A Note on Noncancellable Varieties

Given two affine algebraic varieties X and Y such that X × ? ? Y × ?, it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1.     

Cleft Extensions of Hopf Algebroids

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid ℋ = (ℋ _L , ℋ _R )) is cleft if and only if it is ℋ _R -Galois and has a normal basis property relative to the base ring L of ℋ _L . Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern–Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined. Dedicated to Stef Caenepeel on the occasion of his 50th birthday.     

Faithfully Flat Hopf Bi-Galois Extensions

This article is devoted to faithfully flat Hopf bi-Galois extensions defined by Fischman, Montgomery, and Schneider. Let H be a Hopf algebra with bijective antipode. Given a faithfully flat right H-Galois extension A/R and a right H-comodule subalgebra C ? A such that A is faithfully flat over C, we provide necessary and sufficient conditions for the existence of a Hopf algebra W so that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra. As a consequence, we prove that if R = k, there is a Hopf algebra W such that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra if and only if C is an H-submodule of A with respect to the Miyashita–Ulbrich action.     

Transfer Results for the FIP and FCP Properties of Ring Extensions

For an extension E: R ? S of (commutative) rings and the induced extension F: R(X) ? S(X) of Nagata rings, the transfer of the FCP and FIP properties between E and F is studied. Then F has FCP ? E has FCP. The extensions E for which F has FIP are characterized. While E has FIP whenever F has FIP, the converse fails for certain subintegral extensions; it does hold if E is integrally closed, seminormal, or subintegral with R quasi-local having infinite residue field. If F has FIP, conditions are given for the sets of intermediate rings of E and F to be order-isomorphic.     

On Smash Products Of Hopf Algebras

Let H = X? _R A denote an R-smash product of the two bialgebras X and A. We prove that (X,A) is a pair of matched bialgebras, if the R-smash product H has a braiding structure. When X is an associative algebra and A is a Hopf algebra, we investigate the global dimension and the weak dimension of the smash product H = X? _R A and show that lD(H) ≤ rD(A) + lD(X) and wD(H) ≤ wD(A) + wD(X). As an application, we get lD(H ₄) = ∞ for Sweedler's four dimensional Hopf algebra H ₄. We also study the associativity of smash products and the relations between smash products and factorization for algebras.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

On Commutativity and Centrality in Infinite Rings

We show that if R is an infinite ring such that XY ∩ YX ≠ ? for all infinite subsets X and Y, then R is commutative. We also prove that in an infinite ring R, an element a ∈ R is central if and only if aX ∩ Xa ≠ ? for all infinite subsets X.     

The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Let X* be a subset of an affine space 𝔸 ^s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x → [x] and x → [(x, 1)], respectively, where [x] and [(x, 1)] are points in the projective spaces ? ^s?1 and ? ^s , respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud–Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.     

Twisted Smash Products and L-R Smash Products for Biquasimodule Hopf Quasigroups

We consider two new algebras from an H-biquasimodule algebra A and a Hopf quasigroup H: twisted smash product A ? H and L-R smash product A?H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, if H is a cocommutative Hopf quasigroup, we prove that A ? H is isomorphic to A?H as Hopf quasigroups.     

Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.     

Cohomological Obstructions to Cleft Extensions over Cocommutative Hopf Algebras

We extend the cohomological treatment of cleft extensions over cocommutative Hopf algebras by giving an interpretation of degree three Sweedler cohomology classes as obstructions to extensions. We show that every twisted action of a cocommutative Hopf algebra on an algebra R gives rise to an obstruction, a degree three Sweedler cohomology class of H with values in the center of R. The obstruction vanishes if and only if the twisted action belongs to a crossed product extension. We also show that every Sweedler three-cocycle can be realized as an obstruction.     

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ? ?^N be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimY_sing ≤ min {d + h ? 1, n ? 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H_2(d+h)(Y; ?) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

On Minimal Extensions of Rings

Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions.     

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Module-Theoretic Characterizations of Generalized GCD Domains

We give several module-theoretic characterizations of generalized GCD domains. For example, we show that an integral domain R is a generalized GCD domain if and only if semi-divisoriality and flatness are equivalent for torsion-free R-modules if and only if every w-finite w-module is projective if and only if R is w-Prüfer (in the sense of Zafrullah). We also characterize when a pullback R of a certain type is a generalized GCD domain. As an application, we characterize when R = D + XE[X] (here, D ? E is an extension of domains and X is an indeterminate) is a generalized GCD domain.     

Explicitly Extending Frobenius Splittings over Finite Maps

Suppose that π: Y → X is a finite map of normal varieties over a perfect field of characteristic p > 0. Previous work of the authors gave a criterion for when Frobenius splittings on X (or more generally any p ^?e -linear map) extend to Y. In this paper we give an alternate and highly explicit proof of this criterion (checking term by term) when π is tamely ramified in codimension 1. Some additional examples are also explored.     

Normal Sections and Direct Product Decompositions

Abstract

In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s: Y → X of an arrow f: X → Y produces a direct product decomposition of the form X ? Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories.