Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and difference–differential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchberger-type algorithm for constructing relative Gröbner bases of filtered free modules.

In the last fifteen years the Gröbner basis approach was applied to bifiltered and multifiltered modules over polynomial rings, rings of differential, difference and difference-differential polynomials, and Weyl algebras. The corresponding techniques use different types of reduction with respect to several term orderings; the resulting Gröbner-type bases are called Gröbner bases with respect to several term orderings ( [13][14][15] and [5]) and relative Gröbner bases ( [20,22] and [4]). As applications of the generalized Gröbner basis techniques, these works present proofs of the existence and methods of computation of multivariate difference-differential dimension polynomials, as well as bivariate Bernstein-type dimension polynomials of modules over Weyl algebras. Furthermore, a generalization of the relative Gröbner basis technique to the case of difference-differential modules with weighted basic operators obtained by Dönch in [3] allowed him to prove the existence and obtain methods of computation of Ehrhart-type dimension quasi-polynomials associated with filtrations of such modules.
In this paper we unify the theories of relative Gröbner bases and Gröbner bases with respect to several term orderings (including their "weighted" versions) by developing a generalized relative Gröbner basis method for a wide class of filtered modules that includes modules over rings of differential, difference, inversive difference and difference-differential operators, Weyl algebras, and also multiparameter twisted Weyl algebras introduced and studied in [9]. (Note that the last class of algebras includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras that play an important role in the quantum group covariant differential calculus, see, for example, [19].) In the next section we describe basic settings, give a characterization of ring filtrations considered in the rest of the paper and present a concept of Gröbner reduction in a free module over a filtered ring with monomial filtration. The main results are presented in Sect. 3, where we introduce concepts of admissible orders and set-relative reduction in free modules over rings with monomial filtrations, define the notion of Gröbner basis in this setting and obtain a Buchberger-type algorithm for its construction.

Algebraic Setup
Throughout the paper Z, N and Q denote the sets of integers, nonnegative integers and rational numbers, respectively. If p is a positive integer, then N p is treated as a commutative semigroup with componentwise addition and as a partially ordered set with the product order ≤ π such that In what follows, R denotes an arbitrary, left noetherian (non-commutative) ring containing a commutative ring K ⊆ R as a subring. Unless the contrary is indicated, by an R-module, we always mean a left R-module.
Let M be a left R-module. A subset M of this module is said to be a set of monomials of M, if M is a free K -module with basis M, that is, every element f ∈ M has a unique representation of the form We denote by the support of f , i.e. the set of monomials that appear in f with a non-zero coefficient. For example, if K is a field and is a basis of R as a K -vector space (or K is a commutative ring and R is a free K -module with basis ), then is a set of monomials of R. If F is a free R-module with basis E, then it is easy to see that E = {λe : λ ∈ , e ∈ E} is the set of monomials of F. In this case we write R = K ( ) and F = R (E) = K ( E) . Definition 2.1 A family of K -submodules {R r : r ∈ N p } of R is called a p-fold filtration of a ring R if 1. R r ⊆ R s , whenever r ≤ π s ∈ N p ; 2. R r · R s ⊆ R r +s , r, s ∈ N p ; 3. R = r ∈N p R r ; 4. 1 ∈ R 0 where R 0 stands for R (0,...,0) .
In this case we say that R is a p-filtered ring. (If p = 2 we also use the term bifiltration). A p-fold filtration of R is called monomial (and R is said to be a monomially p-filtered ring) if and only if the inclusion f ∈ R r implies the inclusion T( f ) ⊆ R r .
An R-module equipped with a p-fold filtration is said to be a p-filtered module.
Note that if R is a ring with p-fold filtration {R r : r ∈ N p } and M is a finitely generated R-module with generators {h 1 , . . . , h q }, then M can be naturally treated as a p-filtered module with the p-fold filtration Clearly, this filtration is monomial if the filtration of R is monomial.
Given any mapping u : R → N, one can consider a family of additive subgroups {R The following statement tells when a family of this kind is a (onefold) filtration of R.  Proof Clearly, if u : R → N is a mapping satisfying the above conditions and R (u) k is a K -module.). For the converse, suppose that u is a mapping from R to N such that the family R The other properties of the map u follow from the fact that every R (u) k is a K -module and from the first two conditions of Definition 2.1.
In order to prove the last part of the statement, consider a onefold filtration {R r : r ∈ N} of R and define the mapping u : R → N by setting u(x) = min{k : x ∈ R k }. It is easy to check that u satisfies conditions (i)-(iii). Indeed, since R 0 = K , we have that u(a) = 0 for any a ∈ K and, conversely, the equality u(x) = 0 (x ∈ R) implies that x ∈ R 0 = K . Furthermore, the fact that every R k is a K -module and the first two properties of a filtration (see Definition 2.1) imply that the mapping u satisfies conditions (ii) and (iii).
It remains to show that R (u) Remark 2.4 The first part of Lemma 2.3 can be generalized to p-fold filtrations ( p > 1) as follows. Let us consider a mapping u : R → N p and let Then, one can mimic the corresponding part of the proof of Lemma 2.3 to obtain that {R (u) r : r ∈ N p } is a p-fold filtration of R if and only if the mapping u satisfies the following conditions: At the same time, if p > 1, then not every p-fold filtration is of the form {R (u) r : r ∈ N p } with a mapping u : R → N p satisfying the above conditions. It follows from the fact that the same element of R can belong to different components R r and R s with incomparable (with respect to ≤ π ) p-tuples r, s ∈ N p . For example, let be a polynomial ring in two variables over a field K , equipped with a natural twofold filtration and let the factor ring R r 1 ,r 2 for every r = (r 1 , r 2 ) ∈ N 2 , then one would have u 1 (t) ≤ 0 and u 2 (t) ≤ 0 (we use the notation of Remark 2.4). These inequalities imply that t ∈ K , contrary to the obvious fact that Let R be a p-fold filtered ring with a p-fold filtration {R r : r ∈ N p }, and let F be a free R-module with set of free generators E. Assume that R is a free K -module with basis (we still use the notation and conventions introduced at the beginning of this section), then the support T( f ) of an element f ∈ F is defined as its support with respect to the set of monomials In what follows, a binary relation ρ ⊆ F × F is said to be a reduction and the inclusion ( f, g) ∈ ρ is written as Finally, I ρ will denote the set of ρ-irreducible elements of F, that is With the above notation, one can consider the following concept of Gröbner reduction first introduced in [7]. Definition 2.5 (Gröbner Reduction) With the above notation, let the p-fold filtration of the ring R be monomial and let N be an R-submodule of the free R-module F. A reduction ρ ⊆ F × F is said to be a Gröbner reduction for N if and only if it satisfies the following conditions

Remark 2.6
Examples of Gröbner reductions can be found in [6, Chapter 3] and [8]. If K is a field of zero characteristic, then the theory of Gröbner reduction provides an algorithmic computation of the dimensions of components of p-fold filtrations of finitely generated R-modules. In the cases of p-dimensional filtrations of differential, difference and difference-differential modules (where R is the ring of the corresponding operators), the dimensions of the components are expressed by multivariate polynomials in p variables with rational coefficients (see [12,Theorem 4.3.39] and [17,Theorems 3.3.16]). Similar result for modules over Weyl algebras and rings of Ore polynomials are obtained in [14] and [16]. (Note that by a ring of Ore polynomials we mean a ring defined as follows. Let K be a field and = {δ 1 , . . . , δ n }, σ = {α 1 , . . . , α n } sets of derivations and injective endomorphisms of K , respectively, such that any two mappings from the set ∪ σ commute. Let = (X ) be a free commutative semigroup generated by a set X = {x 1 , . . . , x n } and let O denote the vector K -space with the basis (elements of O are of the form θ∈ a θ θ where a θ ∈ K and only finitely many coefficients a θ are different from zero). Then O can be treated as a ring if one introduces the multiplication according to the rule x i a = α i (a)x i + δ i (a) (a ∈ K , 1 ≤ i ≤ n) and the distributive laws. Then we say that O is a ring of Ore polynomials in the variables x 1 , . . . , x n over K .) In all these cases the results on the multivariate dimension polynomials were proved with the use of certain types of Gröbner bases whose constructions are based on the corresponding reductions. All these reductions satisfy the conditions of Definition 2.5 and therefore are Gröbner reductions. Note that the reduction with respect to several orderings defined in [16], which is a special instance of the Gröbner reduction, can be naturally applied to algebras of a certain subclass of the class of algebras of solvable type (algebras of solvable type were introduced and studied in [11]). This subclass consists of algebras of solvable type R = K {X 1 , . . . , X n } (K is a field) where the term ordering < is degree-respecting (that is, for any two power products t = X k 1 1 . . . X k n n and t = X l 1 1 . . . X l n n , the inequality deg t = n i=1 k i < deg t = n i=1 l i implies that t < t ) and the axiom 1.2(3) in [11, Section 1] is strengthened by the requirement that for all 1 ≤ i ≤ j ≤ n, there exist 0 = c i j ∈ K and p i j ∈ R such that X j * X i = c i j X i X j + p i j and deg p i j ≤ 1. (Algebras of this kind are considered in [6,Section 1.4] where, however, the above requirements on the term order and commutation of the generators are not explicitly formulated. Note also that one can mimic the proof of [16,Theorem 4.2] and obtain a theorem on a multivariate dimension polynomials for finitely generated modules over such algebras.) Finally, using the weight relative Gröbner basis technique developed in [3, Section 2.3] (the corresponding reduction satisfies the conditions of Definition 2.5 as well) C. Dönch showed that the dimensions of components of a multidimensional filtration of a finitely generated difference-skew-differential module over a difference-skew-differential field (that is a field K with the action of finitely many mutually commuting injective endomorphisms and skew derivations of K , see [

Relative Reduction and Buchberger's Algorithm
Buchberger's algorithm, introduced in [1] and formulated for multivariate polynomial rings over a field of characteristic zero, is an algorithm for computing a generating set of an ideal, with the property that every non-zero element contained in the ideal can be reduced to zero in finitely many steps.
This was the starting point for generalizations of this algorithm towards many (non-commutative) ground domains, for example, to the ring of multivariate Ore-polynomials in [15], Weyl-Algebras in [14] and [5], or to the ring of difference-differential operators (see [12,Chapter 4], [13,20] and [22]).
With our considerations, we want to cover modules over rings, whose elements are K -linear combinations of monomials. The monomials should reflect the commutation properties of the considered operators, for example, generalized versions of derivations and automorphisms. Therefore, it is reasonable and appropriate to restrict our view to monomials in finite sets of symbols A := {a 1 , . . . , a m } and B := {b 1 , . . . , b n } where all elements in A ∪ B are pairwise commutative (which is not necessarily the case for elements in K ). Then the monomials are defined as power products of the form = A k · B l where k ∈ N m and l ∈ Z n , using obvious multi-index notation.

22, Definition 2.3]) that a family of subsets {Z
As we have seen, E = {λe i : λ ∈ , 1 ≤ i ≤ t} is a set of monomials of F, which is in natural one-to-one correspondence with the set N m × Z n × E (obviously, a k 1 1 . . . a k m m b l 1 1 . . . b l n n e i ↔ (k 1 , . . . , k m , l 1 , . . . , l n , e i )). A total order ≺ of the set of monomials E is called a generalized term order on E if the corresponding order of the set N m × Z n × E is a generalized term order in the above sense.
Let μ = A k · B l and ν = A k · B l , where k, k ∈ N m and l, l ∈ Z n , and let j be such that l ∈ Z Since E is a free basis of F as a K -module, every element f ∈ F has a unique representation of the form consisting of derivations and automorphisms of K respectively, such that every two operators from the set ∪ commute. The corresponding ring of difference-differential operators D is defined as a free K -module whose free basis consists of all monomials of the form δ k 1 1 . . . δ k m m σ l 1 1 . . . σ l n n , where (k 1 , . . . , k m ) ∈ N m and (l 1 , . . . , l n ) ∈ Z n . The set of all such monomials is denoted by m,n . The multiplication in D is defined by the relationships extended by distributivity. Furthermore, for any ν := δ k 1 1 . . . δ k m m σ l 1 1 . . . σ l n n ∈ m,n , we define the orders of ν relative to and as |ν| 1 = k 1 + · · · + k m , |ν| 2 = |l 1 | + · · · + |l n |, respectively.
Let F be the free D-module generated by a finite set E := {e 1 , . . . , e t }. Then, the orders of a monomial νe i ∈ F (ν ∈ m,n , 1 ≤ i ≤ t) with respect to and are defined as |ν| 1 and |ν| 2 respectively. In this case F can be considered as a bifiltered D-module with twofold filtration {F r,s : (r, s) ∈ N 2 } defined as follows: where for any f ∈ The proof of the following theorem can be obtained by mimicking the proof of [22, Theorem 3.1] that states a similar result for rings of difference-differential operators. Theorem 3.1 (Relative reduction) Let R = K ( ) be a ring, E := {e 1 , . . . , e t } be a set of free generators of the free R-module F = R (E) , and let ≺ 1 and ≺ 2 be two generalized term orders on E. Let G := {g 1 , . . . , g q } ⊆ F\{0} and f ∈ F. Then f can be represented as f = h 1 g 1 + · · · + h q g q + r, (3.1) for some elements h 1 , . . . , h q ∈ R and r ∈ F such that We say that f ≺ 1 -reduces modulo G relative to ≺ 2 to r , and the transition from f to r is said to be the relative reduction.
With this modified reduction, we can now consider the notion of relative Gröbner basis.

Definition 3.2 (Relative Gröbner basis [22, Definition 3.3])
As above, let F be a finitely generated free R-module, and N ⊆ F a submodule. Further, let ≺ 1 and ≺ 2 be a pair of generalized term orders, G := {g 1 , . . . , g q } ⊆ N \{0}. The set G is called a ≺ 1 -Gröbner basis relative to ≺ 2 if and only if every f ∈ N \{0} can be ≺ 1 -reduced modulo G relative to ≺ 2 to zero. Then, if no confusion is possible, G is called a relative Gröbner basis for N .
We consider two generalized term orders ≺ 1 and ≺ 2 and relative reduction for monomially filtered rings R. It is obvious that G is a (relative) Gröbner basis for N if and only if N = R G and every non-zero element in N can be ≺ 1 -reduced modulo G relative to ≺ 2 to zero, which is equivalent to I ρ ∩ N = 0, i.e. axiom 4 of Gröbner reduction.
It is shown in [7,Section 3], that for the bivariate filtration F r,s , using an appropriate choice of term orders ≺ 1 and ≺ 2 it is possible to ensure condition f ∈ F r,s and f −→ h ⇒ h ∈ F r,s . In particular, this property holds with respect to the orders given in [22,Section 3].
To ensure that every non-zero element can be reduced to zero, we need to exploit Buchberger's algorithm, as presented in [22,Theorem 3.4]. At the same time, Dönch [4] gave an example of a certain pair of term-orders such that the Buchberger's algorithm, which takes into account relative reduction, does not terminate. This termination property was further investigated in [10] where one can find a condition on the considered generalized orders that guarantees the termination of the Buchberger's algorithm (it is called the difference-differential degree compatibility condition, see [10,Definition 3.1]).
In what follows, a ring R is assumed to be a free K -module with a basis . Furthermore, we suppose that R is equipped with a bivariate filtration induced by a mapping u : R → N 2 satisfying conditions (i)-(iii) of Remark 2.4. In other words, where the mappings u i := π i • u (i = 1, 2) satisfy the conditions of Lemma 2.3. Let F be a free R-module with basis E and let {F r,s = R r,s E : (r, s) ∈ N 2 } be the induced filtration of F. Definition 3.3 (Admissible orders) With the above notation, a pair of generalized term orders ≺ 1 and ≺ 2 on the set E is said to be admissible, if for any two terms t 1 , t 2 ∈ E • t 1 ≺ 1 t 2 when u 1 (t 1 ) < u 1 (t 2 ), or u 1 (t 1 ) = u 1 (t 2 ) and u 2 (t 1 ) < u 2 (t 2 ); • t 1 ≺ 2 t 2 when u 2 (t 1 ) < u 2 (t 2 ), or u 2 (t 1 ) = u 2 (t 2 ) and u 1 (t 1 ) < u 1 (t 2 ); and u 2 (t 1 ) = u 2 (t 2 ).
Example Let ϕ : E → N s (s is some appropriate chosen positive integer) uniquely identify a monomial. An example is given in [8, Paragraph after Proposition 6.]. The pair is an example for a pair of admissible orders on the monomials E in F.
The following Lemma is proven in [13,Lemma 4.1].
Proof Since for all λ ∈ and h ∈ G i we have LT ≺ 1 (r i+1 ) = LT ≺ 1 (λ · h), the element r i+1 is irreducible with respect to G i . The second condition (3.2) involving the order ≺ 2 would apply only if r i+1 would be reducible, hence, we are considering the usual notion of reduction with respect to G i for the order ≺ 1 . If the chain G 1 ⊆ G 2 ⊆ · · · does not stabilize, then there would be an infinite sequence of monomials {LT(r i ) : i = 1, 2, . . . } in E such that LT(r i ) does not divide LT(r i+1 ) for all i, contrary to the statement of Lemma 3.4.
Assume given a fixed orthant decomposition of N m × Z n consisting of k orthants and j (1 ≤ j ≤ k) is a subset of (we use the notation of Lemma 3.4) consisting of all power products whose exponent vectors lie in the j-th orthant. Furthermore, K [ j ] will denote the subring of R generated by the set j . If f and g are non-zero elements of a free R-module F with a finite set of free generators E and ≺ a generalized term order of E, then V ( j, f, g) will denote a finite system of generators of the K [ j ]-module (The idea of considering such modules and their systems of generators is due to F. Winkler and M. Zhou, see [22].) As it is shown in [21,Lemma 3.5], for any h ∈ F and j = 1, . . . , k, there exists some λ ∈ and a monomial u j in h such that LT ≺ (λh) = λu j ∈ j E. Moreover, this term u j in h is unique; it is denoted by LT j,≺ (h). With the above notation, for every generator v ∈ V ( j, f, g), the element is said to be an S-polynomial of f and g with respect to j, v and ≺. The following algorithm is applied for each orthant Z where u(t i ) = u 1 (t i ) + · · · + u p (t i )(1 ≤ m, n ≤ p) and ϕ is a fixed map ϕ : → N s that uniquely identifies the monomials (as demonstrated in [8, Paragraph after Proposition 6.]). Obviously, the pair of generalized term orders ≺ m n and ≺ n m are admissible.
Theorem 3.7 (Set-relative reduction) Let F be the free R-module, where R is a (not necessarily commutative) noetherian ring and the fixed commutative subring K of R is a field. Let f ∈ F and G = {g 1 , . . . , g q } ⊆ F\{0}. Let A be a subset of the orders {≺ 1 , . . . , ≺ p }, and an order ≺ n m (1 ≤ m, n ≤ p) defined above be fixed. Then there exist elements h 1 , . . . , h q ∈ R and r ∈ F such that f = h 1 g 1 + · · · + h q g q + r and In this case we say that the element f A-reduces to r by G relative to ≺ n m .
The termination of the algorithm is justified by the following facts. First, a generalized term order is a well-order and the first loop terminates by Buchberger's Theorem (it is proven in [1]). For the second loop, in each step we have a pair of admissible orders ≺ l −1 and ≺ −1 , and hence one can apply Theorem 3.6. The set-relative reduction ensures condition 5 of Definition 2.5. Finally, property 4 in Definition 2.5 follows from a straight-forward generalization of [22,Theorem 3.3]. An implementation of a the algorithm for the ring of difference-differential operators and a bi-filtered ring has been obtained in [2].