Classical Buchberger theory is generalized to a new family of rings. The family includes all subalgebras of the polynomial algebra in one variable. Some subalgebras of polynomial algebras in several variables are included. The new rings are integral domains and have a number of other properties in common with polynomial rings. The rings sit in a field F in appropriate position relative to a val...

Using the Log minimal model program we construct nice models of del Pezzo fibrations which are relevant to the inductive study of del Pezzo fibrations. To investigate birational maps of Fano fibrations, we study elements in anticanonical linear systems of Fano varieties, which have the “worst” singularities. We also investigate certain special points on smooth hypersurfaces of degree n ≥ 3 in P...

Kaplansky proved in 1942 that among ail fields with a valuation having a given divisible value group G, a given algebraically closed residue field R, and a given restriction to the minimal subfield (either the trivial valuation on Qor Fp , or the /?-adic valuation on Q), there is one that is maximal in the strong sensé that every other can be embedded in it. In this paper, we construct this fie...

Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information abou...

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 ...

CARL FAITH All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective . SISI rings include Noetherian rings, Morita rings, and almost maximal valuation rings ([Vil) . In [F3] we raised the question of whether a polynomial ring R[-1 over a SISI ring R is again SISI . In this paper we show this is not...

This paper is part of the program of reverse mathematics. We assume the reader is familiar with this program as well as with RCA,, and WKL,, the two weak subsystems of second-order arithmetic we are going to work with here. (If not, a good place to start is [2].) In [2], [3], [4], many well-known theorems about countable rings, countable fields, etc. were studied in the context of reverse mathe...

We show that an algebraic immediate valuation ring extension of characteristic $$p>0$$ is a filtered union complete intersection algebras finite type.

In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring A. We give two versions of the construction: the first, ...