I)$\mathrm{I}$k-basis ofof$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$$\mathrm{R}/$$\mathrm{R}/\mathrm{I}$(Irad(I )$\sim$$\{\mathrm{z}+1,\mathrm{y}+1,\mathrm{x}-1\}$$\{1\}$$”=\{\mathrm{z}+1 .\mathrm{y}+1, \mathrm{x}-1\}$$=$$\{1\}$$\approx$$\{0\}$. of Loevy factors:

, 1
, 0

upper $(>):1$ $(1)$
—–2

———-

$=$

$\{\mathrm{z}-1.\mathrm{y}-1,\mathrm{x}-1\}$$\epsilon$$\{\mathrm{z}-1,\mathrm{y}-1,\mathrm{x}-1\}$$\mathrm{I}$$\mathrm{R}/\mathrm{I}$$\mathrm{R}/$$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$.

$\mathrm{k}$$\mathrm{I}$$1-\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{l}\sim\triangleleft \mathrm{f}$$\sim$–$\mathrm{R}/$$\mathrm{C}>$

flfI

.

( :rad(I))$\mathrm{I}$) $:1$ . 3 3

True$\{\mathrm{z}4,\mathrm{y}\mathrm{z}2,\mathrm{y}2\mathrm{z},\mathrm{y}4,\mathrm{x}\mathrm{z}-\mathrm{y}3,\mathrm{x}\mathrm{y}-\mathrm{z}3,\mathrm{x}3-\mathrm{y}\mathrm{z}\}$of rad(I)k-basis of$\mathrm{R}/$

—–6
True
$\sim$$\{\mathrm{z}+1.\mathrm{y}-1,\mathrm{x}+1\}$$=$$\{1\}$$\sim\{\mathrm{z}+1,\mathrm{y}-1,\mathrm{x}+1\}$.

, 1{0}. 0$=\{1\}$$\mathrm{R}/\mathrm{I}$( :rad(I))$\mathrm{I}$

upper
—–7

$\mathrm{R}/\mathrm{I}$$\mathrm{R}/$..

$\{\mathrm{z}-1,\mathrm{y}+1,\mathrm{x}+1\}$$\{\mathrm{z}-1.\mathrm{y}+1 ,\mathrm{x}+1\}$$\cdot$$\Leftrightarrow\{0\}$$\mathrm{I}$$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$

upper

$(>)$

:

1(1)

9
Local QF ?
Groebner Basis of I$\cdot$———$\mathrm{R}/\mathrm{I}$$\mathrm{R}/\mathrm{I}$$\mathrm{k}$–$\mathrm{R}/$$\mathrm{I}$.

$\{1\rangle$$\{\mathrm{z}+1,\mathrm{y}2+1.\mathrm{x}-\mathrm{y}\}$$\{\mathrm{y}, 1\}$

Local QF ?
True
Groebner Basis of I.
{1}
of (I rad(I))..

–

of rad(I)k-basis of–$\cdot$$\mathrm{R}/\mathrm{I}$–$\{\mathrm{z}-1,\mathrm{y}2+1,\mathrm{x}+\mathrm{y}\}$$\{\mathrm{z}-1.\mathrm{y}2+1,\mathrm{x}+\mathrm{y}\}$$\{\mathrm{y}, 1\}$

, 2

, 0
or (Irad(I ). of Loevy factors:$\Leftrightarrow\{0\}$$\mathrm{R}/$$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$upper $(>)$ : 2(1)

10 $–\sim--$
Local QF ?
Groebner Basis of—–$\cdot$of (I:rad(I))
of rad(I)k-basis of–$\{\mathrm{z}+1,\mathrm{y}2+1,\mathrm{x}-\mathrm{y}\}$———-Local QF
True
Groebner Basis of I.
of CI:rad(I){1}
rad(I)
of{1}, 1
k-basis of( :rad(I)), 0of. of Loevy factors:
$)$

..
.
$\sim$

upper $(>)$ : 2(1)—–8$\cdot$?

Local QF ?
True
Groebner Basis of I
of (I:rad(I))
of rad(I), 2-basis of:rad(I))(of{0}. 0of Loevy factors:$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$: 1(1)

———

———-

–

. of Loevy factors:
$(>)$

.

, ii
10

$\{\mathrm{z}4,\mathrm{z}3,\mathrm{z}2,\mathrm{J}\mathrm{z},\mathrm{z},\mathrm{y}3,fl\cdot \mathrm{y},\mathrm{S}_{1}\mathrm{z},1\}$$=\{\mathrm{z}3,\mathrm{z}2,\mathrm{y}\mathrm{z},\mathrm{z},\mathrm{y}3,\mathrm{y}2.\mathrm{y}.\mathrm{x}2,\mathrm{x}, 1\}$3 1 (5)(I:rad(I))$o\mathrm{f}$—–.

$=\{\mathrm{z},\mathrm{y},\mathrm{x}\}$$\cdot$Local QF ?
Groebner Basis of I

of$\{\mathrm{z}, 1\}$$\sim\{\mathrm{z}5,\mathrm{y}\mathrm{z}2,\mathrm{y}2\mathrm{z}, \mathrm{y}4-\mathrm{z}4.\mathrm{x}\mathrm{z}-\mathrm{y}3,\mathrm{x}\mathrm{y}-23,\mathrm{x}3-\mathrm{y}\mathrm{z}\}$—–5 $——arrow—$$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$$\{\mathrm{z}2+1,\mathrm{y}+1,\mathrm{x}-\mathrm{z}\}$$=$$\mathrm{R}/$. of Loevy factors :

upper

.
$=$

$\mathrm{R}/\mathrm{I}$upper $(>)$ : 2(1)of ( :rad(I))
of rad(I)
of

–

———-

Local QF ?
Groebner Basis of I$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$True$=\{\mathrm{z}2+1,\mathrm{y}+1.\mathrm{x}-\mathrm{z}\}$of (I rad(I ) $=\{1\}$of rad(I), 2-basis of$\{0\}$ ,(Irad(I )0ofof Loevy factors:–$\cdot$upper $(>)$ : 1(1)

—–4

—–3
Local QF ?
Groebner Basis of I———-

Local QF ?
True
Groebner Basis of I
of ( :rad(I)) ’ {1}of rad(I)k-basis of{1}, 1$\{0\}$ ,(Irad(I )of0of Loevy factors:

True$\cdot$.
$=$$\{\mathrm{z}2+1,\mathrm{y}-1,\mathrm{x}+\mathrm{z}\}$$\{1\}$.

–$\{\mathrm{z}2+1,\mathrm{y}-1,\mathrm{x}+\mathrm{z}\}$$=\{\mathrm{z}$$1\rangle$, 2–$\mathrm{k}$True$\mathrm{I}=\{\mathrm{z}2+1,\mathrm{y}-\mathrm{z},\mathrm{x}+1\}$of (I:rad(D))
of rad(I)Basis of$\mathrm{R}/\mathrm{I}$$\Leftrightarrow\{1\rangle$$=\{\mathrm{z}2+1,\mathrm{y}-\mathrm{z}.\mathrm{x}+1\}$$\cdot$$\{\mathrm{z}, 1\}$

, 2

27
of
$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$$\mathrm{R}/$( :rad(I))$=$$\mathrm{I}$$\{0\}$. of Loewy factors :upper $(>):2(1)$
11

—–

, 0

of
$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$$\mathrm{R}/$( :rad(I))$\mathrm{I}$$\sim$$\{0\}$. of Loewy factors:, 0

upper $(>):2(1)$

———-

.

Local QF ?

Groebner Basis ofTrue$\mathrm{I}\approx$$\{\mathrm{z}2+1,\mathrm{y}+\mathrm{z},\mathrm{x}-1\}$of (I rad(I)). –{z2+1,
of rad(I)k-basis of–{z, 1}, 2(Irad(I)) –{0}, 0
ofof Loewy factors :upper $(>):2(1)$$=$$\{1\}$$\mathrm{y}+\mathrm{z}.\mathrm{x}-1$}$\mathrm{R}/\mathrm{I}$$\mathrm{R}/$$\mathrm{k}-\mathrm{d}\mathrm{i}\mathrm{m}$.

QF :True

This output, which is arranged for want of space, shows not only to be a(nonlocal QF algebra but also further information about this algebra. For example,, and the number of local components ofis 11; four components
have dimension 1, six components have dimension 2and one ‘exceptional’ comp0nent has dimension 11 and Loewy length 5.
We also run our program in case of the algebras and over
. It is thenremarkable thatand
as well as have one exceptional local component, say
and , of dimension 131 and 1829, respectively. Omitting to show the output,
we here give only atable showing the dimension and the number of componentsof the algebras and the series of dimensions of the factors in Loewy series of thecomponentsand .
$T_{3}$$\dim_{\mathbb{Q}}T_{3}=27$$T_{3}$$T_{4}$$T_{4}$$C_{4}$$T_{5}$$k=\mathrm{Q}$$T_{5}$$T_{3}$$C_{5}$$C_{4}$$C_{5}$$T_{4}$

dimension$T_{5}$

the number ofcomponents

dimension

the number ofcomponents
16

111
11
431
2640
131
1
11829
the number of components $=33$ the number of components $=657$$\dim T_{5}=3125$$\dim T_{4}=256$

the series of dimensions of the factors in Loewy series of$-C_{4}:14101622252216$$C_{4}$

and$C_{5}$:$\mathrm{I}\mathrm{O}41$$C_{5}:151535$ $65105$ $151195225$ $235225195151105$ $6535$ $1551$

Table 1: the dimension and the number of components of$T_{4}$

and$T_{5}$

28
Classification of QF algebras of type (1, 2, 1)

2

For the sake of completeness, we give abrief proof of the following known result.

Lemma 2.1 For a commutative ring , the following conditions are equivalent:$\Lambda$(1) Ais a -algebra with $\dim_{k}\Lambda/Rad(\Lambda)=1$ ;
$k$

(2)

where I is an ideal of the polynomial ring $R=k[x_{1}, \ldots.x_{n}]$ suchfor some integers $m$ , $n\geq 1$ .

$\mathrm{A}\underline{\simeq}R/I$

that$\langle x_{1}, \ldots, x_{n}\rangle^{m}\subset I$ $\subset\langle x_{1}, \ldots,x_{n}\rangle$

In particular, if A is not a field, then the ideal I of (2) can be chosen to satisfy.
I$\subset\langle x_{1}, \ldots,x_{n}\rangle^{2}$. Assume that Ais a -algebra with $J=Rad(\mathrm{A})$ such that$\dim_{k}\Lambda/J=1$ and $J^{m}=0$ for some integer $m\geq 1$ . Since $\dim_{k}\Lambda/J=1$ , it follows. Then it is shownthat $\Lambda=k+J$ . Let $\{u:+J|i=1, \ldots, n\}$ be a -basis of

Proof.

(1)$k$$\Rightarrow(2)$$J/J^{2}$

$k$

that
$\Lambda=k+\sum_{1\underline{<}_{\dot{1}1}\leq n}ku:_{1}+\sum_{1\leq:_{1}\underline{<}_{2}\leq n}ku_{i_{1}}u_{\dot{1}2}+\cdots+\sum_{1\leq:_{1}\leq\cdots\underline{<}_{m-1}\leq n}ku_{\dot{1}1}\ldots u:_{m-1}$,

which implies the following epimorphism:$\varphi:R=k[x_{1}, \ldots, x_{n}]arrow\Lambda$, $f(x_{1}, \ldots, x_{n})|\mapsto f(u_{1}, \ldots, u_{n})$ .

, we will obtain the result.
Now, setting(2). This is easily shown. @$I$$=\mathrm{K}\mathrm{e}\mathrm{r}\varphi$$\Rightarrow(1)$Let Abe a -algebra with $J=Rad(\mathrm{A})$ andthat Ais of type $(1, n, 1)$ if$k$d.

$\mathrm{i}\mathrm{m}_{k}\Lambda/J=1$,$\dim_{k}J/J^{2}=n$,$n$apositive integer. Then we say$\dim_{k}J^{2}=1$

and$\dim_{k}J^{3}=0$.

To determine the isomorphism classes of QF -algebras Awith J $=Rad(\Lambda)$
satisfying the condition$\mathrm{f}\mathrm{c}$$\Lambda/J\underline{\simeq}k^{(m)}$and$J^{3}=0$

for some $m\geq 1$ , we need to consider the -algebras of type $(1, n, 1)$ , for the local
components of-algebras satisfying this condition are the field , the serial-algebra of length 2, or the -algebras of type $(1, n, 1)$ .
The following lemma gives the “canonical forms” of -algebras of type $(1, n, 1)$ .
$k$

$k$

$\mathrm{Q}\mathrm{F}k$

$k$

$k$

$k$

29
Lemma 2.2 Let Abe a -algebra. Then the following conditions are equivalent:
$k$

(1)

is of type (1, n, 1);$\Lambda$(2) There exist

(i) ,
$p$

(ii)

$q$

with $1\leq p\leq q\leq n$ , and$a_{\dot{|}j}\in k$for$1\leq i\leq j\leq n$with $(i,j)\neq(p, q)$such that$\Lambda\cong k[x_{1},$$x_{n}]/\langle x_{i}x_{j}-a_{j}x_{p}x_{q}$\ldots ,. Set R $=k[x_{1},$(ii) of (2), set$Pro\mathrm{o}/$\ldots ,I$=\langle x:x_{j}-a_{\dot{|}j}x_{p}x_{q}$and J$x_{n}]$, $x_{p}^{2}x_{q}|1\leq i\leq j\leq n$ with $(i,j)\neq(p,q)\rangle$ .

$=\langle x_{1},$\ldots ,$x_{n}\rangle$. For p, q and, $x_{p}^{2}x_{q}|1\leq i\leq j\leq n$ with $(i,j)\neq(p,$$a_{\dot{|}j}’ \mathrm{s}$$q)\rangle$

as in (i) and$\subset R$.

$(\#)$. Assume that $\mathrm{A}=R/I$ . For $f\in R$ , set $\overline{f}=f+I$. Since it
is easy to see that, it follows from Lemma 2.1 that Ais a -algebra with
Rad(A) $=J/I$ . Furthermore, it can be easily seen that
(2)$\Rightarrow(1)$$\in \mathrm{A}$$J^{3}\subset I$$k$$\{\overline{x_{1}}+Rad(\Lambda)^{2}, \ldots,\overline{x_{n}}+Rad(\Lambda)^{2}\}$are -bases of Rad(\Lambda )/Rad \Lambda
$k$

$($

$)^{}$

and Rad$(\Lambda)^{2}$and$\{\overline{x_{p}x_{q}}\}$, respectively. Therefore,$\Lambda$

is ak-

algebra of type $(1, n, 1)$ .
(1). Let Abe a -algebra of type $(1, n, 1)$ . By Lemma 2.1, we may
assume that $\mathrm{A}=R/H$ where $H$ is an ideal of such that(;and$\dim_{k}J^{2}/H=1$ . Then we can choose an
where $1\leq p\leq q\leq n$ . Since$\dim_{k}J^{2}/H=1$ , it follows that for $1\leq i\leq j\leq n$ with $(i,j)\neq(p, q)$ , there exist
such that $x:x_{j}-a_{ij}x_{p}x_{q}\in H$. For the $(p, q)$ and the, define an idealof as in $(\#)$ . Since
and $\dim_{k}R/I=\dim_{k}R/H=n+2$ , w\‘e have $H=I$ ,which shows (2).
$k$$\Rightarrow(2)$$J^{3}\subset H$$R$$J^{2}$$x_{p}x_{q}\not\in H$$a_{ij}\in k$$a_{j}’ \mathrm{s}$

$R$

$I$

$I$ $\subset H$$\mathrm{s}\mathrm{g}$It is easy to see that in case $n=1$ , any -algebra of type (1, 1, 1) is isomorphic
to, aserial -algebra of length 3. In the rest of this paper, we shallconsider anon-trivial case $n=2$ , i.e., -algebras of type (1, 2, 1). To this, wedefine -algebras and $\Sigma_{ab}(a, b\in k)$ by
$k$

$k$

$k[x]/\langle x^{3}\rangle$

$k$

$k$

$\Gamma$and$\Gamma$$=$$k[x, y]/\langle x^{2}, y^{2}\rangle$$\Sigma_{ab}$$=$$k[x, y]/\langle xy-ay^{2}, x^{2}-by^{2}, y^{3}\rangle$

Then we have the following.

30
Lemma 2,3 (1) Any -algebraA of type (1, 2, 1) is isomorphic to.
some a, bor$\Sigma_{ab}$forfor some a, b$\in k$$k$$\Gamma$$\in k$(2) Any QF -algebra of type (1, 2, 1) is isomor phic to
with
.
$k$

$\Gamma$or$\Sigma_{ab}$$a^{2}-b\neq 0^{\cdot}$

Proof. (1) It suffices to consider the factor algebras of $k[x, y]$ (where $x=x_{1}$ and$y=x_{2})$ in (2) of Lemma 2.2 for $(p, q)\in\{(1,2), (2,2)\}$ . Then it can be checkedthat
if $(p, q)=(1,2)$ andfor some , $b\in k$ otherwise.
(2) Since it is easily seen that is, it suffices to show that Ea&is not QF if
and only if $a^{2}-b=0$ . Set $J=Rad(\Sigma_{ab})$ and $S=Soc(\Sigma_{ab})$ . Then we see thatandare -bases of $J/J^{2}$ and , respectively, and that$\mathrm{A}\cong\Gamma$$a_{11}=a_{22}=0;\mathrm{A}\cong\Sigma_{ab}$$\Gamma$$\{\overline{x}+J^{2},\overline{y}+J^{2}\}$$\mathrm{Q}\mathrm{F}$$k$$\{\overline{y}^{2}\}$$a$$J^{2}$.

$J^{2}=k\overline{y}^{2}\subset S=\{f\in\Sigma_{ab}|f\overline{x}=f\overline{y}=0\}\subset J$

It follows from Lemma 1.1 that, which means
is not QF if and only ifthat there exists $0\neq(s, t)\in k^{(2)}$ such that
. By the note above on ,we see that this is also equivalent to the condition $a^{2}-b=0$ .
$J^{2}\subsetneq S$$\Sigma_{ab}$$S$$s\overline{x}+t\overline{y}\in S$$\blacksquare$

Lemma 2.4 Let , , ,conditions are equivalent:
$a$

(1)

$b$

$c$

$d\in kw\dot{\iota}th$$a^{2}-b\neq 0$

and $c^{2}-d\neq 0$ . Then the following$\Sigma_{ab}\underline{\simeq}\Sigma_{cd;}$(2) $\sqrt{(c^{2}-d)(a^{2}-b)^{-1}}\in k$ .

Proof. (1)$\Rightarrow(2)$. Let

be a -isomorphism from$k$$\varphi$$\Sigma_{ab}$to$\Sigma_{d}$$\varphi(\overline{x})$$=$$s\overline{x}+t\overline{y}+u\overline{y}^{2}$$(s, t, u\in k)$$\varphi(\overline{y})$$=p\overline{x}+q\overline{y}+r\overline{y}^{2}$$(p, q, r\in k)$

and

.

Then,

.

$0\neq\varphi(\overline{y}^{2})=\varphi(\overline{y})^{2}=p^{2}\overline{x}^{2}+2pq\overline{xy}+q^{2}\overline{y}^{2}=(dp^{2}+2\varphi q +q^{2})\overline{y}^{2}$

Set $w=dp^{2}+2\varphi q+q^{2}$ . Sinceof relative to -bases$\varphi$$k$$\varphi$

is an isomorphism, the determinant of the matrix
of bothandis$\{\overline{1},\overline{x},\overline{y},\overline{y}^{2}\}$$\det$$\Sigma_{ab}$$\Sigma_{cd}$$(\begin{array}{llll}\mathrm{l} 0 0 00 s t u0 p q r0 0 0 w\end{array})=w(qs-pt)\neq 0$

.

31

0

$=\varphi(\overline{xy}-a\overline{y}^{2})=\varphi(\overline{x})\varphi(\overline{y})-a\varphi(\overline{y})^{2}$$=$$arrow$$\{dsp+c(.sq+tp)+tq-aw\}\overline{y}^{2}$.

Similarly,$0=\varphi(\overline{x}^{2}-b\overline{y}^{2})=(ds^{2}+2cst+t^{2}-bw)\overline{y}^{2}$.

It follows thatNow we substitute the

$a$

$=$

$\{dsp+c(sq+tp)+tq\}w^{-1}$

$b$

$=$

$(ds^{2}+2cst+t^{2})w^{-1}$

and

$a$

$b$

.

above for $(c^{2}-d)(a^{2}-b)^{-1}$ to obtain$(c^{2}-d)(a^{2}-b)^{-1}=w^{2}(qs-pt)^{-2}$ ,which implies (2).
(2). Assume that $u:=\sqrt{(c^{2}-d)(a^{2}-b)^{-1}}\in k$ . Then the following
correspondence gives a -isomorphism fromto:$\Rightarrow(1)$$k$$\overline{1}$a ,$\overline{1}$$\Sigma_{ab}$$\overline{x}\succ+\overline{x}+(au-c)\overline{y}$,$\Sigma_{\mathrm{c}d}$$\overline{y}\mapsto*u\overline{y}$,$\overline{y}^{2}\vdasharrow u^{2}\overline{y}^{2}$,

which completes the proof. @Now, consider the following algebra:$\Lambda_{a}:=k[x, y]/\langle x^{2}-ay^{2}, xy\rangle$$(a\in k)$.

Then we can show the following.

Theorem 2.5 Any QF -algebra of type (1, 2, 1) is isomorphic to$k$or$\Gamma$for some a$\in k^{*}$$\Lambda_{a}$. Moreover(1) For a, b$\in k^{*}$(2) (i) chk$\neq 2\Leftrightarrow$,$\sqrt{ab^{-1}}\in k\Leftrightarrow\Lambda_{a}\underline{\simeq}\Lambda_{b}$

(ii)

.

$\Gamma\cong\Lambda_{-1}\Leftrightarrow(\mathrm{i}\mathrm{i}\mathrm{i})\Gamma\cong\Lambda_{c}$Proof. Note from Lemma 2.4 that forfor some c$\in k^{*}$, $b\in k$ with $a^{2}-b\neq 0$ ,. Then the first assertion follows from Lemma 2.3.
(1) This follows immediately fromand Lemma 2.4$a$$\Lambda_{b-a^{2}}$$\Lambda_{a}=\Sigma_{0a}$.

$\Sigma_{ab}\cong\Sigma_{0,b-a^{2}}=$32(2) (i) \Rightarrow (\"u). If
isomorphism from to$\mathrm{c}\mathrm{h}k\neq 2$$\Lambda_{-1}$$\Gamma$$\overline{1}.\mapsto\overline{1}$

(ii)
(iii)

$\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$\Rightarrow(\mathrm{i})$,, then the following correspondence gives ak-:$\overline{x}\vdash*\overline{x}+\overline{y};\overline{y}\vdasharrow\overline{x}-\overline{y}$,$\overline{xy}|\mapsto-2\overline{y}^{2}$.

. Obvious.

. Let$\varphi:\Gammaarrow\Lambda_{c}(c\in k^{*})$

be a -isomorphism and$k$$\varphi(\overline{x})$$=s\overline{x}+t\overline{y}+u\overline{y}^{2}(s,t,u\in k)$$\varphi(\overline{y})$$=p\overline{x}+q\overline{y}+r\overline{y}^{2}(p, q, r\in k)$

.

Then,
$0=\varphi(\overline{x}^{2})=\varphi(\overline{x})^{2}=s^{2}\overline{x}^{2}+t^{2}\overline{y}^{2}=(cs^{2}+t^{2})\overline{y}^{2}$from which $cs^{2}+t^{2}=0$ , i.e., $t^{2}=-cs^{2}$ . Similarly,$q^{2}=-\varphi^{2}$,. On the other hand,.

$0\neq\varphi(\overline{xy})=\varphi(\overline{x})\varphi(\overline{y})=(csp+tq)\overline{y}^{2}$

Hence, $csp$ $+tq\neq 0$ . Therefore we have$0\neq(csp+tq)^{2}=c^{2}s^{2}p^{2}+2cstpq$ $+t^{2}q^{2}=2(c^{2}s^{2}p^{2}+cstpq)$ ,which implies that$\mathrm{c}\mathrm{h}\neq 2$. si

Let $\{a:|(a:\in I)\}$ be aset of representative elements of the multiplicative groupof square elements, i.e., $k^{*}/(k^{*})^{2}=\{a:(k^{*})^{2}|i\in I\}$ .
modulo the subgroup-algebras of type (1, 2, 1)
Then, Theorem 2.5 gives the isomorphism classes ofas follows.
$(k^{*})^{2}$$k^{*}$$\mathrm{Q}\mathrm{F}k$, then the isomorphism classes of QF -algebras ofCorollary 2.6 (1) If chk.
(i $\in I)$ . In particular, I
type (1, 2, 1) are$k$$\neq 2$$\cong\Lambda_{-1}$$\Lambda_{a:}$(2) If chk $=2$ , then the isomorphism classes of QF -algebras of type (1, 2, 1) are(i. In particular,
andfor all a$k$$\Gamma$$\Lambda_{a:}$$\in I)$$\Gamma\not\cong\Lambda_{a}$$\in k^{*}$

Remark 2.7 (1) For agiven field , the number $N$ of the isomorphism classes-algebras of type (1, 2, 1) is given as follows.
of$k$$\mathrm{Q}\mathrm{F}k$$N=\{\begin{array}{l}2’ (e\geq 0)\mathrm{o}\mathrm{r}\infty(\mathrm{c}\mathrm{h}k\neq 2)2\mathrm{o}\mathrm{r}\infty(\mathrm{c}\mathrm{h}k=2)\end{array}$(2) For any $e\geq 0$ , there exists afield$k$such that $N=2^{e}$ (cf. [5, Chapter 4, \S 7]).

The following table shows the isomorphism classes of QF algebras of type (1, 2, 1)
over some typical fields .
$k$

33

Table 2: the isomorphism classes of QF -algebras of type (1, 2, 1)$k$References[1] H. Bass: On the ubiquity of Gorenstein rings, Math. Z. 82(1963), 8-28.
[2] T. Becker and V. Weispfenning: Grobner Bases, GTM 141, Springer-Verlag,
New York, 1993.
[3] D. Cox, J. Little and D. O’Shea: Ideals, Varieties, and Algorithms, 2nd ed.,
UTM, Springer, New York, 1997.

[4] G.-M. Greuel, G. Pfister, and H. Schonemann: Singular version 1.2 User Manual . In Reports On Computer Algebra, number 21. Centre for Computer Algebra, University of Kaiserslautern, June 1998.
http:$//\mathrm{w}\mathrm{w}\mathrm{w}$.mathemat ik.uni-kl.

$\mathrm{d}\mathrm{e}/-\mathrm{z}\mathrm{c}\mathrm{a}/\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$[5] G. Karpilovsky: Field Theory, Monographs and Textbooks in Pure and Applied Math. 120, Marcel Dekker, New York-Basel, 1988.
[6] H. Tachikawa: QF algebras and Grobner bases of pure binomials, Preprint
00-014, SFB 343, Bielefeld, 2000