Infinite Transcendence Degree Research Articles

Subgroups of free proalgebraic groups and Matzat’s conjecture for function fields

We show that a closed finite index subgroup of a free proalgebraic group is itself a free proalgebraic group. As an application, we prove that the absolute differential Galois group of a one-variable function field over an algebraically closed characteristic zero field of infinite transcendence degree is a free proalgebraic group.

Local cohomology bounds and the weak implies strong conjecture in dimension 4

We verify the weak implies strong conjecture for 4-dimensional finitely generated algebras over a field of prime characteristic p>5 which has infinite transcendence degree over Fp.

Free differential Galois groups

We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat’s freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over Q \mathbb {Q} . This is the first known case of the twenty year old conjecture.

A few remarks on invariable generation in infinite groups

A subset [Formula: see text] of a group [Formula: see text] invariably generates [Formula: see text] if [Formula: see text] is generated by [Formula: see text] for any choice of [Formula: see text]. A topological group [Formula: see text] is said to be [Formula: see text] if it is invariably generated by some subset [Formula: see text], and [Formula: see text] if it is topologically invariably generated by some subset [Formula: see text]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [Formula: see text] and the automorphism group of a regular tree are [Formula: see text], and that the groups [Formula: see text] are not [Formula: see text] for countable fields of infinite transcendence degree over a prime field.

On the Representation of Fields as Finite Sums of Proper Subfields

We study which fields F can be represented as finite sums of proper subfields. We prove that for any n ge 2 every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field F = K(x_1, ldots , x_n) can be represented as an unshortenable sum of n + 1 subfields. We also show that no subfield of the algebraic closure of a finite field is a finite sum of proper subfields, and no finite extension of the field {mathbb {Q}} of rationals can be decomposed into a sum of two proper subfields.

LARGE FIELDS IN DIFFERENTIAL GALOIS THEORY

AbstractWe solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over $\mathbb{Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over$\mathbb{Q}_{p}(x)$.

Reidemeister spectrum of special and general linear groups over some fields contains 1

We prove that if [Formula: see text] is an algebraically closed field of zero characteristic which has infinite transcendence degree over [Formula: see text], then there exists a field automorphism [Formula: see text] of [Formula: see text] and [Formula: see text] such that [Formula: see text]. This fact implies that [Formula: see text] and [Formula: see text] do not possess the [Formula: see text]-property. However, if the transcendece degree of [Formula: see text] over [Formula: see text] is finite, then [Formula: see text] and [Formula: see text] are known to possess the [Formula: see text]-property [13].

ON the D-saturation property

In this paper we study the notion of a D-saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D-saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D-saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of Dsaturation imply D-saturation.

Computable valued fields

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension $$\omega $$ .

On algebraic independence of certain multizeta values in characteristic p

In this paper, we study multizeta values over function fields in characteristic p. For each d≥2, we show that when the constant field has cardinality >2, the field generated by all multizeta values of depth d is of infinite transcendence degree over the field generated by all single zeta values.

Computable dimension for ordered fields

The computable dimension of a structure counts the number of computable copies up to computable isomorphism. In this paper, we consider the possible computable dimensions for various classes of computable ordered fields. We show that computable ordered fields with finite transcendence degree are computably stable, and thus have computable dimension 1. We then build computable ordered fields of infinite transcendence degree which have infinite computable dimension, but also such fields which are computably categorical. Finally, we show that 1 is the only possible finite computable dimension for any computable archimedean field.

Algebraic independence of elements in immediate extensions of valued fields

Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the field has countable cofinality, then these criteria give the same information for the completions of the field. The criteria have applications to the classification of valuations on rational function fields. We also apply the criteria to the question of which extensions of a maximal valued field, algebraic or of finite transcendence degree, are again maximal. In the case of valued fields of infinite p-degree, we obtain the worst possible examples of nonuniqueness of maximal immediate extensions: fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree.

Computably Categorical Fields via Fermat’s Last Theorem

We construct a computable, computably categorical field of infinite transcendence degree over the rational numbers, using the Fermat polynomials and assorted results from algebraic geometry. We also show that this field has an intrinsically computable (infinite) transcendence basis.

A finiteness condition for algebras over coordinate rings of varieties

Let A be a finitely presented k[X]-algebra, where k[X] is the algebra of regular functions on a variety X defined over an algebraically closed field k. The following problem arises in the study of degenerations of algebras (Hajduk and Kasjan, in press [7]). Assume that the specialization of A at x has finite dimension over k for every x from a dense subset of X. Is there an open dense subset U of X such that the localization of A with respect to U is a finitely generated k[U]-module? We prove that this is the case if k has infinite transcendence degree over its prime subfield. We provide some applications to the concept of generalized CB-degenerations.

A negative answer to a question of Bass

We address Bass’ question, on whether $K_n(R)=K_n(R[t])$ implies $K_n(R)=K_n(R[t_1,t_2])$. In a companion paper, we establish a positive answer to this question when $R$ is of finite type over a field of infinite transcendence degree over the rationals. Here we provide an example of an isolated surface singularity over a number field for which the answer the Bass’ question is “no” when $n=0$.

Kronecker Function Rings of Transcendental Field Extensions

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all valuation rings of the extension F/D and investigate the ideal structure of Kr(F/D) in the case where D is an affine algebra over a subfield K of F and the extension F/K has countably infinite transcendence degree, by using the topological structure of the Zariski–Riemann space. We show that for any pair of nonnegative integers d and h, there are infinitely many prime ideals of dimension d and height h that are minimal over any proper nonzero finitely generated ideal of Kr(F/D).

Smooth Representations and Sheaves

The paper is concerned with `geometrization' of smooth (i.e. with open stabilizers) representations of the automorphism group of universal domains, and with the properties of `geometric' representations of such groups. As an application, we calculate the cohomology groups of several classes of smooth representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of an algebraically closed field.

Extended Centres of Finitely Generated Prime Algebras

Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) −2 over K. As a consequence, we are able to show that if A is a prime K-algebra of quadratic growth, then either the extended centre is algebraic over K or A is PI. Finally, we give an example of a finitely generated non-PI prime K-algebra of GK dimension 2 with a locally nilpotent ideal such that the extended centre has infinite transcendence degree over K.

Generic representation theory of quivers with relations

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra Λ are explored, with regard to both their geometry and the structure of the modules they encode. Provided that the base field is algebraically closed and has infinite transcendence degree over its prime field, we establish the existence and uniqueness (not up to isomorphism, but in a strong sense to be specified) of a generic module for any irreducible component C , that is, of a module which displays all categorically defined generic properties of the modules parametrized by C ; the crucial point of the existence statement – a priori almost obvious – lies in the description of such a module in a format accessible to representation-theoretic techniques. Our approach to generic modules over path algebras modulo relations, by way of minimal projective resolutions, is largely constructive. It is explicit for large classes of algebras of wild type. We follow with an investigation of the properties of such generic modules in terms of quiver and relations. The sharpest specific results on all fronts are obtained for truncated path algebras, that is, for path algebras of quivers modulo ideals generated by all paths of a fixed length; this class of algebras extends the comparatively thoroughly studied hereditary case, displaying many novel features.

GROWTH OF MODULES OVER GENERIC GOLOD-SHAFAREVICH-ALGEBRAS

Let K be a field of infinite transcendence degree and let A be a finitely generated K-algebra. Suppose that the number of homogeneous generic relations in A of degrees smaller than n grows exponentially with n. Then all infinitely-dimensional finitely generated A-modules have exponential growth. In particular there are Golod-Shafarevich algebras all of whose finitely generated modules either have exponential growth or are finite-dimensional.