Integral Domain Of Characteristic Zero Research Articles

Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields

We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all mathbb {Z}-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety.

Urata's theorem in the logarithmic case and applications to integral points

Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field K $K$ with only finitely many O L , T $\mathcal {O}_{L,T}$ -points for any number field L / K $L/K$ and any finite set of finite places T $T$ of L $L$ has, in fact, only finitely many points in any given Z $\mathbb {Z}$ -finitely generated integral domain of characteristic zero. We use this latter result in combination with Green's criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green's criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over C $\mathbb {C}$ which is not hyperbolically embeddable.

Based quasi-hereditary algebras

A notion of a split quasi-hereditary algebra has been defined by Cline, Parshall and Scott. Du and Rui describe a based approach to split quasi-hereditary algebras. We develop this approach further to show that over a complete local Noetherian ring, one can achieve even stronger basis properties. This is important for ‘schurifying’ quasi-hereditary algebras as developed in our subsequent work. The schurification procedure associates to an algebra A a new algebra, which is the classical Schur algebra if A is a field. Schurification produces interesting new quasi-hereditary and cellular algebras. It is important to work over an integral domain of characteristic zero, taking into account a super-structure on the input algebra A. So we pay attention to super-structures on quasi-hereditary algebras and investigate a subtle conforming property of heredity data which is crucial to guarantee that the schurification of A is quasi-hereditary if so is A. We establish a Morita equivalence result which allows us to pass to basic quasi-hereditary algebras preserving conformity.

Twisted conjugacy classes in unitriangular groups

Abstract Let R be an integral domain of characteristic zero. In this note we study the Reidemeister spectrum of the group UT n ⁢ ( R ) {{\rm UT}_{n}(R)} of unitriangular matrices over R. We prove that if R + {R^{+}} is finitely generated and n > 2 ⁢ | R * | {n>2|R^{*}|} , then UT n ⁢ ( R ) {{\rm UT}_{n}(R)} possesses the R ∞ {R_{\infty}} -property, i.e. the Reidemeister spectrum of UT n ⁢ ( R ) {{\rm UT}_{n}(R)} contains only ∞ {\infty} , however, if n ≤ | R * | {n\leq|R^{*}|} , then the Reidemeister spectrum of UT n ⁢ ( R ) {{\rm UT}_{n}(R)} has nonempty intersection with ℕ {\mathbb{N}} . If R is a field and n ≥ 3 {n\geq 3} , then we prove that the Reidemeister spectrum of UT n ⁢ ( R ) {{\rm UT}_{n}(R)} coincides with { 1 , ∞ } {\{1,\infty\}} , i.e. in this case UT n ⁢ ( R ) {{\rm UT}_{n}(R)} does not possess the R ∞ {R_{\infty}} -property.

Some open problems on locally finite or locally nilpotent derivations and ℰ-derivations

Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-algebra. An [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is an [Formula: see text]-linear map of the form [Formula: see text] for some [Formula: see text]-algebra endomorphism [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the identity map of [Formula: see text]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [Formula: see text], and whether or not a locally nilpotent (LN) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] maps every ideal of [Formula: see text] to a Mathieu subspace of [Formula: see text]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [Formula: see text] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [Formula: see text]-[Formula: see text]-derivations of integral domains of characteristic zero.

Formula omitted] of a p-adic group ring II. The determinantal kernel [formula omitted

We describe the group SK1(R[G]) for group rings R[G] where G is an arbitrary finite group and where the coefficient ring R is a p-adically complete Noetherian integral domain of characteristic zero which admits a lift of Frobenius and which also satisfies a number of further mild conditions. Our results extend previous work of R. Oliver who obtained such results for the valuation rings of finite extensions of the p-adic field.

Characterizing ring derivations of all orders via functional equations: results and open problems

We provide a unifying framework for the treatment of equations of the form $$\sum_{k=1}^n x^{p_k} f_k (x^{q_k}) = 0$$ for additive maps fk and integers pk, qk (1 ≤ k ≤ n). We show how to solve many equations of this type, and we present some open problems. In general our unknown functions map an integral domain of characteristic zero into itself. When negative exponents appear, we restrict our attention to fields of characteristic zero. All of the results could be formulated for integral domains or fields of sufficiently large characteristic as well.

A classification of bisymmetric polynomial functions over integral domains of characteristic zero

We describe the class of n-variable polynomial functions that satisfy Aczél’s bisymmetry property over an arbitrary integral domain of characteristic zero with identity.

On One Class of Divisors of Polynomial Matrices over Integral Domains

We establish conditions for the existence of a unital divisor for a polynomial matrix over an integral domain of characteristic zero in the case where its eigenvalues are known.

Isomorphic Group Rings Over Domains

AbstractLet R and S be rings, G and H abelian groups, and RG and SH the goup rings of G and H over R and S respectively. In this note we consider what relations must hold between G and H or between R and S if the group rings RG and SH are isomorphic. For example, it is shown that if R and S are integral domains of characteristic zero, G and H torsion abelian groups such that if G has an element of order p then p is not invertible in R, and RG and SH are isomorphic, then the rings R and 5 are isomorphic and the groups G and H are isomorphic.

A Diophantine problem for Laurent polynomial rings

Let R R be an integral domain of characteristic zero. We prove that the diophantine problem for the Laurent polynomial ring R [ T , T − 1 ] R[T,{T^{ - 1}}] with coefficients in Z [ T ] {\mathbf {Z}}[T] is unsolvable. Under suitable conditions on R R we then show that either Z {\mathbf {Z}} or Z [ i ] {\mathbf {Z}}[i] is diophantine over R [ T , T − 1 ] R[T,{T^{ - 1}}] .

Localized group rings, the invariant basis property and Euler characteristics

The technique of embedding a complex group algebra into the von Neumann regular ring associated with the corresponding W ∗ {W^ * } group algebra is exploited to prove that certain localizations of a group ring K G KG possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain K G KG -modules have zero Euler characteristic. The assumptions are that K K is a commutative integral domain of characteristic zero, and that G G has a nontrivial, torsion-free, abelian normal subgroup A A . The main result of the paper is that the localization of K G KG obtained by inverting all elements of the form α − a \alpha - a , where α \alpha is a nonzero element of K K and a a is a nontrivial element of A A , has the invariant basis property; more generally, this localization and all its matrix rings are directly finite. (M. Smith has extended the methods of this paper to cover the localization of K G KG obtained by inverting all nonzero elements of K A KA .) Given a K G KG -module M M which has a finite free resolution, such that M M is finitely generated over K K , it is proved that the Euler characteristic of M M is zero. This verifies an unpublished result of Rosset.

Groupings of Metabelian Groups and Extension Categories

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall proveTHEOREM. If RG ≌ RH as R-algebras, then G ≌ HThe question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

The Diophantine Problem for Polynomial Rings and Fields of Rational Functions

We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.

The Diophantine problem for polynomial rings and fields of rational functions

We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.

Homomorphisms of integral domains of characteristic zero

Every category of universal algebras is isomorphic to a full subcategory of the category of all integral domains of characteristic zero and all their 1-preserving homomorphisms. Consequently, every monoid is isomorphic to the monoid of all 1-preserving endomorphisms of an integral domain of characteristic zero.

Homomorphisms of Integral Domains of Characteristic Zero

Homomorphisms of Integral Domains of Characteristic Zero

The free centre-by-metabelian groups

Let be the free centre-by-metabelian group of rank n. In this paper, our main result is the followingTheorem. For n ≧ 4, Gn has a finite elementary abelian subgroup Hn of rank . More precisely, Hn is a minimal fully invariant subgroup contained in the centre of Gn and Gn/Hn is isomorphic to a group of 3 x 3 matrices over a finitely generated integral domain of characteristic zero.