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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.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 : we consider a finite set 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 : Let and a new variable. If has total degree then the polynomial given by is a homogeneous polynomial in the extended polynomial algebra, called the homogenization of . For an ideal define to be the ideal of given by . For any , define .
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 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].
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 we have or .
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 :
- 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
Since is a -basis of , for each , there is a unique family such that , where , except for a finite number of indices.
If , we will say that occurs in if . We define now the notion of tip, which is also called in the literature by leading term.
DEFINITION. 2.2. [3] If is a -basis of , as a vector space, well ordered by in , and if is non zero, we will call tip of a and denote by the largest basis element in the support of and its coefficient is denoted by .
If is a subset of , we define
- for some
So, both and 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 is a Gröbner basis for with respect to the order , if
DEFINITION. 2.4. [3] Let , we will say that divides (in ) if there exist such that , or .
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 the length of as the smaller such that with If define the length of f by occurs in .
We say that an element , with and is homogeneous if for every . An ideal is homogeneous if can be generated by homogeneous elements.
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 , 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 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 , as above, we define , where has the same vertices of and for each vertex of , we add a loop in , and we consider the -algebra , where as a two-sided ideal of , with and . We call 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 and . Both and are finitely generated as -algebras, moreover and are finitely generated as semigroups.
For each generator set of , as a semigroup, we associate the following generator for , . It is not hard to see that is minimal if and only if is minimal.
Define in the order
We show now that this is, in fact, an admissible order.
Let , 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 , we define the homogenization of f in by
.
Observe that, for every , the homogenization of is an homogeneous element.
LEMMA 3.3. For every , we have , with .
PROOF. Let and , with and . Since , consider the natural number . Then,
∎
Now, we define the following application, between the algebras and :
To simplify the notation, we call for every .
LEMMA 3.4. For every we have .
PROOF. Let , with and . Observe that
∎
LEMMA 3.5. Let homogeneous of length and let . Then and .
PROOF. The inequality follows from the definition of . Let , , with . Then the monomial correspondent to is . As the monomial in correspondent to is . Then . ∎
DEFINITION. 3.6. Let and , we define by
PROOF. Consider with and , where is a basis of , .
By definition we have that . For every summand of we have:
Then, ∎
LEMMA 3.8. Let be a subset of , not necessarily finite, and . If and . Then .
PROOF. Consider , , by 3.3 we have
Let
, by lemma 3.3. So, and is homogeneous (by construction ) with . Moreover, using lemma 3.3 and lemma 3.4, we have
Using Lemma 3.5, we can conclude that
As , finally we have:
∎
LEMMA 3.9. Let be a subset of . Then .
PROOF. Let . By Lemma 3.8, , for some , and then
By the other hand, if , say with and for , we have
So, . ∎
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 a length-lexicographic order defined in the basis of paths of . Let be a maximal arrow with respect to in .
We define the quiver in the following way: and .
For a set of indices, we define the following application . Let a ( right) projective in -Mod.
We define a right projective module in
-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 , , with , e . We call the such that for every . For , we will call by .
Following Green, we say that is right a Gröbner basis for , with respect to the order , if generates as a right module.
Here is Green's Elimination Theorem, found in [2].
THEOREM 3.10. [2] Let be a quiver and let be a length-lexicographic order in , where is the set of paths in . Let be a maximal arrow with respect to in and a projective in -Mod. Let be an ordered basis ( as defined above ) for . If is a right uniform (reduced ) Gröbner basis for P, then is a right uniform (reduced ) Gröbner basis for .
As a consequence of the Elimination Theory, Green find a new algebra , that we will call added by loops.
This algebra 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 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 , is not necessarily hereditary.
THEOREM 3.11. Let be a subset of and let be homogeneous. If is a Gröbner basis for , then is a Gröbner basis for .
PROOF. Suppose that is a Gröbner basis for . We will prove the theorem, using the definition of Gröbner basis.
As , then , and we only need to verify that , that is, if given there exists such that divides .
Let , we can write , where and for .
Without lost of generality, assume that , then
it follows by the given order that .
By lemma 3.8, there exists such that . By the above observation, .
As is a Gröbner basis for , there exists such that , for some . , so for some and . By the definition of order in , for every that occurs in , , then , or and , but this cannot occur, because is homogeneous, so .
Then, . So, we have
Then , and s is a Gröbner basis for . ∎
[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