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In 93, we describe a filtered-graded transfer of the Groebner basis computation in solvable polynomial algebras which gives a complete answer to the above question

# Filtered-Graded Transfer Of Groebner Basis Computation In Solvable Polynomial Algebras

COMMUNICATIONS IN ALGEBRA, no. 1 (2000): 15-32

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Abstract

Let A = k[a(1),..., a(n)] be an affine algebra over a field k of characteristic 0, and let FA = (F(n)A}(n)greater than or equal to(0) be the standard filtration on A. Consider the graded algebras associated with A: G(A) = circle plus(p)greater than or equal to(0)(F(p)A/Fp-1 A), the associated graded algebra of A, and (A) over tilde = circ...More

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Introduction
• A solvable polynomial algebra with >grlex if and only if is a solvable polynomial algebra with >grlex' Suppose that A is a solvable polynomial algebra with.
• In 93, the authors describe a filtered-graded transfer of the Groebner basis computation in solvable polynomial algebras which gives a complete answer to the above question
• Lemma Let A be a solvable polynomial algebra with the standard filtration
Highlights
• A solvable polynomial algebra with >grlex if and only if is a solvable polynomial algebra with >grlex' Suppose that A is a solvable polynomial algebra with
• A ideal of L in with respect to the filtration FL induced by FA on L, and the a(li)'s resp. h's are corresponding homogeneous elements of the Ii's in G(L)
• [B2] 1985) has shown to be very powerful in commutative algebra and commutative algebraic geometry, in particular, all well known algorithms given in commutative theory have been realized in computer algebra systems (e.g. Macaulay)
• 1990), the Groebner basis computation has been successfully extended to a large class of noncommutative algebras, namely, the class of solvable polynomial algebras, and the ideal membership problem, word problem and the computation of generators for modules of syzygies in a solvable polynomial algebra have been solved
• In 93, we describe a filtered-graded transfer of the Groebner basis computation in solvable polynomial algebras which gives a complete answer to the above question
• If we start with a homogeneous generating set of G (L) a minimal generating set of Ker

Results
• Theorem Let L be a nonzero left ideal of A with Groebner basis F.
• The authors point out that to obtain a Groebner basis from a given finite set of elements, it is enough to define the "S-polynomial" for any two nonzero elements in A and use a noncommutative analogue to the Buchberger algorithm (see the algorithm LGROBNER given in [K-RWD.
• (i) Let R be a filtered ring with filtration F Rand L a left ideal of R generated by {II, ..., Is}.
• Let them consider the left ideal L in the second Weyl algebra A2 (k) generated by 11 = X181, 12 = X28r - 81.
• Do not generate (ii) The filtered-graded transfer of Groebner bases used in the present section has motivated the study of the associated homogeneous defining relations for G(A) and A by using a very noncommutative Groebner basis theory in the sense of [Mor2].
• Let A be a solvable polynomial algebra with the standard filtration F A.
• Example In the following two examples, after finding a Groebner basis for the given left ideal L the authors use Macaulay to do calculations by passing to G(L).
Conclusion
• Another application of the filtered-graded transfer of Groebner basis computation to A is to calculate the syzygy modules and to give finte free resolutions.
• If the authors start with a homogeneous generating set of G (L) a minimal generating set of Ker

• This yields the syzygy matrices and afree resolution of L as follows: let them point out that the method the authors used in this paper has some more possible applications, for example: (1) From the definition it is clear that the Rees algebra A and the associated graded algebra G(A) of a solvable polynomial algebra A with the standard filtration FA are graded quadratic algebras; and in many cases they are regular algebras in the sense of [AS] (e.g. when A is the Weyl algebra An(k) or the enveloping algebra U(g) of a finite dimensional Lie algebra g).
Reference
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