Valuations and filtrations

The classical theory of Gröbner bases, as developed by Bruno Buchberger, can be expanded to utilize objects more general than term orders. Each term order on the polynomial ring k[x] produces a filtration of k[x] and a valuation ring of the rational function field k(x). The algorithms developed by Buchberger can be performed by using directly the induced valuation or filtration in place of the term order. There are many valuations and filtrations that are suitable for this general computational framework that are not derived from term orders, even after a change of variables. Here we study how to translate between properties of filtrations and properties in valuation theory, and give a characterization of which valuations and filtrations are derived from a term order after a change of variables. This characterization illuminates the properties of valuations and filtrations that are desirable for use in a generalized Gröbner basis theory

Good Reduction of Good Filtrations at Places

We consider filtered or graded algebras $A$ over a field $K$. Assume that there is a discrete valuation $O_v$ of $K$ with $m_v$ its maximal ideal and $k_v:=O_v/m_v$ its residue field. Let $\Lambda$ be $O_v$-order such that $\Lambda K=A$ and $\bar{\Lambda}:=k_v\otimes_{O_v}\Lambda$ the $\Lambda$-reduction of $A$ at the place $K\leadsto k_v$. Using the filtration of $A$ induced by $\Lambda$ we shall prove that for certain algebras $A$ their properties are related to $\bar{\Lambda}$.Comment: 17 page

Galois cohomology of a number field is Koszul

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures that were proposed in our previous papers. The proofs are based on the Class Field Theory and computations with quadratic commutative Groebner bases (commutative PBW-bases).Comment: LaTeX 2e, 25 pages; v.2: minor grammatic changes; v.3: classical references added, remark inserted in subsection 1.6, details of arguments added in subsections 1.4, 1.7 and sections 5 and 6; v.4: still more misprints corrected, acknowledgement updated, a sentence inserted in section 4, a reference added -- this is intended as the final versio