Table Of ContentGr¨obner bases of modules
5 σ PBW
over − extensions
1
0
2
∗
Haydee Jim´enez & Oswaldo Lezama
n
a Seminario de A´lgebra Constructiva - SAC2
J
Departamento de Matem´aticas
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Universidad Nacional de Colombia, Bogot´a, COLOMBIA
3
]
A
R Abstract
h. For σ−PWB extensions, we extend to modules the theory of Gr¨obner
t bases of left ideals presented in [5]. As an application, if A is a bijec-
a
m tive quasi-commutative σ−PWB extension, we compute the module of
syzygies of a submodule of thefree module Am.
[
1
v Key words and phrases. PBW extensions, noncommutative Gr¨obner
2 bases, Buchberger’s Algorithm, module of syzygies.
8
8
7
2010 Mathematics Subject Classification. Primary: 16Z05. Secondary:
0
. 18G10.
1
0
5
1
:
v
i
X
r
a
∗Graduatestudent, UniversidaddeSevilla,Spain.
1
Gr¨obner Bases for Ideals of σ−PBW Extensions 2
1 Introduction
In this paper we present the theory of Gr¨obner bases for submodules of Am,
m ≥ 1, where A = σ(R)hx ,...,x i is a σ−PBW extension of R, with R a
1 n
LGS ring (see Definition 12) and Mon(A) endowed with some monomial order
(see Definition 9). Am is the left free A-module of column vectors of length
m≥1; if A is bijective, A is a left Noetherian ring (see [8]), then A is an IBN
ring(InvariantBasisNumber),andhence,allbasesofthe freemodule Am have
m elements. Note moreover that Am is a left Noetherian, and hence, any sub-
moduleofAm isfinitely generated. Themainpurposeistodefineandcalculate
Gr¨obner bases for submodules of Am, thus, we will define the monomials in
Am, orders onthe monomials,the concept of reduction, we will constructa Di-
vision Algorithm, we will give equivalent conditions in order to define Gr¨obner
bases, and finally, we will compute Gr¨obner bases using a procedure similar to
Buchberger’s Algorithm in the particular case of quasi-commutative bijective
σ−PBW extensions. The results presented here generalize those of [5] where
σ-PBW extensions were defined and the theory of Gr¨obner bases for the left
idealswasconstructed. Mostofproofsareeasilyadaptedfrom[5]andhencewe
will omit them. As an application, the final section of the paper concerns with
the computation of the module of syzygies of a given submodule of Am for the
particular case when A is bijective quasi-commutative.
Definition 1. Let R and A be rings, we say that A is a σ−PBW extension
of R (or skew PBW extension), if the following conditions hold:
(i) R⊆A.
(ii) There exist finite elements x ,...,x ∈ A− R such A is a left R-free
1 n
module with basis
Mon(A):={xα =xα11···xαnn|α=(α1,...,αn)∈Nn}.
In this case we say also that A is a left polynomial ring over R with
respect to {x ,...,x } and Mon(A) is the set of standard monomials of
1 n
A. Moreover, x0···x0 :=1∈Mon(A).
1 n
(iii) For every 1≤i≤n and r ∈R−{0} there exists c ∈R−{0} such that
i,r
x r−c x ∈R. (1.1)
i i,r i
(iv) For every 1≤i,j ≤n there exists c ∈R−{0} such that
i,j
x x −c x x ∈R+Rx +···+Rx . (1.2)
j i i,j i j 1 n
Under these conditions we will write A=σ(R)hx ,...,x i.
1 n
The following propositionjustifies the notationthat we haveintroduced for the
skew PBW extensions.
Gr¨obner Bases for Ideals of σ−PBW Extensions 3
Proposition2. LetAbeaσ−PBW extensionofR. Then,forevery1≤i≤n,
there exist an injective ring endomorphism σ : R → R and a σ -derivation
i i
δ :R→R such that
i
x r =σ (r)x +δ (r),
i i i i
for each r ∈R.
Proof. See [5].
A particular case of σ −PBW extension is when all derivations δ are zero.
i
Another interesting case is when all σ are bijective. We have the following
i
definition.
Definition 3. Let A be a σ−PBW extension.
(a) A is quasi-commutative if the conditions (iii) and (iv) in the Definition 1
are replaced by
(iii′) For every 1≤i≤n and r ∈R−{0} there exists c ∈R−{0} such
i,r
that
x r =c x . (1.3)
i i,r i
(iv′) For every 1≤i,j ≤n there exists c ∈R−{0} such that
i,j
x x =c x x . (1.4)
j i i,j i j
(b) A is bijective if σ is bijective for every 1≤i≤n and c is invertible for
i i,j
any 1≤i<j ≤n.
Someinterestingexamplesofσ−PBW extensionsweregivenin[5]. Werepeat
next some of them without details.
Example4. (i)AnyPBW extension(see[2])isabijectiveσ−PBW extension.
(ii) Any skew polynomial ring R[x;σ,δ], with σ injective, is a σ −PBW ex-
tension; in this case we have R[x;σ,δ] = σ(R)hxi. If additionally δ = 0, then
R[x;σ] is quasi-commutative.
(iii)AnyiteratedskewpolynomialringR[x ;σ ,δ ]···[x ;σ ,δ ]isaσ−PBW
1 1 1 n n n
extension if it satisfies the following conditions:
For 1≤i≤n, σ is injective.
i
For every r ∈R and 1≤i≤n, σ (r),δ (r)∈R.
i i
For i<j, σ (x )=cx +d, with c,d∈R, and c has a left inverse.
j i i
For i<j, δ (x )∈R+Rx +···+Rx .
j i 1 i
Under these conditions we have
R[x ;σ ,δ ]···[x ;σ ,δ ]=σ(R)hx ,...,x i.
1 1 1 n n n 1 n
In particular,any Ore algebraK[t ,...,t ][x ;σ ,δ ]···[x ;σ ,δ ] (K a field)
1 m 1 1 1 n n n
is a σ−PBW extension if it satisfies the following condition:
Gr¨obner Bases for Ideals of σ−PBW Extensions 4
For 1≤i≤n, σ is injective.
i
Some concrete examples of Ore algebras of injective type are the following.
The algebra of shift operators: let h ∈ K, then the algebra of shift operators
is defined by S := K[t][x ;σ ,δ ], where σ (p(t)) := p(t−h), and δ := 0
h h h h h h
(observe that S can be considered also as a skew polynomial ring of injective
h
type). Thus, S is a quasi-commutative bijective σ−PBW extension.
h
The mixed algebra D : let again h ∈K, then the mixed algebra D is defined
h h
by D := K[t][x;i , d][x ;σ ,δ ], where σ (x) := x. Then, D is a quasi-
h K[t] dt h h h h h
commutative bijective σ−PBW extension.
The algebra for multidimensional discrete linear systems is defined by D :=
K[t ,...,t ][x ,σ ,0]···[x ;σ ,0], where
1 n 1 1 n n
σ (p(t ,...,t )):=p(t ,...,t ,t +1,t ,...,t ), σ (x )=x , 1≤i≤n.
i 1 n 1 i−1 i i+1 n i i i
D is a quasi-commutative bijective σ−PBW extension.
(iv) Additive analogue of the Weyl algebra: let K be a field, the K-algebra
A (q ,...,q )isgeneratedbyx ,...,x ,y ,...,y andsubjecttotherelations:
n 1 n 1 n 1 n
x x =x x ,y y =y y , 1≤i,j ≤n,
j i i j j i i j
y x =x y , i6=j,
i j j i
y x =q x y +1, 1≤i≤n,
i i i i i
whereq ∈K−{0}. A (q ,...,q )satisfiestheconditionsof(iii)andisbijective;
i n 1 n
we have
A (q ,...,q )=σ(K[x ,...,x ])hy ,...,y i.
n 1 n 1 n 1 n
(v)Multiplicativeanalogueofthe Weylalgebra: letK be afield,the K-algebra
O (λ ) is generated by x ,...,x and subject to the relations:
n ji 1 n
x x =λ x x , 1≤i<j ≤n,
j i ji i j
where λ ∈K−{0}. O (λ ) satisfies the conditions of (iii), and hence
ji n ji
O (λ )=σ(K[x ])hx ,...,x i.
n ji 1 2 n
Note that O (λ ) is quasi-commutative and bijective.
n ji
(vi) q-Heisenberg algebra: let K be a field , the K-algebra h (q) is generated
n
by x ,...,x ,y ,...,y ,z ,...,z and subject to the relations:
1 n 1 n 1 n
x x =x x ,z z =z z ,y y =y y , 1≤i,j ≤n,
j i i j j i i j j i i j
z y =y z ,z x =x z ,y x =x y , i6=j,
j i i j j i i j j i i j
z y =qy z ,z x =q−1x z +y ,y x =qx y , 1≤i≤n,
i i i i i i i i i i i i i
with q ∈K−{0}. h (q) is a bijective σ−PBW extension of K:
n
h (q)=σ(K)hx ,...,x ;y ,...,y ;z ,...,z i.
n 1 n 1 n 1 n
(vi) Many other examples are presented in [8].
Gr¨obner Bases for Ideals of σ−PBW Extensions 5
Definition 5. Let A be a σ−PBW extension of R with endomorphisms σ ,
i
1≤i≤n, as in Proposition 2.
(i) For α = (α1,...,αn) ∈ Nn, σα := σ1α1···σnαn, |α| := α1 +···+αn. If
β =(β1,...,βn)∈Nn, then α+β :=(α1+β1,...,αn+βn).
(ii) For X =xα ∈Mon(A), exp(X):=α and deg(X):=|α|.
(iii) Let 0 6= f ∈ A, t(f) is the finite set of terms that conform f, i.e., if
f = c X + ··· + c X , with X ∈ Mon(A) and c ∈ R − {0}, then
1 1 t t i i
t(f):={c X ,...,c X }.
1 1 t t
(iv) Let f be as in (iii), then deg(f):=max{deg(X )}t .
i i=1
The σ−PBW extensions can be characterizedin a similar way as was done in
[3] for PBW rings.
Theorem 6. Let A be a left polynomial ring over R w.r.t {x ,...,x }. A is a
1 n
σ−PBW extension of R if and only if the following conditions hold:
(a) For every xα ∈Mon(A) and every 06=r ∈R there exists unique elements
r :=σα(r)∈R−{0} and p ∈A such that
α α,r
xαr =r xα+p , (1.5)
α α,r
where p = 0 or deg(p ) < |α| if p 6= 0. Moreover, if r is left
α,r α,r α,r
invertible, then r is left invertible.
α
(b) For every xα,xβ ∈ Mon(A) there exist unique elements c ∈ R and
α,β
p ∈A such that
α,β
xαxβ =c xα+β +p , (1.6)
α,β α,β
where c is left invertible, p =0 or deg(p )<|α+β| if p 6=0.
α,β α,β α,β α,β
Proof. See [5].
Remark 7. (i) A left inverse of c will be denoted by c′ . We observe that
α,β α,β
if α=0 or β =0, then c =1 and hence c′ =1.
α,β α,β
(ii)Letθ,γ,β ∈Nnandc∈R,thenweitiseasytocheckthefollowingidentities:
σθ(c )c =c c ,
γ,β θ,γ+β θ,γ θ+γ,β
σθ(σγ(c))c =c σθ+γ(c).
θ,γ θ,γ
(iii)WeobserveifAisaσ−PBW extensionquasi-commutative,thenfromthe
proof of Theorem 6 (see [5]) we conclude that p =0 and p = 0, for every
α,r α,β
06=r ∈R and every α,β ∈Nn.
(iv) We have also that if A is a bijective σ − PBW extension, then c is
α,β
invertible for any α,β ∈Nn.
A key property of σ-PBW extensions is the content of the following theorem.
Gr¨obner Bases for Ideals of σ−PBW Extensions 6
Theorem 8. Let A be a bijective skew PBW extension of R. If R is a left
Noetherian ring then A is also a left Noetherian ring.
Proof. See [8].
Let A = σ(R)hx ,...,x i be a σ−PBW extension of R and let (cid:23) be a total
1 n
order defined on Mon(A). If xα (cid:23)xβ but xα 6=xβ we will write xα ≻xβ. Let
f 6=0 be a polynomial of A, if
f =c X +···+c X ,
1 1 t t
withc ∈R−{0}andX ≻···≻X arethemonomialsoff,thenlm(f):=X
i 1 t 1
is the leading monomial of f, lc(f) := c is the leading coefficient of f and
1
lt(f):= c X is the leading term of f. If f = 0, we define lm(0) := 0,lc(0):=
1 1
0,lt(0) := 0, and we set X ≻ 0 for any X ∈ Mon(A). Thus, we extend (cid:23) to
Mon(A)∪{0}.
Definition 9. Let (cid:23) be a total order on Mon(A), we say that (cid:23) is a monomial
order on Mon(A) if the following conditions hold:
(i) For every xβ,xα,xγ,xλ ∈Mon(A)
xβ (cid:23)xα ⇒ lm(xγxβxλ)(cid:23)lm(xγxαxλ).
(ii) xα (cid:23)1, for every xα ∈Mon(A).
(iii) (cid:23) is degree compatible, i.e., |β|≥|α|⇒xβ (cid:23)xα.
Monomialordersarealsocalledadmissible orders. Fromnowonwewillassume
that Mon(A) is endowed with some monomial order.
Definition 10. Let xα,xβ ∈ Mon(A), we say that xα divides xβ, denoted by
xα|xβ, if there exists xγ,xλ ∈Mon(A) such that xβ =lm(xγxαxλ).
Proposition 11. Let xα,xβ ∈Mon(A) and f,g ∈A−{0}. Then,
(a) lm(xαg)=lm(xαlm(g))=xα+exp(lm(g)). In particular,
lm(lm(f)lm(g))=xexp(lm(f))+exp(lm(g))
and
lm(xαxβ)=xα+β. (1.7)
(b) The following conditions are equivalent:
(i) xα|xβ.
(ii) Thereexistsauniquexθ ∈Mon(A)suchthatxβ =lm(xθxα)=xθ+α
and hence β =θ+α.
(iii) Thereexistsauniquexθ ∈Mon(A)suchthatxβ =lm(xαxθ)=xα+θ
and hence β =α+θ.
Gr¨obner Bases for Ideals of σ−PBW Extensions 7
(iv) β ≥α for 1≤i≤n, with β :=(β ,...,β ) and α:=(α ,...,α ).
i i 1 n 1 n
Proof. See [5].
We note that a least common multiple of monomials of Mon(A) there exists:
in fact, let xα,xβ ∈ Mon(A), then lcm(xα,xβ) = xγ ∈ Mon(A), where γ =
(γ ,...,γ ) with γ :=max{α ,β } for each 1≤i≤n.
1 n i i i
Some naturalcomputationalconditionsonR willbe assumedinthe restofthis
paper (compare with [7]).
Definition 12. A ring R is left Gr¨obner soluble (LGS) if the following condi-
tions hold:
(i) R is left Noetherian.
(ii) Given a,r ,...,r ∈R there exists an algorithm which decides whether a
1 m
is in the left ideal Rr +···+Rr , and if so, find b ,...,b ∈ R such
1 m 1 m
that a=b r +···+b r .
1 1 m m
(iii) Given r ,...,r ∈ R there exists an algorithm which finds a finite set of
1 m
generators of the left R-module
Syz [r ··· r ]:={(b ,...,b )∈Rm|b r +···+b r =0}.
R 1 m 1 m 1 1 m m
The three above conditions imposed to R are needed in order to guarantee a
Gr¨obner theory in the rings of coefficients, in particular, to have an effective
solutionofthemembershipprobleminR(see(ii)inDefinition20below). From
now on we will assume that A = σ(R)hx ,...,x i is a σ−PBW extension of
1 n
R, where R is a LGS ring and Mon(A) is endowedwith some monomialorder.
We conclude this chapter with a remark about some other classes of noncom-
mutative rings of polynomial type close related with σ-PBW extensions.
Remark 13. (i) Viktor Levandovskyy has defined in [6] the G-algebras and
has constructed the theory of Gr¨obner bases for them. Let K be a field, a K-
algebra A is called a G-algebra if K ⊂ Z(A) (center of A) and A is generated
byafinite set{x ,...,x }ofelementsthatsatisfythe followingconditions: (a)
1 n
the collection of standard monomials of A, Mon(A)=Mon({x ,...,x }), is a
1 n
K-basis of A. (b) x x = c x x +d , for 1 ≤ i < j ≤ n, with c ∈ K∗ and
j i ij i j ij ij
d ∈ A. (c) There exists a total order < on Mon(A) such that for i < j,
ij A
lm(d ) < x x . (d) For 1 ≤ i < j < k ≤ n, c c d x −x d +c x d −
ij A i j ik jk ij k k ij jk j ik
c d x +d x −c c x d =0. Accordingtothisdefinition,thecoefficientsof
ij ik j jk i ij ik i jk
a polynomial in a G-algebra are in a field andthey commute with the variables
x ,...,x . From this, and also from (c) and (d), we conclude that the class
1 n
of G-algebras does not coincide with the class of σ-PBW extensions. However,
the intersection of these two classesof rings is not empty. In fact, the universal
enveloping algebra of a finite dimensional Lie algebra, Weyl algebras and the
Gr¨obner Bases for Ideals of σ−PBW Extensions 8
additive or multiplicative analogue of a Weyl algebra, are G-algebras and also
σ-PBW extensions.
(ii) A similar remark can be done with respect to PBW rings and algebras
defined by Bueso, G´omez-Torrecillas and Verschoren in [4].
2 Monomial orders on Mon(Am)
We will often write the elements of Am also as row vectors if this not represent
confusion. We recall that the canonical basis of Am is
e =(1,0,...,0),e =(0,1,0,...,0),...,e =(0,0,...,1).
1 2 m
Definition 14. A monomial in Am is a vector X = Xe , where X = xα ∈
i
Mon(A) and 1≤i≤m, i.e.,
X=Xe =(0,...,X,...,0),
i
where X is in the i-th position, named the index of X, ind(X) := i. A term
is a vector cX, where c ∈ R. The set of monomials of Am will be denoted
by Mon(Am). Let Y = Ye ∈ Mon(Am), we say that X divides Y if i = j
j
and X divides Y. We will say that any monomial X ∈ Mon(Am) divides the
null vector 0. The least common multiple of X and Y, denoted by lcm(X,Y),
is 0 if i 6= j, and Ue , where U = lcm(X,Y), if i = j. Finally, we define
i
exp(X):=exp(X)=α and deg(X):=deg(X)=|α|.
We now define monomials orders on Mon(Am).
Definition 15. A monomial order on Mon(Am) is a total order (cid:23) satisfying
the following three conditions:
(i) lm(xβxα)e (cid:23) xαe , for every monomial X = xαe ∈Mon(Am) and any
i i i
monomial xβ in Mon(A).
(ii) If Y =xβe (cid:23)X =xαe , then lm(xγxβ)e (cid:23)lm(xγxα)e for all X,Y ∈
j i j i
Mon(Am) and every xγ ∈Mon(A).
(iii) (cid:23) is degree compatible, i.e., deg(X)≥deg(Y)⇒X(cid:23)Y.
If X(cid:23)Y but X6=Y we will write X≻Y. Y(cid:22)X means that X(cid:23)Y.
Proposition 16. Every monomial order on Mon(Am) is a well order.
Proof. We can easy adapt the proof for left ideals presented in [5].
Given a monomial order (cid:23) on Mon(A), we can define two natural orders on
Mon(Am).
Definition 17. Let X=Xe and Y=Ye ∈Mon(Am).
i j
(i) The TOP (term over position) order is defined by
Gr¨obner Bases for Ideals of σ−PBW Extensions 9
X (cid:23)Y
X(cid:23)Y⇐⇒ or
X =Yand i>j.
(ii) The TOPREV order is defined by
X (cid:23)Y
X(cid:23)Y⇐⇒ or
X =Yand i<j.
Remark 18. (i) Note that with TOPwe have
e ≻e ≻···≻e
m m−1 1
and
e ≻e ≻···≻e
1 2 m
for TOPREV.
(ii)ThePOT(positionoverterm)andPOTREVordersdefinedin[1]and[7]for
modulesoverclassicalpolynomialcommutativeringsarenotdegreecompatible.
(iii) Other examples of monomial orders in Mon(Am) are considered in [4].
We fix monomial orders on Mon(A) and Mon(Am); let f 6= 0 be a vector of
Am, then we may write f as a sum of terms in the following way
f =c X +···+c X ,
1 1 t t
where c ,...,c ∈ R − {0} and X ≻ X ≻ ··· ≻ X are monomials of
1 t 1 2 t
Mon(Am).
Definition 19. With the above notation, we say that
(i) lt(f):=c X is the leading term of f.
1 1
(ii) lc(f):=c is the leading coefficient of f.
1
(iii) lm(f):=X is the leading monomial of f.
1
For f = 0 we define lm(0) = 0,lc(0) = 0,lt(0) = 0, and if (cid:23) is a monomial
order on Mon(Am), then we define X ≻ 0 for any X ∈ Mon(Am). So, we
extend (cid:23) to Mon(Am)∪{0}.
3 Reduction in Am
The reduction process in Am is defined as follows.
Definition 20. Let F be a finite set of non-zero vectors of Am, and let f,h ∈
Am, we say that f reduces to h by F in one step, denoted f−−F→h, if there exist
elements f ,...,f ∈F and r ,...,r ∈R such that
1 t 1 t
Gr¨obner Bases for Ideals of σ−PBW Extensions 10
(i) lm(f )|lm(f), 1 ≤ i ≤ t, i.e., ind(lm(f )) = ind(lm(f)) and there exists
i i
xαi ∈Mon(A) such that αi+exp(lm(fi))=exp(lm(f)).
(ii) lc(f) = r1σα1(lc(f1))cα1,f1 + ··· + rtσαt(lc(ft))cαt,ft, with cαi,fi :=
c .
αi,exp(lm(fi))
(iii) h=f− ti=1rixαifi.
We say that fPreduces to h by F, denoted f −−F→ h, if and only if there exist
+
vectors h ,...,h ∈Am such that
1 t−1
F F F F F
f −−−−→ h −−−−→ h −−−−→ ··· −−−−→ h −−−−→ h.
1 2 t−1
f is reduced (also called minimal) w.r.t. F if f = 0 or there is no one step
reduction of f by F, i.e., one of the first two conditions of Definition 20 fails.
F
Otherwise, we will say that f is reducible w.r.t. F. If f−−→ h and h is reduced
+
w.r.t. F, then we say that h is a remainder for f w.r.t. F.
Remark 21. Relatedto the previousdefinitionwehavethe followingremarks:
(i) By Theorem 6, the coefficients c are unique and satisfy
αi,fi
xαixexp(lm(fi)) =cαi,fixαi+exp(lm(fi))+pαi,fi,
where p =0 or deg(lm(p ))<|α +exp(lm(f ))|, 1≤i≤t.
αi,fi αi,fi i i
(ii) lm(f) ≻ lm(h) and f − h ∈ hFi, where hFi is the submodule of Am
generated by F.
(iii) The remainder of f is not unique.
F
(iv) By definition we will assume that 0−→0.
(v)
t
lt(f)= r lt(xαilt(f )),
i i
i=1
X
The proofs of the next technical proposition and theorem can be also adapted
from [5].
Proposition 22. Let A be a σ-PBW extension such that c is invertible for
α,β
each α,β ∈ Nn. Let f,h ∈ Am, θ ∈ Nn and F = {f ,...,f } be a finite set of
1 t
non-zero vectors of Am. Then,
(i) If f −−F→ h, then there exists p ∈ Am with p = 0 or lm(xθf) ≻ lm(p)
such that xθf+p−−F→xθh. In particular, if A is quasi-commutative, then
p=0.
(ii) If f −−F→ h and p ∈ Am is such that p = 0 or lm(h) ≻ lm(p), then
+
F
f+p−−→ h+p.
+
(iii) If f −−F→ h, then there exists p ∈ Am with p = 0 or lm(xθf) ≻ lm(p)
+
such that xθf+p−−F→ xθh. If A is quasi-commutative, then p=0.
+
