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A remark on Hodge algebras and Gröbner bases. (English) Zbl 0795.13009
Useful properties for computations in a finitely generated commutative \(K\)-algebra \(A\) \((K\) a field) given by a finite set \(E\) of generators and their ideal \(I\) of relations (i.e. \(A=K[E]/I\), where \(K[E]\) is the commutative polynomial ring over \(K)\) are considered. The properties useful to have, are
(1) a good \(K\)-vector space basis of \(A\), consisting of the residue classes of all power products outside a monomial ideal \(\Sigma\) (“standard monomials”) and
(2) “rewriting rules” making it possible to write any monomial modulo \(I\) in a finite number of steps as a finite \(K\)-linear combination of standard monomials.
The Hodge algebra concept and the Gröbner basis method are two known concepts of such rewriting rules. – In this note the authors show how to use Gröbner bases to verify condition (1). This is shown in a proposition and an algorithmic version is also presented. The paper is concluded by two new example where Hodge algebras do exist, but behave rather badly because \(A\) fails to be affine graded.
MSC:
13F50 Rings with straightening laws, Hodge algebras
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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References:
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