Gröbner Basis in Algebras Extended by Loops

Chalom, G.; Marcos, E.; Oliveira, P.

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Revista de la Unión Matemática Argentina

Print version ISSN 0041-6932On-line version ISSN 1669-9637

Rev. Unión Mat. Argent. vol.48 no.3 Bahía Blanca 2007

Gröbner Basis in Algebras Extended by Loops

G. Chalom, E. Marcos, P. Oliveira

We dedicate this work to the 60th birthday of Maria Ines Platzeck and the 70th birthday of Hector Merklen.

Abstract. In this work we extend, to the path algebras context, some results obtained in the commutative context, [2]. The main result is that one can extend the Gröbner bases of an ungraded ideal to one possible definition of homogenization for the non commutative case.

The second author thanks CNPq for support in the form of a research grant, and the third one for partially financing his master degree. The three authors take the opportunity to thank Prof. Ed. Green for suggesting the subject.

We will introduce very briefly the homogenization process in the commutative case, just to explain the main motivation of our work. In the commutative context, the Buchberger Algorithm give us a very direct strategy for computing Gröbner Basis for a given ideal I ∈ k[x1,x2,...,xn ] : we consider a finite set {f1,f2,...fk} of generators of I, compute the S polynomials, for any pair i,j, reduce them, and if the remainder is non zero, add this remainder to the list of the given polynomials, to make all the S polynomials reduce to zero.

Although this process always finish, in the commutative case, it can be very inefficient and time consuming, by instance getting S polynomials of much higher degree that the ones we begin with. It is easy to see ( see [1]) that if we begin with a set of homogeneos polynomials this problem does not occur and the S polynomials we obtain are again homogeneous.

So, lets define this process for Λ = k[x1,x2,...,xn] : Let f ∈ Λ and a new variable. If has total degree then the polynomial given by f * = wdf (x1 ∕w,x2 ∕w,... ,xn∕w ) ∈ k [x1, x2,...,xn,w ] is a homogeneous polynomial in the extended polynomial algebra, called the homogenization of . For an ideal I ∈ k [x ,x ,...,x ] 1 2 n define to be the ideal of k[x1,x2,...,xn, w] given by * * I = < f |f ∈ I > . For any h ∈ k [x1,x2, ...,xn,w ] , define h * = h(x1,x2,...,xn, 1) ∈ k [x1,x2, ...,xn] .

As we will prove, in the last section, if G is a Gröbner basis for I with respect to a certain order, then the set G * = {g*|g ∈ G} is a Gröbner Basis for the ideal with respect to the extended order.

For the non commutative case, this process has been extended in many contexts, and most computer programs devoted to non commutative Grobner Basis work only with homogeneous ideals [4].

2. Preliminaries

In this section, we define some concepts that will be used in the following sections. All these concepts can be found in [3], with a detailed description of the theory of Gröbner basis.

In order to have a Gröebner basis theory in an algebra we need a multiplicative basis with an admissible order. We define, in the sequence, these concepts.

A -basis is called a multiplicative basis of . if for every ′ b,b ∈ B we have ′ bb ∈ B or ′ bb = 0 .

We also will need the multiplicative basis to be completely ordered. We stress that we are not interested in an arbitrary order in , but we want an order that preserves the multiplicative structure of .

DEFINITION. 2.1. [3] We will say that a well order in , is admissible, if it satisfies the following conditions, for every p,q,r,s ∈ B :

If then , if both are non zero;
If then , if both are non zero;
If then .

Let be a field and a -algebra with a fixed -basis B = {bi}i∈I.

Since is a -basis of , for each a ∈ Λ , there is a unique family (λi)i∈I such that ∑ a = i∈I λibi , where λi = 0 , except for a finite number of indices.

If ∑ a = i∈I λibi , we will say that occurs in if λi ⁄= 0 . We define now the notion of tip, which is also called in the literature by leading term.

DEFINITION. 2.2. [3] If B = {bi}i∈I is a -basis of , as a vector space, well ordered by in , and if ∑ a = i∈I λibi is non zero, we will call tip of a and denote by Tip(a) the largest basis element in the support of and its coefficient is denoted by CT ip(a) .

If is a subset of , we define

for some
$N onT ip(X ) = B \ T ip(X )$

So, both T ip(X ) and N onT ip(X ) are subsets of depending on the choice of the well order of .

DEFINITION. 2.3. [3] Let be a two sided ideal of , we will say that a set G ⊂ I is a Gröbner basis for with respect to the order , if

⟨T ip(G)⟩ = ⟨Tip(I)⟩

as two-sided ideals.

DEFINITION. 2.4. [3] Let b1,b2 ∈ X ⊂ Λ , we will say that divides (in ) if there exist c,d ∈ X such that b2 = b1d , b2 = cb1 or b2 = cb1d .

We will say that a -algebra has Gröbner basis theory if has a multiplicative basis with an admissible order in this basis.
From this point on, we assume that the -algebra has a Gröbner basis theory. Moreover will always denote a two-sided ideal in .

Given a multiplicative basis of an arbitrary algebra and and a fixed minimal set of generators of , as a semigroup, we define ℓ(b) the length of b ∈ B as the smaller n ∈ ℕ such that b = b1b2⋅⋅⋅bn with b ∈ U. i If f ∈ Λ f ⁄= 0 define the length of f by ℓ(f) = max {ℓ(b) : b ∈ B occurs in .
We say that an element ∑n f = i=1λibi , with λi ∈ K and bi ∈ B, is homogeneous if ℓ(f) = ℓ(bi) for every 1 ≤ i ≤ n . An ideal is homogeneous if can be generated by homogeneous elements.

3. Homogenization

In this section, we present our main results, which extend the algorithms used in commutative algebra, and also some results obtained in [2].

Since the polynomial ring on commutative variables is a special case of a quotient of a path algebra and, as it was proved in [2], any algebra with 1 that admits Gröbner basis theory is isomorphic to a quotient of a path algebra, we asked ourselves if the same process ( that is, the homogenization process) can be extended, and which results remain true in the general case of quotient path algebras. In this work, we consider the non commutative version of the homogenization process, for path algebras KQ ∕I , where is a two-sided ideal in .

In [2], Green used a similar technic of the extension by loops, to construct Gröbner basis to some indecomposable projectives in Mod-, based on a special admissible order, where the loops where always maximal elements.

In our work, we start with a quotient of a path algebra with an admissible order, and we define another quotient of path algebra, also with an admissible order, which we will call the extended by loops algebra.

Let be a field and a finite quiver. Let Λ = KQ ∕I be the path algebra associated to and a two-sided ideal of . Consider in the multiplicative basis and an admissible order in .

DEFINITION. 3.1. Let Λ = KQ ∕I , as above, we define , where has the same vertices of and for each vertex of , we add a loop in ˜Q1 , and we consider the -algebra Λ′ = KQ˜∕I˜ , where ˜I = ⟨I,αZ - Zα ⟩ as a two-sided ideal of KQ˜ , with α ∈ Q 1 and ∑ Z = i∈|Q0|li . We call ′ Λ the extended by loops algebra of

Observe that is the sum of all the new loops that were added to the quiver. For we consider the following basis ˜B = {Znb : b ∈ B and n ≥ 0} . Both and Λ ′ are finitely generated as -algebras, moreover and ˜ B are finitely generated as semigroups.
For each generator set of , as a semigroup, we associate the following generator for , ˜U = U ∪ {li : i ∈ |Q0|} . It is not hard to see that is minimal if and only if is minimal.

Define in the order

ei ≺ lj ≺ b, f or every i,j ∈ |Q0 | and b ∈ B and

( { if b1 < b2 in B or Znb1 ≺ Zmb2 if : if b1 = b2 in B and n < m (

We show now that this is, in fact, an admissible order.
Let Znb ,Zmb ,Zrb ,Zsb ∈ ˜B 1 2 3 4 , then:

if and and are non zero, we have that, if then . Now, if , and so and . Then, .
in the same way, if , then , if the products are non zero.
if , we have that and so .

Therefore, the order given above is an admissible order.

DEFINITION. 3.2. For ∑ f = mi=1 λibi ∈ Λ , we define the homogenization of f in by

f* = ∑m λ Z ℓ(f)-ℓ(bi)b i=1 i i .

Observe that, for every f ∈ Λ , the homogenization of is an homogeneous element.

LEMMA 3.3. For every f,g ∈ Λ , we have k * * * Z (fg) = f g , with k = ℓ(f ) + ℓ(g) - ℓ(fg) .

PROOF. Let ∑ f = ni=1 λibi and ∑ g = mj=1 βjbj , with λi,βj ∈ Λ and bi,bj ∈ B . Since ℓ(f) + ℓ(g) ≥ ℓ(f g) , consider the natural number k = ℓ(f ) + ℓ(g) - ℓ(fg) . Then,

∑n m∑ f*g* = ( λ b )*( β b )* ii j j i=1 j=1 ∑n ℓ(f)-ℓ(bi) ∑m ℓ(g)-ℓ(bj) = ( λiZ bi)( βjZ bj) i=1 j=1 ∑ (ℓ(f)+ℓ(g))- (ℓ(bi)+ ℓ(bj)) = λiβjZ bibj ∑i,j = λ β Zℓ(fg)-ℓ(bibj)Zkb b i j ij i,j∑ = Zk( λiβjbibj)* i,j n m k ∑ ∑ * = Z (( λibi)( βjbj)) i=1 j=1 = Zk(f g)*

∎

Now, we define the following application, between the algebras ′ Λ and :

φ : Λ ′ → Λ

that associates to each element n Z b ∈ ˜B

the element b ∈ B

, for every n ∈ ℕ

. Observe that, for every b ∈ B

, there exists Zb ∈ ˜B

such that

. In this way, we have that

extended by linearity to every element in Λ ′

is, in fact, an epimorphism of algebras. Also, observe that ker (φ ) = ⟨Z - 1⟩

To simplify the notation, we call g* = φ (g ) for every ′ g ∈ Λ .

LEMMA 3.4. For every f ∈ Λ we have * (f )* = f .

PROOF. Let ∑n f = i=1 λibi , with λi ∈ Λ and bi ∈ B . Observe that

n * ∑ ℓ(f)-ℓ(bi) (f )* = ( λiZ bi)* i=1 ∑n = (λiZ ℓ(f)-ℓ(bi)bi)* i=1 ∑n = λi(Zℓ(f)-ℓ(bi)))*(bi)* i=1 n = ∑ λ b i i i=1 = f

∎

LEMMA 3.5. Let g ∈ Λ ′ homogeneous of length and let d′ = ℓ(g ) * . Then d′ ≤ d and g = Zd -d′(g )* * .

PROOF. The inequality ′ d ≤ d follows from the definition of . Let m ∈ supp ˜B(g) , m = Zit , with t ∈ B . Then the monomial m * ∈ suppB (g*) correspondent to is . As ℓ(t) = d - i the monomial in supp ((g*)*) correspondent to m * is tZd′-(d-i) . Then Zd -d′(g )* = g * . ∎

DEFINITION. 3.6. Let F ⊂ Λ and ′ G ⊂ Λ , we define by

F * = {f* : f ∈ F }

G = {g : g ∈ G } * *

LEMMA 3.7. Let f ∈ Λ . Then * ℓ(f) = ℓ(f ) .

PROOF. Consider ∑m f = i=1 λibi with λi ∈ K and bi ∈ B , where is a basis of , 1 ≤ i ≤ m .

By definition we have that f* = ∑m λiZl(f)-l(bi)bi i=1 . For every summand of we have:

ℓ(f)-ℓ(bi) ℓ(f)- ℓ(bi) ℓ(Z bi) = ℓ(Z ) + ℓ(bi) = (ℓ(f) - ℓ(bi)) + ℓ(bi) = ℓ(f)

Then, ℓ(f*) = max { ℓ(Z ℓ(f)-ℓ(bi)bi) : 1 ≤ i ≤ m } = ℓ(f) ∎

LEMMA 3.8. Let F = {f } i i∈I be a subset of , not necessarily finite, and ∑m f = i=1rifisi ∈ ⟨F ⟩ . If * * * d = max { ℓ((ri) (fi) (si)) : 1 ≤ i ≤ m } and d′ = ℓ(f) . Then ′ Zd -df * ∈ ⟨F *⟩ .

PROOF. Consider ki = ℓ(ri) + ℓ(fi) + ℓ(si) - ℓ(rifisi) , 1 ≤ i ≤ m , by 3.3 we have ki * * * * z (rifisi) = (ri) (fi)(si)

Let -- ∑m ∑m f = ( i=1(ri)*(fi)*(si)*)* = i=1Zd -ℓ(ri)+l(fi)+l(si)(ri)*(fi)*(si)* =

∑ ( mi=1 Zd- ℓ(ri)+l(fi)+l(si)Zki(rifisi)* , by lemma 3.3. So, -- f ∈ ⟨F *⟩ and is homogeneous (by construction ) with d ′′ = ℓ(f-) ≤ d . Moreover, using lemma 3.3 and lemma 3.4, we have

-- ∑m d-ℓ(r)+ℓ(f )+ℓ(s) k * f * = ( Z i i i Z i(rifisi) )* i=1 ∑m = Zd*- ℓ(ri)+ℓ(fi)+ ℓ(si)(r*i)*(f *i )*(s*i)* i=1 ∑m = rf s = f ii i i=1

Using Lemma 3.5, we can conclude that

-- d′′-d′-- * d′′-d′ * f = Z (f*) = Z f

As d′′ ≤ d , finally we have:

d-d′ * d-d′′ d′′-d′ * d- d′′-- * Z f = Z Z f = Z f ∈ ⟨F ⟩

∎

LEMMA 3.9. Let be a subset of . Then (⟨F*⟩)* = ⟨F ⟩ .

PROOF. Let f ∈ ⟨F ⟩ . By Lemma 3.8, Zkf * ∈ ⟨F *⟩ , for some k ∈ ℕ , and then

f = (f*)* = (Zkf *)* ∈ (⟨F*⟩)*

By the other hand, if * g ∈ ⟨F ⟩ , say ∑m * g = i=1 ri(fi) si with fi ∈ F and ri,si ∈ Λ′ for 1 ≤ i ≤ m , we have

m∑ * g* = ( ri(fi)si)* i=1 ∑m = (ri)*[(fi)*]*(si)* i=1 ∑m = (r )f (s ) i* i i* i=1

So, g* ∈ ⟨F⟩ . ∎

We reproduce here the Elimination Theorem, found in [2], to discuss and compare the two results. For that, we define some new concepts.

Let be a quiver and <ll a length-lexicographic order defined in the basis of paths of . Let be a maximal arrow with respect to <ll in .

We define the quiver in the following way: (Q α)0 = Q0 and $(Q α)1 = Q1 \ {α}$ .

For a set of indices, we define the following application V : T -→ Q0 . Let ∐ P = V (i)KQ i∈T a ( right) projective in -Mod.

We define ∐ Pα = V (i)KQ α i∈T a right projective module in

KQ α -Mod.

Let be a -basis of with order such that:

For every and every , if , then , if and are non zero.
For every and every , if , then , if both are non zero.
For every and every , or .

Let m ∈ P , ∑ m = λimibi i∈T , with mi ∈ BP , bi ∈ B e λi ∈ K . We call tip(m ) = the m i such that m ≼ m i j for every j ∈ T . For X ⊂ P , we will call by tip(X ) = {tip(x) : x ⁄= 0,x ∈ X } .

Following Green, we say that G ⊂ P is right a Gröbner basis for , with respect to the order , if tip(G ) generates tip(P ) as a right module.

Here is Green's Elimination Theorem, found in [2].

THEOREM 3.10. [2] Let be a quiver and let <ll be a length-lexicographic order in , where is the set of paths in . Let be a maximal arrow with respect to <ll in and ∐ P = V (i)KQ i∈T a projective in -Mod. Let B P be an ordered basis ( as defined above ) for . If is a right uniform (reduced ) Gröbner basis for P, then Gα = G ∩ Pα is a right uniform (reduced ) Gröbner basis for .

As a consequence of the Elimination Theory, Green find a new algebra KQ [T] , that we will call added by loops.

This algebra KQ [T] is an hereditary algebra, obtained adding loops to , as above, but without adding any relation. Observe that both are hereditary algebras and the basis of KQ [T ] is ordered in such a way that the new loops are maximal elements. In this situation, given two ideals and generators sets (Gröbner basis), we can find, as described in [2], a generators set (Gröbner basis), of the intersection of these ideals, constructed by the Elimination Theorem (that can be found, with more details, in [2], section 8).

In our work, there are no additional hypothesis over the given order, the only additional assumption is that the extra loops must be between the vertices and the arrows.

Moreover, we consider the more general case, where Λ = KQ ∕I , is not necessarily hereditary.

THEOREM 3.11. Let be a subset of and let ′ G ⊂ Λ be homogeneous. If is a Gröbner basis for ⟨F *⟩ , then is a Gröbner basis for ⟨F ⟩ .

PROOF. Suppose that is a Gröbner basis for * ⟨F ⟩ . We will prove the theorem, using the definition of Gröbner basis.

As G* ⊂ ⟨F ⟩ , then ⟨T ip(G*)⟩ ⊆ ⟨Tip(⟨F ⟩)⟩ , and we only need to verify that ⟨T ip (⟨F ⟩)⟩ ⊆ ⟨Tip(G*)⟩ , that is, if given f ∈ ⟨F⟩ there exists g ∈ G * * such that T ip(g ) * divides T ip(f) .

Let f ∈ ⟨F⟩ , we can write ∑m f = i=1 λibi , where λi ∈ K and bi ∈ B for 1 ≤ i ≤ m .

Without lost of generality, assume that Tip(f) = b1 , then

m * ℓ(f)- ℓ(b1) ∑ ℓ(f)-ℓ(bi) f = λ1Z b1 + λiZ bi i=2

it follows by the given order that T ip (f *) = Zℓ(f)-ℓ(b1)b1 = ZnT ip(f) .

By lemma 3.8, there exists k ∈ ℕ such that h = Zkf * ∈ ⟨F *⟩ . By the above observation, T ip (h ) = ZkZnT ip (f ) .

As is a Gröbner basis for ⟨F *⟩ , there exists g ∈ G such that kr ks T ip(h) = Z rT ip(g)Z s , for some kr ks ˜ Z r,Z s ∈ B . ˜ T ip (g ) ∈ B , so T ip(g) = Zkgb for some b ∈ B and kg ∈ ℕ . By the definition of order in Λ ′ , for every Ztbt ⁄= Tip(g) that occurs in , Zkgb > Ztbt , then b > bt , or b = bt and kg > t , but this cannot occur, because is homogeneous, so T ip(g) = b * .

Then, k n kr ks kr kg ks kr+kg+ks T ip(h ) = Z Z b1 = Z rT ip(g)Z s = Z rZ bZ s = Z rbs . So, we have
b1 = (T ip(h ))* = (ZkZnb1 )* = (Zkr rTip(g)Zkss)* = (Zkr rZkgbZkss )* =

(Zkr+kg+ksrbs )* = rbs.

Then T ip(f) = rT ip (g *)s , and s G * is a Gröbner basis for ⟨F⟩ . ∎

References

[1] Becker, T., Weispfenning, V., Gröbner Bases, A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics, Springer-Verlag. 1993. [ Links ]

[2] Green, E. L., Multiplicative Bases,Gröbner Bases, and Right Grbner Bases, J. Symbolic Computation, 29, 2000, n.4-5, 601-623. [ Links ]

[3] Green, E. L., Non commutative Gröbner Bases and Projectives Resolutions, In Michler and Schneider, eds, Proceedings of the Euroconference Computational Methods for Representations of Groups and Algebras, Essen, 1997, vol. 173 of Progress in Mathematics, 29-60. Basel, Bikhaser Verlag. [ Links ]

[4] Nordbeck,P. On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings Gröbner Basis and Applications, London Mathematical Society Lecture Note Series, Vol. 251, Edited by B. Buchberger and Franz Winkler. [ Links ]

Gladys Chalom
Departamento de Matemática - IME,
Universidade de São Paulo,
CP 66281, 05315-970, São Paulo, Brasil
agchalom@ime.usp.br

Eduardo do Nascimento Marcos
Departamento de Matemática - IME,
Universidade de São Paulo,
CP 66281, 05315-970, São Paulo, Brasil
enmarcos@ime.usp.br

P. Oliveira
Departamento de Matemática - IME,
Universidade de São Paulo,
CP 66281, 05315-970, São Paulo, Brasil

Recibido: 31 de enero de 2007
Aceptado: 20 de diciembre de 2007