## 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)

### Keywords:

straightening; Hodge algebra; Gröbner basis
Full Text:

### References:

 [1] Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Ideal. Dissertation, Universität Innsbruck, Austria, 1965. [2] Buchberger, B.: Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. Multidimensional Systems Theory, Bose, N. (ed.), Reidel Publ. Comp., Dordrecht, 1985, pp. 184-232. · Zbl 0587.13009 [3] Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Math. vol. 1327, Springer, Berlin, 1988. · Zbl 1079.14533 [4] DeConcini, C., Eisenbud, D., Procesi, C.: Hodge Algebras. Astérisque 91, 1982. [5] Eisenbud, D.: Introduction to Algebras with Straightening Laws. Ring Theory and Algebra III, McDonald, R. (ed.), Marcel Dekker, New York, 1980, pp. 243-268. · Zbl 0448.13010 [6] Gräbe, H.-G.: Über Streckungsringe. Beiträge zur Algebra und Geometrie 23 (1986), 85-100. · Zbl 0646.13010 [7] Gräbe, H.-G.: Streckungsringe. Dissertation B. Pädagogische Hochschule “Dr. Theodor Neubauer”, Erfurt, DDR, 1988. [8] Gräbe, H.-G.: Moduln über Streckungsringen. Results in Math. 15 (1989), 202-220. · Zbl 0694.13006 [9] Pauer, F., Pfeifhofer, M.: The Theory of Gröbner Bases. L’Enseignement Math. 34 (1988), 215-232. · Zbl 0702.13019 [10] Sturmfels, B., White, N.: Gröbner Bases and Invariant Theory. Advances Math. 76 (1988 1989), 245-259. · Zbl 0695.13001 [11] Trung, N. V.: Questions on the Presentation of Hodge Algebras and the Existence of Hodge Algebra Structures. (1990).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.