Search

We give an algorithm for computing the V-saturation of any finitely-generated submodule of a power of V[X], where V is a valuation domain. Our algorithm is based on a notion of "echelon form" which ensures its correctness. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of a power of V[X]., 4 figures

A multiplicative subset of a commutative ring contains the zero element precisely if the set in question meets every prime ideal. While this form of Krull's Lemma takes recourse to transfinite reasoning, it has recently allowed for a crucial reduction to the integral case in Kemper and the third author's novel characterization of the valuative dimension. We present a dynamical solution by which transfinite reasoning can be avoided, and illustrate this constructive method with concrete examples. We further give a combinatorial explanation by relating the Zariski lattice to a certain inductively generated class of finite binary trees. In particular, we make explicit the computational content of Krull's Lemma.

The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's constructive characterization of the Krull dimension. While Lombardi's characterization uses the lexicographic monomial order, ours uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order. Apart from this, the paper contains some related results and some examples which readers may find illuminating., 7 pages

We prove that a valuation domain V has Krull dimension ≤ 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Grobner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.

We give an algorithm for the well-known result asserting that if R is a polynomial ring in a finite number of variables over a Noetherian ring A of Krull dimension d ∞ , then for n ⩾ max ( 3 , d + 2 ) , SL n ( R ) acts transitively on Um n ( R ) . For technical reasons we demand that the Noetherian ring A has a theory of Grobner bases and contains an infinite set E = { y 1 , y 2 , … } such that y i − y j ∈ A × for each i ≠ j . The most important guiding examples are affine rings K [ x 1 , … , x m ] / I and localizations of polynomial rings S −1 K [ x 1 , … , x m ] , with K an infinite field. Moreover, we give an algorithmic proof of Suslin's stability theorem over these rings. For the purpose to prepare the ground for this algorithmic generalizations of the Quillen–Suslin theorem (corresponding to the particular case A is a field), we will give in the first section a constructive proof of an important lemma of Suslin which is the only nonconstructive step in Suslin's second elementary solution of Serre's conjecture. This lemma says that for a commutative ring A, if 〈 v 1 ( X ) , … , v n ( X ) 〉 = A [ X ] where v 1 is monic and n ⩾ 3 , then there exist γ 1 , … , γ l ∈ E n − 1 ( A [ X ] ) such that 〈 Res ( v 1 , e 1 . γ 1 ( v 2 , … , v n ) ) t , … , Res ( v 1 , e 1 . γ l ( v 2 , … , v n ) ) t 〉 = A . Thanks to this constructive proof, Suslin's second proof of Serre's conjecture becomes fully constructive.

We give a constructive deciphering for a generalization of the Quillen–Suslin theorem due to Maroscia and Brewer & Costa stating that finitely generated projective modules over R[X1,…,Xn], where R is a Prüfer domain with Krull dimension ≤1, are extended from R.