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finiteness of leading monomial ideals and critical cones of characteristic varieties


Boldini, Roberto. finiteness of leading monomial ideals and critical cones of characteristic varieties. 2012, University of Zurich, Faculty of Science.

Abstract

This dissertation consists of two parts. In the first part we give a self-contained and unified approach to some finiteness results on leading monomial ideals of a polynomial ring with respect to various types of total orderings of the monomials. To a large extent the results of this part are not new and may be found in the work of other authors either relying on different approaches or applied in different contexts. We generalize a part of these results to vector spaces isomorphic to a polynomial ring, a part to the large class of admissible algebras, which comprehends at least the class of algebras of solvable type. In the literature leading monomial ideals with respect to monoid orderings of Nt 0 with t ∈ N are the main object of study because these orderings induce a fruitful division theory. In this context Macaulay’s Basis Theorem is the key to finiteness results on leading monomial ideals. We consider leading monomial ideals with respect to total orderings, degree orderings, semigroup orderings, monoid orderings, and degree-compatible monoid orderings. It turns out that an ideal of a polynomial ring admits at most finitely many minimal leading monomial ideals arising from total orderings, of course minimal with respect to inclusion. Furthermore an ideal possesses at most finitely many minimal leading monomial ideals with respect to degree orderings. Due to a slightly generalized version of Macaulay’s Basis Theorem shown here, leading monomial ideals induced by monoid orderings are minimal, thus an ideal has only finitely many leading monomial ideals with respect to monoid orderings.
Initially, inspired by the classical proof for polynomial rings, we planned to show the existence of universal Gröbner bases in admissible algebras by the finiteness results mentioned above. This was our original motivation for investigating these finiteness properties. But, indeed, the existence of universal Gr¨obner bases already follows from the fact that the set of all total orderings on any given set builds a compact topological space and that admissible algebras are noetherian. With this topic we conclude the first part of our work.
The second and innovative part of this dissertation is a slightly more detailed version of our article [13], which in December 2010 was accepted for publication on the Transactions of the American Mathematical Society. Here we dedicate ourselves to the characteristic varieties of modules over Weyl algebras. These affine varieties are constructed by providing a finitely generated module over a Weyl algebra with weighted filtrations and forming their associated graded modules. Therefore we first recall some facts over filtered modules and their associated graded modules. For any filtered module over a filtered commutative ring we show that the annihilator of the associated graded module is equal (up to taking radicals) to the associated graded ideal of the filtered annihilator.
A classical theorem of Bernstein states that the characteristic varieties by degree and by order of a given module have the same Krull dimension. Indeed all characteristic varieties of a module have the same dimension. This is usually proved by homological methods.
We embed the mentioned dimension theorem in the wider context of a deformation theory of weighted degree filtrations and monomial orderings. Our deformation-theoretic approach applies universal Gr¨obner bases, and the mentioned equality of dimensions follows as a corollary of a deeper result. Namely, characteristic varieties denote a remarkable behaviour when one deforms their defining filtrations by certain adjustments of the weights.
More precisely, by such adjustments a characteristic variety is stepwise deformed into its own critical cone. This permits to deform a nonfinite filtration in such a manner that the resulting filtration becomes finite and the characteristic variety associated to it is the critical cone of the original variety. From this follows the wanted dimension equality. A reason is that a variety has the same Krull dimension as its own critical cone. A further reason is that the Krull and Gelfand–Kirillov dimension of a finitely generated module over a finitely generated K-algebra agree. A third reason is that the Gelfand–Kirillov dimension of a finitely filtered module is preserved when passing to the associated graded module.
Our result represents also a first step in trying to classify characteristic varieties. We were not able to perform such a classification in full generality. Therefore we have focused on characteristic varieties of cyclic modules over the first Weyl algebra and have calculated an approximated classification by a computer experiment. The experiment shows that the weight space N20 r {(0, 0)} of the filtrations can be subdivided in semicone-shaped regions such that each region corresponds to the same characteristic variety. On the basis of this experiment we can also conjecture a higher bound for the number of these characteristic varieties in terms of total degree of elements of a universal Gr¨obner basis. In view of a work of Aschenbrenner and Leykin [2], this higher bound can be given also in terms of total degrees of generators of the ideal that defines the considered cyclic module.
We end the second part with a result of ˇ Skoda on localizations of filtered modules. By means of an easy lemma we give a geometric interpretation to ˇ Skoda’s results in our context.
In the first appendix we furnish a more direct proof of the existence of universal Gröbner bases in Weyl algebras using the division properties of these algebras together with the compactness of the topological space of monoid orderings. In the second appendix we list the computer program that we wrote to perform the mentioned computer experiment.

Abstract

This dissertation consists of two parts. In the first part we give a self-contained and unified approach to some finiteness results on leading monomial ideals of a polynomial ring with respect to various types of total orderings of the monomials. To a large extent the results of this part are not new and may be found in the work of other authors either relying on different approaches or applied in different contexts. We generalize a part of these results to vector spaces isomorphic to a polynomial ring, a part to the large class of admissible algebras, which comprehends at least the class of algebras of solvable type. In the literature leading monomial ideals with respect to monoid orderings of Nt 0 with t ∈ N are the main object of study because these orderings induce a fruitful division theory. In this context Macaulay’s Basis Theorem is the key to finiteness results on leading monomial ideals. We consider leading monomial ideals with respect to total orderings, degree orderings, semigroup orderings, monoid orderings, and degree-compatible monoid orderings. It turns out that an ideal of a polynomial ring admits at most finitely many minimal leading monomial ideals arising from total orderings, of course minimal with respect to inclusion. Furthermore an ideal possesses at most finitely many minimal leading monomial ideals with respect to degree orderings. Due to a slightly generalized version of Macaulay’s Basis Theorem shown here, leading monomial ideals induced by monoid orderings are minimal, thus an ideal has only finitely many leading monomial ideals with respect to monoid orderings.
Initially, inspired by the classical proof for polynomial rings, we planned to show the existence of universal Gröbner bases in admissible algebras by the finiteness results mentioned above. This was our original motivation for investigating these finiteness properties. But, indeed, the existence of universal Gr¨obner bases already follows from the fact that the set of all total orderings on any given set builds a compact topological space and that admissible algebras are noetherian. With this topic we conclude the first part of our work.
The second and innovative part of this dissertation is a slightly more detailed version of our article [13], which in December 2010 was accepted for publication on the Transactions of the American Mathematical Society. Here we dedicate ourselves to the characteristic varieties of modules over Weyl algebras. These affine varieties are constructed by providing a finitely generated module over a Weyl algebra with weighted filtrations and forming their associated graded modules. Therefore we first recall some facts over filtered modules and their associated graded modules. For any filtered module over a filtered commutative ring we show that the annihilator of the associated graded module is equal (up to taking radicals) to the associated graded ideal of the filtered annihilator.
A classical theorem of Bernstein states that the characteristic varieties by degree and by order of a given module have the same Krull dimension. Indeed all characteristic varieties of a module have the same dimension. This is usually proved by homological methods.
We embed the mentioned dimension theorem in the wider context of a deformation theory of weighted degree filtrations and monomial orderings. Our deformation-theoretic approach applies universal Gr¨obner bases, and the mentioned equality of dimensions follows as a corollary of a deeper result. Namely, characteristic varieties denote a remarkable behaviour when one deforms their defining filtrations by certain adjustments of the weights.
More precisely, by such adjustments a characteristic variety is stepwise deformed into its own critical cone. This permits to deform a nonfinite filtration in such a manner that the resulting filtration becomes finite and the characteristic variety associated to it is the critical cone of the original variety. From this follows the wanted dimension equality. A reason is that a variety has the same Krull dimension as its own critical cone. A further reason is that the Krull and Gelfand–Kirillov dimension of a finitely generated module over a finitely generated K-algebra agree. A third reason is that the Gelfand–Kirillov dimension of a finitely filtered module is preserved when passing to the associated graded module.
Our result represents also a first step in trying to classify characteristic varieties. We were not able to perform such a classification in full generality. Therefore we have focused on characteristic varieties of cyclic modules over the first Weyl algebra and have calculated an approximated classification by a computer experiment. The experiment shows that the weight space N20 r {(0, 0)} of the filtrations can be subdivided in semicone-shaped regions such that each region corresponds to the same characteristic variety. On the basis of this experiment we can also conjecture a higher bound for the number of these characteristic varieties in terms of total degree of elements of a universal Gr¨obner basis. In view of a work of Aschenbrenner and Leykin [2], this higher bound can be given also in terms of total degrees of generators of the ideal that defines the considered cyclic module.
We end the second part with a result of ˇ Skoda on localizations of filtered modules. By means of an easy lemma we give a geometric interpretation to ˇ Skoda’s results in our context.
In the first appendix we furnish a more direct proof of the existence of universal Gröbner bases in Weyl algebras using the division properties of these algebras together with the compactness of the topological space of monoid orderings. In the second appendix we list the computer program that we wrote to perform the mentioned computer experiment.

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Item Type:Dissertation
Referees:Brodmann Markus, Ayoub Joseph, Rosenthal Joachim
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2012
Deposited On:17 Aug 2012 09:09
Last Modified:05 Apr 2016 15:56
Number of Pages:82
Related URLs:http://opac.nebis.ch/F/?local_base=NEBIS&CON_LNG=GER&func=find-b&find_code=SYS&request=007325926 (Organisation)

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