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Gröbner bases of modules 5 σ PBW over − extensions 1 0 2 ∗ Haydee Jiménez & Oswaldo Lezama n a Seminario de A´lgebra Constructiva - SAC2 J Departamento de Matemáticas 0 Universidad Nacional de Colombia, Bogotá, COLOMBIA 3 ] A R Abstract h. For σ−PWB extensions, we extend to modules the theory of Gröbner 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 Gröbner 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 Gröbner Bases for Ideals of σ−PBW Extensions 2 1 Introduction In this paper we present the theory of Gröbner 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 Gröbner 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 Gröbner bases, and finally, we will compute Gröbner 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 Gröbner 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. Gröbner 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 extension; 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: Gröbner 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]. Gröbner 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. Gröbner 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. Gröbner 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 Gröbner soluble (LGS) if the following conditions 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 Gröbner 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 noncommutative 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 Gröbner 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 Gröbner 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, Gómez-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 Gröbner 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 by X (cid:23)Y X(cid:23)Y⇐⇒ or  X =Yand i<j. Remark 18. (i) Note that with TOPwe 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 Gröbner 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. +