Serial factorizations of right ideals

In a Dedekind domain D, every non-zero proper ideal A factors as a product $A=P_1^{t_1}?P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain D, the D-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal A of an arbitrary ring R as a product of proper right ideals $A_1,\dots ,A_n$ with all the modules $R/A_i$ uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of h-local Prüfer domains and that of semirigid commutative GCD domains.     

The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

Dedekind–Mertens lemma and content formulas in power series rings

Let $R?X?$ be the power series ring over a commutative ring R with identity. For $f\in R?X?$, let $A_f$ denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if $g\in R?X?$ such that $A_g$ is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely $A_f^2{A}_g=A_f{A}_{fg}$ for all $f\in R?X?$. More generally for a Prüfer domain R, we prove the content formula ${(A_f{A}_g)}^2=(A_f{A}_g)A_{fg}$ for all $f,g\in R?X?$. As a consequence it is shown that an integral domain R is completely integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R?X?$, which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R[X]$, where $R[X]$ is the polynomial ring over R.For a ring R and $g\in R?X?$, if $A_g$ is not locally finitely generated, then there may be no positive integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. Assuming that the locally minimal number of generators of $A_g$ is $k+1$, Epstein and Shapiro posed a question about the validation of the formula $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of $A_g$ is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of $A_g$ is $k+1$, then $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. As a consequence we show that if $A_g$ is finitely generated (in particular if $g\in R[X]$), then there exists a nonnegative integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$.     

Decomposition of integer-valued polynomial algebras

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is $\mathrm{Int}(A)=\{f\in B[X]|f(A)?A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module.A D-algebra A which is free as a D-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a D-module basis for A is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\mathrm{Int}}_K$-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)$ is isomorphic to ${\mathrm{Int}}_K(A){?}_{D}A$. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an ${\mathrm{Int}}_K$-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of K.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

Non-level semi-standard graded Cohen–Macaulay domain with h-vector (h0,h1,h2)

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0, and $A={?}_{i\in \mathbb{N}}{A}_i$ a Cohen–Macaulay graded domain with $A_0=\mathbb{k}$. If A is semi-standard graded (i.e., A is finitely generated as a $\mathbb{k}[A_1]$-module), it has the h-vector$(h_0,h_1,\dots ,h_s)$, which encodes the Hilbert function of A. From now on, assume that $s=2$. It is known that if A is standard graded (i.e., $A=\mathbb{k}[A_1]$), then A is level. We will show that, in the semi-standard case, if A is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers h and n, there is a non-level A with the h-vector $(1,h,(h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).     

Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

On Shimura subvarieties generated by families of abelian covers of P1

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of ${\mathbb{P}}^1$ in $A_g$ and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus $T_g$ inside $A_g$. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.     

Embeddings of rank-2 tori in algebraic groups

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type $A_2$, $G_2$ and $F_4$ defined over k, via their rank-2 k-tori. Simple, simply connected groups of type $A_2$ play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of $A_n$ type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type $A_2$, $G_2$ and $F_4$ in terms of the mod-2 Galois cohomological invariants attached with these groups. The results in this paper and our earlier paper ([6]) show that the mod-2 invariants of groups of type $G_2,F_4$ and $A_2$ are controlled by their k-subgroups of type $A_1$ and $A_2$ as well as the unitary k-tori embedded in them.     

On the topology of valuation-theoretic representations of integrally closed domains

Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let $A(X)={?}_{V\in X}V$ and $J(X)={?}_{V\in X}{\mathfrak{M}}_V$, where ${\mathfrak{M}}_V$ denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring $A(X)$. We show that if $J(X)\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that $A(X)=A(Y)$. If any countable subset of points is removed from Y, then the resulting set remains a representation of $A(X)$. Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring $A(X)$ with specific focus on topological conditions that guarantee $A(X)$ is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings $A(X)$ where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.