数理解析研究所講究録
第 1604 巻 2008 年 142-150

142

Gr\"obner bases on projective bimodules
and the Hochschild cohomology *Part III. Syzygies

YUJI KOBAYASHI
Department of Information Science, Toho University
Funabashi 274-8510, Japan
This is a continuation of the previous papers [2] and [3]. We develop the theory

of Gr\"obner bases on projective modules over an algebra based on a well-ordered semi-group. We discuss syzygy modules on projective modules and construct generatorsof the modules in terms of Gr\"obner bases. The results can be used to compute theintersection of given submodules (see [1] for the polynomial algebra case).

Derivation graphs

9

Let $S=B\cup\{0\}$ be a well-ordered reflexive semigroup with and $K$ be acommutative ring with 1. Let $F=K\cdot B$ be the K-algebra based on $B$ and let
be a (two-sided) ideal of $F$ . Let $A=F/I$ be the quotient algebra of $F$ by
and : $Farrow A$ be the natural surjection. We fix a reduced Gr\"obner basisofI. For $f\in F,$ $f$ denotes the normal form of modulo .
Let $X$ be an left edged set andbe the projective left F-module generated$X$
by . Letbe a (not necessarily complete) rewriting system on $F\cdot X$ . Set
$0$

$I$

$I$

$G$

$\rho$$f$$G$$F\cdot X$$T$$H=H_{T}=\{s-t|sarrow t\in T\}$ .

The set $H$ is assumed to be uniform and is considered to be a left edged set;
for an element
in $H,$
is defined by $\sigma(h)=\sigma(x)$ , where $1t(h)=x\cdot\xi$$(x\in B, \xi\in X)$ . We consider the projective left F-module $F\cdot H$ generated
by $H$ . For $\in H,$ $[h]$ denotes the formal generator of $F\cdot H$ corresponding to$h\in H$ . An element $f$ of $F\cdot H$ is written as a finite sum$h$$\sigma(h)$$f’,$$f= \sum k_{i}x_{i}[h_{i}|$(9.1)$h_{i}\in H,$
with.
We define a graphcalled the derivation graph as sociated toandas follows. The set of vertices is the projective F-module $F\cdot X$ and for$k_{i}\in K\backslash \{0\},$$x_{i}\in B_{\sigma(h_{1})}$$\mathcal{D}=\mathcal{D}(T, G)$$G$’This is a preliminary report and the details appear elsewhere

$T$

143
$f,$ $g\in F\cdot X$ an (positive) edge
from the source $f=\sigma(e)$ to the target $g=\tau(e)$is a one-step $(T, G)$ -reduction fromto , that is,has a term
with$\xi\in X$ , and
$e$

$f$

$k\in K\backslash O,$$k\cdot x\xi$

$f$

$g$

$x\in B_{\sigma(\xi)},$and$(i)xarrow cx$ ’

or$g=f+k\cdot(x^{l\prime}-x)\xi$ ,$(ii)x=x^{l}z,$ $h=z\xi-t\in H$

and$g=f+k\cdot x’(t-z\xi)$ .

In case (i), is called a G-edge, and in case (ii), is called a T-edge (or an
H-edge). The label of the T-edge in (ii) is the element. For anofedge from to , we have the reverse (negative) edge
from $\sigma(e^{-1})=g$ toand $\tau(e^{-1})=f$ . The label of the reverseof the T-edge in case (ii) above
is $-k\cdot x^{l}[h]$ . A path inis a concatenation
$e$

$e$

$k\cdot x’[h]$

$e$

$f$

$e$

$F\cdot H$$e^{-1}$$g$$e^{-1}$$e$$\mathcal{D}$

$p$

(9.2)

$p=e_{1}\circ e_{2}o\cdots oe_{n}$with $\tau(e_{i})=\sigma(e_{i+1})$ for $i=1,$ $n-1$ . The
path is positive if all the edges
in are positive. Define the source
andthe target
of by $\sigma(p)=\sigma(e_{1})$ and $\tau(p)=\tau(e_{n})$ respectively. Here, isclosed if $\sigma(p)=\tau(p)$ . For two paths and such that $\tau(p)=\sigma(q)$ , we have apathwhich is a concatenation of and at $\tau(p)=\sigma(q)$ .
We define a mapping from the set of all paths in
to F. $H$ as follows.
Let be a path given in (9.2). If is trivial, that is, $n=0$ , then $\int(p)=0$ . If
$n\geq 1$ , let $p^{l}=e_{2}\circ\cdots\circ e_{n}$ . If
is a G-edge, then $\int(p)=\int(p’)$ . If
is a
T-edge with label $k\cdot x’[h]$ , thenof (positive or negative) edges$p$$-e_{i}$$\ldots,$$\tau(p)$$\sigma(p)$$p$$e_{i}$

$p$

$p$

$p$

$p\circ q$

$q$

$p$

$q$

$\mathcal{D}$$\int$$p$$p$$e_{1}$$e_{1}$.

$\int(p)=k\cdot x[h]+\int(p’)$

Thus sums up all the labels of T-edges in . We also define a mapping called
the boundary mapping from the set of paths to the projective module $F\cdot X$ by$\int$$d$$p$$d(p)=\sigma(p)-\tau(p)$ .

It is easy to see that for paths

$p$

and

$q$

with $\tau(p)=\sigma(q)$ ,$\int(poq)=\int(p)+\int(q)$and$d(p\circ q)=d(p)+d(q)$ .

Let$\delta$

be a morphism of left F-modules from F. $H$ to F. $X$ defined by$\delta([h])=h$

for $h\in H$ .
Proposition 9.1. We have$\delta\circ\int(p)\equiv d(p)$for any path

$p$

in

$\mathcal{D}$$(mod G)$. In particular,$\delta\circ\int(p)\equiv 0(mod G)$

for a closed path

$p$

in

$\mathcal{D}$

.

,

(9.3)

144
Let $A\cdot X$ and $A\cdot H$ be the projective left A-modules generated by $X$ and$H$ , respectively. We consider a morphism: $A\cdot Harrow A\cdot X$ of left A-modules
by $\partial([h])=\rho x(h)$ for $h\in H$ . Then we have a commutative diagram$\partial$$F\cdot H$$arrow\delta$$F\cdot X$$A\cdot H$(9.4)$\downarrow\rho x$$\downarrow\rho H$$arrow\partial$$A\cdot X$,

are the canonical surjections. Clearly we have ${\rm Im}(\delta)=L(H)$
where
and
, which are mappings from
andand ${\rm Im}(\partial)=L_{A}(H)$ . Set$A\cdot H$ respectively.
$A\cdot
X$and tothe set of paths to$\rho x$$\beta H$$\overline{d}=\rho H^{\circ d}$$\overline{\int}=\rho x\circ\int$

Corollary 9.2. We have$\partial\circ\overline{\int}(p)=\overline{d}(p)$in $A\cdot X$ for any path

$p$

in

$\mathcal{D}$. In particular,$\partial\circ\overline{\int}(p)=0$

for any closed path

$p$

in

$\mathcal{D}$.

Standard reductions and the linear map

10$\int$

Suppose that a rule $x\cdot\xiarrow t\in T$ is applied to a termof $f\in F\cdot X$ ,$(y’,x\xi)$$farrow\tauf-ky’(x\cdot\xi-t)$
$y=y’
x$
. If
, and we haveis an leftmost
where(resp. rightmost) appearance of Left $(T)$ in, the application is leftmost(resp. rightmost). Since
is reduced only one rule can be applied toatthe leftmost (rightmost) position.
A positive path$f_{1}arrow_{T,G}f_{2}arrow T,G\ldotsarrow_{T,G}f_{n}$(10.1)$ky\cdot\xi$$y\cdot\xi$$T$$y\cdot\xi$$n-1$ ,is standard, if for every $i=1,$$f_{i}arrow\tau,cf_{i+1}$(i) whenG-reducible,isis a G-edge, and(ii) whenis G-irreducible, the edge $f_{i}arrow T,Gf_{i+1}$ is by a leftmost application of a rule from $T$ to the greatest T-reducible term ofwith respect to ,that is, $x\cdot\xiarrow t\in T,$
is the greatest T-reducible$T$$x’\cdot\xiarrow
t’$
of
term
in
and no rulecan be applied toso thatappears at the left of.
If
is reduced tothrough a standard reduction as above, we write as. A standard one-step reduction by a rule from $T$ is denoted by
,
is,
that
if is G-irreducible and is obtained by a leftmost applicationof a rule of $T$ to the greatest T-reducible term of .
Since
is complete, if
is G-irreducible, the standard reduction (10.1)

in$\mathcal{D}$$\ldots,$$f_{i}$$f_{i}$$f_{i}$$x\in\Sigma^{*},$$k\in^{-}K\backslash \{0\},$$\succ$$k\cdot zx\cdot\xi$$x’\cdot\xi$$zx\cdot\xi$$f_{i}$$x\cdot\xi$$f_{n}$$f_{1}$$f_{1}\Rightarrow_{T,G}^{*}f_{n}$$f\Rightarrow\tau g$$\Rightarrow T$

$f$

$g$

$f$

$arrow G$$f_{n}$

can be rewritten as,$f_{1}=g_{1}arrow_{G}^{*}\hat{g}_{1}\Rightarrow\tau g_{2}arrow_{G}^{*}\hat{g}_{2}\Rightarrow T\ldots\Rightarrow\tau g_{m}arrow_{G}\hat{g}_{m}=f_{n}$(10.2)

145
inwhere is the G-normal form of $gi$ Since is reduced, in the step
$T$ is applicable to the greatest
the above reduction sequence, only one rule fromat a unique leftmost position. In this sense, a standard
T-reducible term ofis $(T, G)$ -irreducible, it isis unique. In particular, if
reduction from to.
is called the standard form of $f_{\sim}denoted$ by
unique. This unique element$f$coincides with the normal form of .
If $T$ is complete modulo $G,$$T$$\hat{g}i$$\hat{g}_{\tau}\Rightarrow\tau g_{i+1}$$\hat{g}_{i}$$f$$f_{n}$$f_{n}$$f^{s}$$f_{n}$$f$$f^{s}$$g^{l}\in F\cdot X\cdot F,$
Proposition 10.1. Let $f,$andare standard reductions(1) There is a standard reduction$f^{l},$$f\Rightarrow_{T,G}^{*}f^{l}$$l\in K$

$k,$

$g,$

$g\Rightarrow_{T,G}^{*}g’$.

$k\cdot f+\ell\cdot g\Rightarrow_{T}^{*},c^{k\cdot f’+\ell\cdot g’}$and $g’$ are the standard forms of;is the standard form of(2) If$f^{l}$

$f$

and

$g$

and assume that there.

respectively, then$k\cdot f’+\ell\cdot g^{l}$$k\cdot f+.\ell\cdot g$$(k\cdot f+\ell\cdot g)^{s}=k\cdot f^{s}+\ell,$$\cdot g^{s}$.

Now we define a K-linear map $\int=\int_{H}:F\cdot Xarrow F\cdot H$ by$\int(f)=\int(p(f))$.
for $f\in F\cdot X$ , where $p(f)$ is a standard path from to the standard form
The reader should not be confused by using the same symbol for the mappingfrom the module $F\cdot X$ and for the mapping from the set of paths in . Clearly,$\int(f)$ does not depend on the choice of the standard reduction of
to . So we
can choose the standard reduction$f$$f^{s}$$\int$$\mathcal{D}$$f$. $=f^{s}$$p:f=g_{1}arrow_{G}^{*}\hat{g}_{1}\Rightarrow\tau g_{2}arrow_{G}^{*}\hat{g}\Rightarrow T\ldots\Rightarrow\tau g_{m}arrow c\hat{g}_{m}$$f^{S}$(10.3)

like (10.2) to define $\int(f)$ . Precisely, for $f\in F\cdot X$ let $\pi(f)$ be the uniquestandard path given in (10.3), then$p$$\int(f)=\int(\pi(f))$.

Proposition 10.2. (1) $\int(f)=\int(\hat{f})$ for $f\in F\cdot X$ , whereform of with respect to .
(2)
is a morphism of K-moIules, that is,
$G$

$f$

$\int$$\int(k_{1}f_{1}+k_{2}f_{2})=k_{1}\int(f_{1})+k_{2}\int(f_{2})$for$k_{1},$$k_{2}\in K$and$f_{1},$$f_{2}\in F\cdot X$.

By the definition, we have$d(\pi(f))=f-f^{s}$ .

Thus, by Proposition 9.1 we obtain$\hat{f}$is the normal

146
Proposition 10.3. For $f\in F\cdot X$ ,$(mod G)$ ,$\delta\circ\int(f)\equiv f-f^{s}$in $F\cdot X$ .

The assertion (1) in Proposition 10.2 means that $\int(f)=\int(g)$ follows from$\rho x(f)=\rho x(g)$ . Thus,: $A\cdot Xarrow F\cdot H$ such thatinduced a K-linear map. The composition
with the surjection
is a
K-linear map from $A\cdot X$ to $A\cdot H$ . Thus,$\int’$$\int$$\int=\int^{l}\circ\rho x$$\overline{\int}=\overline{\int}_{H}=\beta H^{O}\int’$$\rho H$

Proposition 10.4. The K-linear map induces a K-linear map
and we have a commutative diagmm$\int$$\overline{\int};A\cdot Xarrow$$A\cdot H$$F\cdot X$$arrow\int$$F\cdot H$$\rho x\downarrow$$\downarrow\rho H$$arrow\overline{\int}$$A\cdot X$$A\cdot H$.

Since $\rho x(f)=\rho x(g)$ if and only if, we sometimes regard a G$F\cdot
X$$A\cdot
X$ .
irreducible element ofas an element ofThus, a G-irreducible
element and its standard form , which is also G-irreducible, are considered
to be an element of $A\cdot X$ . With this convention, Proposition 10.3 means$\hat{f}=\hat{g}$$f$$f^{\theta}$

Corollary 10.5. For $f\in A\cdot X$ we have$\partial\circ\overline{\int}(f)=f-f^{s}$

11.

Cycles made from critical pairs and z-elements

Let be the kernel of the morphismin (9.4). We are interested
in finding generators of .
Let $h=x\cdot\xi-t,$ $h=x^{l}\cdot\xi-t^{l}(t, t’\in F\cdot X, \xi\in X, x, x’\in_{\tau(\xi)}B)$ be rules
in $H$ and $u-v(u\in B, v\in F)$ be a rule in $G$ .
First, we consider a critical pair of the first kind. Suppose that $zx=z^{l}x’\neq 0$
for some $z^{l}\in B$ , where the appearance $(z, x\cdot\xi)$ of
is at the right of theappearance $(z’, x’\cdot\xi)$ ofin $zx\xi=z^{l}x’\xi$ , and andare left coprime.
Then we have critical pair$\mathcal{K}$$\rho x\circ\delta=\partial\circ\rho H$$\mathcal{K}$$z,$$x\cdot\xi$$x^{l}\cdot\xi$$z^{l}$$z$$(zx\cdot\xiarrow_{H}zt, z’x’\cdot\xiarrow_{H}z’t’)$of reductions. For this critical pair define an element$c_{1}$

of $F\cdot H$ by$c_{1}=z[h|-z^{l}[h’|+ \int(z\cdot t)-\int(z’t’)$.

(11.1)

Next, we consider a critical pair of the second kind. Suppose that $zx=$”$\neq 0$ for some $z,$ $z’,$$\in B$ , whereis G-irreducible, the appearance$z’uz$ ”

$z$

$z$

147
of and the appearance $(z, x\xi)$ ofinare left coprime. Then we have a critical pair$(z^{l},u, z’\cdot\xi)$

and

$z’$

$u$

$x\xi$$zx\cdot\xi$

are rightmost and$z$$(zx\cdot\xiarrow_{H}zt, z’uz" \cdot\xiarrow cz’vz" \cdot\xi)$of reductions. For this critical pair we define an element$c_{2}=z[h]+ \int(z\cdot t)-\int(z’vz"\xi)$$c_{2}$

of F. $H$ by.

(11.2)

Lemma 11.1. If the critical pair $(zt, z’t’)$ resp. $(zt,$ $z’vz”\xi))$ is resolvable, the(resp.
elements) above is in .
$($$c_{1}$$c_{2}$$\mathcal{K}$Consider a z-pair $(z, h)$ , that is, $h=x\xi-t\in H,$ $z\in B$ and $zx=0$ . We
have an z-element $zt$ and for this z-pair we define an elementof F. $H$ by$c_{3}$$c_{3}^{-}=z[h|+ \int(z\cdot t)$.

(11.3)

Lemma 11.2. If the z-element $zt$ is resolvable, the element
Let $C$ be the collection of all the elements$c_{1},$$c_{2}$

and$c_{3}$$c_{3}$

above is in$\mathcal{K}$.

above. If $H$ is

a Gr\"obner basis, then all the critical pairs and the z-elements are resolvable(Theorem 7.2). Hence, $C$ is contained inby Lemmas 11.1 and 11.2,. More$\mathcal{K}$strongly we have

Theorem 11.3. If $H$ is a Grobner basis, $C$ generates

12$\mathcal{K}$.

Syzygies

Let $Y$ be a left edged set and let
be a sequence of left uniform
elements of $F\cdot X$ indexed by $Y$ with
. A sequenceof right uniform elements of $F$ withis a syzygy of modulo $G$ if$f_{\eta}=0$ for all but a finite number$Y$of in , and$h=(h_{\eta})_{\eta\in Y}$$\sigma(h_{\eta})=\sigma(\eta)$$\tau(f_{\eta})=\sigma(\eta)$$f=(f_{\eta})_{\eta\in Y}$$h$$\eta$$\sum_{\eta\in Y}f_{\eta}h_{\eta}=0$in. The set Syz(h) of all syzygies of forms a submodule of the projective
left F-module $F\cdot Y$ generated by . We call it the syzygy module of . Let: $F\cdot Yarrow F\cdot X$ be the morphism defined by$A\cdot X$$h$$Y^{r}$$h$$\delta$$\delta(\eta)=h_{\eta}$

for $\eta\in Y$ . Then, Syz(h) is nothing but the kernel of the morphism$F\cdot Yarrow A\cdot X$ .
Let : $A\cdot Yarrow A\cdot X$ be the morphism defined by$\partial$$\partial(\eta)=\rho x(h_{\eta})$,$\rho x\circ\delta$:

148then we have a commutative diagram$F\cdot Y$$F\cdot X$$arrow\delta$$\rho Y\downarrow$$\downarrow$$A\cdot Y$$arrow\partial$$\beta X$$A\cdot X$.

Thus, $Ker(\partial)=\rho Y(Syz(h))$ .

By Theorem 11.3 we have

Theorem 12.1. If $H=\{h_{\eta}|\eta\in Y\}$ forms a Grobner basis on $F\cdot X$ , theby the set $C$ of elements (11.1),
syzygy module $Syz(h)$ is genemted modulo(11.2) and (11.3) made from the critical pairs and the z-pairs with respect to $H$
and $G$ .
$G$

If $H$ is not a Gr\"obner basis then we take a Gr\"obner basisof the submonoid generated by $H$ modulo . We may apply the completion
defined byprocedure to obtain . We have a morphism 6 :. Sinceare projective. Then,
and
for
:
and
left F-modules we have a morphisms :;andsuch that$\overline{H}=\{h_{\overline{\eta}}|\overline{\eta}\in\overline{Y}\}$$G$$F\cdot\overline{Y}arrow F\cdot X$$\overline{H}$$\overline{\delta}(\overline{\eta})=h_{\overline{\eta}}$$\phi$$F\cdot Y$$F\cdot\overline{Y}$${\rm Im}(6)={\rm Im}(\overline{\delta})$$\overline{\eta}\in\overline{Y}$$F\cdot\overline{Y}arrow F\cdot Y$$\psi$$F\cdot Yarrow F\cdot\overline{Y}$$6=\overline{6}\circ\psi$$\overline{6}=6\circ\phi$$arrow\delta$$F\cdot Y$$F\cdot X$$\phi\uparrow\downarrow\psi$$||$$arrow\overline{\delta}$$F\cdot\overline{Y}$$F\cdot X$

Let $C=\{c_{\zeta}|\zeta\in Z\}$ be the set of elements ofand . We haveand z-pairs with respect to$\overline{H}$

Theorem 12.2. The set.
modulo.
made from critical pairs$F\cdot Y$$G$$\phi(C)\cup\{\eta-\phi(\psi(\eta))|\eta\in Y\}$generates$G$

Suppose that$c_{\zeta}(\zeta\in Z),$$h_{\overline{\eta}}(\overline{\eta}\in\overline{Y})$,$c_{\zeta}= \sum_{\overline{\eta}\in\overline{Y}}z_{\zeta}^{\overline{\eta}}\cdot\overline{\eta}$

with$z_{\zeta}^{\overline{\eta}},$$x \frac{\eta}{\eta},$$y_{\eta}^{\overline{\eta}}\in F$

Corollary 12.3.

and $h_{\eta}(\eta\in Y)$ are written as$h_{\overline{\eta}}= \sum_{\eta\in Y}x\frac{\eta}{\eta}\cdot h_{\eta}$,$h_{\eta}= \sum_{\overline{\eta}\in\overline{Y}}y_{\eta}^{\overline{\eta}}\cdot h_{\overline{\eta}}$. In this situation we have

$Syz\mu i$) is generated by the elements$\sum_{\overline{\eta}\in\overline{Y},\eta\in Y}z_{\zeta}^{\overline{\eta}}x\frac{\eta}{\eta}\cdot\eta(\zeta\in Z)$

and$(\eta\in Y)$$\eta-\sum_{\eta’\in Y,\overline{\eta}\in\overline{Y}}y_{\eta}^{\overline{\eta}}x\frac{\eta’}{\eta}\cdot\eta^{l}$.

$Syz(h)$149

Intersections of submodules

13Let
an $dL_{2}$ be a submodules of the projective left A-module. Supposethat they are generated by $H_{1}=\{h_{\eta 1}|\eta_{1}\in Y_{1}\}$ and $H_{2}=\{h_{\eta 2}|\eta_{2}\in Y_{2}\}$ ,respectively. Define a morphism$A\cdot X$$L_{1}$$\partial$:$A$. $X\oplus A\cdot Y_{1}\oplus A\cdot Y_{2}arrow A\cdot X\oplus A\cdot X$

by$\partial(\xi, \eta_{1}, \eta_{2})=(\xi+h_{\eta_{1}}, \xi+h_{\eta 2})$

for $\xi\in X,$$\eta_{1}\in Y_{1}$

and
$\pi$

$\eta_{2}\in Y_{2}$. Let: $A\cdot X\oplus A\cdot Y_{1}\oplus A\cdot Y_{2}arrow A\cdot X$

be the projection onto the first component. Then, $\pi(Ker(\delta))$ is equal to theintersection$L_{1}\cap L_{2}$.

Let $H=\{h_{\eta}|\eta\in Y\}$ be a Grobner basis on $F\cdot X$ of $L_{1}+L_{2}$ , which
could be obtained by completing. Let $C=\{c_{\zeta}|\zeta\in Z\}$ be the set ofelements of $F\cdot Y$ made from critical pairs and z-pairs with respect to $H$ . We
see that $H‘=\{(\xi, \xi)\xi\in X\}\cup\{(h_{\eta}, 0)\eta\in Y\}$ is a Gr\"obner basis of
on $F\cdot X\oplus F\cdot X$ and $(0, C)$ is the set of elements of $F\cdot X\oplus F\cdot Y$ made fromcritical pairs and z-pairs with respect to. We have a morphism$H_{1}\cup H_{2}$${\rm Im}(\partial)$$H^{l}$$\overline{\partial}:A\cdot X\oplus A\cdot Yarrow A\cdot X\oplus A\cdot X$

defined by$\overline{\partial}(\xi, \eta)=(\xi+h_{\eta}, \xi)$

for $\xi\in X$ and $\eta\in Y$ . Then,${\rm Im}(\partial)={\rm Im}(\overline{\partial})$$A\cdot X\oplus A\cdot Y_{1}\oplus A\cdot Y_{2}arrow A\cdot X\oplus A\cdot Y$

such that$\partial 0\phi=\overline{\partial}$and$\overline{\partial}\circ\psi=\partial$

and$\psi$:

and we have morphisms. Thus, We have a commutative diagram$A\cdot Xarrow^{\pi}A\cdot X\oplus A\cdot Y_{1}\oplus A\cdot Y_{2}arrow\partial A\cdot X\oplus A\cdot X$$\phi\uparrow$$\downarrow\psi$$A\cdot X\oplus A\cdot Y$$||$$arrow\overline{\partial}$$A\cdot X\oplus A\cdot X$

Since $H$ is a Gr\"obner basis of $L_{1}+L_{2}$ , we can write as$h_{\eta}= \sum_{\eta’\in Y_{1}\cup Y_{2}}x_{\eta}^{\eta’}h_{\eta’}$with$x_{\eta}^{\eta^{l}}\in A$

in$A\cdot Y_{1}\oplus A\cdot Y_{2}$$\phi$:$A\cdot X\oplus A\cdot YXarrow A\cdot X\oplus A\cdot Y_{1}\oplus A\cdot Y_{2}$for $\eta\in Y$ , and$h_{\eta’}= \sum_{\eta\in Y}y_{\eta}^{\eta},h_{\eta}$

.

150
with

$y_{\eta^{l}}^{\eta}\in A$

for$\eta^{l}\in Y_{1}\cup Y_{2}$in$A\cdot Y$. Then,$\phi$

andare given as$\psi$$\phi(\xi, \eta)=(\xi+\sum_{\eta_{2}\in Y_{2}}x_{\eta}^{\eta 2}h_{\eta 2},\sum_{1\eta\in Y_{1}}x_{\eta^{1}}^{\eta}\eta_{1}, -\sum_{2\eta\in Y_{2}}x_{\eta^{2}}^{\eta}\eta_{2})$and$\psi(\xi, \eta_{1}, \eta 2)=(\xi+h_{\eta 2},\sum_{\eta\in Y}((y_{\eta}^{\eta_{1}}-y_{\eta}^{\eta_{2}})\eta)$.

Theorem 13.1. Under the above situation, let$c_{\zeta}= \sum_{\eta\in Y}z_{\zeta}^{\eta}\eta$with$z_{\zeta}^{\eta}\in A$

for$\zeta\in Z$. Then,$- L_{1}\cap L_{2}$is genemted by the elements$(\zeta\in Z)$$\eta\in Y,\eta\in Y_{2}\sum_{2}z_{\zeta}^{\eta}x_{\eta}^{\eta 2}\cdot h_{\eta 2}$,$(\eta_{1}\in Y_{1})$$\eta\in Y,\eta\in Y_{2}\sum_{2}y_{\eta\iota}^{\eta}x_{\eta^{2}}^{\eta}\cdot h_{\eta 2}$,

and$(\eta_{2}\in Y_{2})$$h_{\eta 2}- \sum_{\eta\in Y,\eta_{2}’\in Y_{2}}y_{\eta_{2}}^{\eta}x_{\eta}^{\eta_{2}’}\cdot h_{\eta_{2}’}$.

Corollary 13.2. If $H_{1}\cup H_{2}$ forms a Grobner basis, then $Y=Y_{1}\cup Y_{2}$ andis generated by the elements$L_{1}\cap L_{2}$$(\zeta\in Z)$$\sum_{\eta 2\in Y_{2}}z_{\zeta}^{\eta 2}\cdot h_{\eta 2}$and$h_{\eta}(\eta\in Y_{1}\cap Y_{2})$.

References[1] T. Becker and V. Weispfenning, Grobner Bases. A computational approachto commutative algebm, Springer, 1993
[2] Y. Kobayashi, Grobner bases on projective bimodules and the Hochschildcohomology I. Rewriting on vector spaces, Kokyuroku 1503, RIMS (Kyoto
University), 2006, 30-40.

[3] Y. Kobayashi, Gr\"obner bases on projective bimodules and the Hochschildcohomology II. Critical pairs, Kokyuroku 1562, RIMS (Kyoto University),
2007, 40-45.