Regular Homomorphisms Research Articles

A functorial approach to regular homomorphisms

Classically, regular homomorphisms have been defined as a replacement for Abel--Jacobi maps for smooth varieties over an algebraically closed field. In this work, we interpret regular homomorphisms as morphisms from the functor of families of algebraically trivial cycles to abelian varieties and thereby define regular homomorphisms in the relative setting, e.g., families of schemes parameterized by a smooth variety over a given field. In that general setting, we establish the existence of an initial regular homomorphism, going by the name of algebraic representative, for codimension-2 cycles on a smooth proper scheme over the base. This extends a result of Murre for codimension-2 cycles on a smooth projective scheme over an algebraically closed field. In addition, we prove base change results for algebraic representatives as well as descent properties for algebraic representatives along separable field extensions. In the case where the base is a smooth variety over a subfield of the complex numbers we identify the algebraic representative for relative codimension-2 cycles with a subtorus of the intermediate Jacobian fibration which was constructed in previous work. At the heart of our descent arguments is a base change result along separable field extensions for Albanese torsors of separated, geometrically integral schemes of finite type over a field.

On the universal regular homomorphism in codimension 2

We point out a gap in Murre’s proof of the existence of a universal regular homomorphism for codimension 2 cycles on a smooth projective variety, and offer two arguments to fill this gap.

The Walker Abel\u2013Jacobi map descends

For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

Necessary and sufficient conditions for the conjugacy of Smale regular homeomorphisms

The class of Smale regular homeomorphisms of closed topological manifolds, with nonwandering set consisting of a finite number of periodic orbits of hyperbolic type, is considered. This class contains the Morse-Smale diffeomorphisms of smooth closed manifolds. For two Smale regular homomorphisms necessary and sufficient conditions for being conjugate are presented. Bibliography: 26 titles.

A remark on a $3$-fold constructed by Colliot–Thélène and Voisin

A classical question asks whether the Abel-Jacobi map is universal among all regular homomorphisms. In this paper, we prove that we can construct a $4$-fold which gives the negative answer in codimension $3$ if the generalized Bloch conjecture holds for a $3$-fold constructed by Colliot-Thelene and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.

More on the Functor Induced by z-Ideals

An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I, then so is the other. For a completely regular frame L we denote by \({{\mathrm{ZId}}}(\mathcal {R}L)\) the lattice of z-ideals of the ring \(\mathcal {R}L\) of continuous real-valued functions on L. It is a coherent frame, and it is known that \(L\mapsto {{\mathrm{ZId}}}(\mathcal {R}L)\) is the object part of a functor \(\mathsf {Z}:\mathbf {CRFrm}\rightarrow \mathbf {CohFrm}\), where \(\mathbf {CRFrm}\) is the category of completely regular frames and frame homomorphisms, and \(\mathbf {CohFrm}\) is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere.

Proximity Biframes and Nachbin Spaces

In our previous paper, in order to develop the pointfree theory of compactifications of ordered spaces, we introduced the concept of a proximity on a biframe as a generalization of the concept of a strong inclusion on a biframe. As a natural next step, we introduce the concept of a proximity morphism between proximity biframes. Like in the case of de Vries algebras and proximity frames, we show that the proximity biframes and proximity morphisms between them form a category PrBFrm in which composition is not function composition. We prove that the category KRBFrm of compact regular biframes and biframe homomorphisms is a proper full subcategory of PrBFrm that is equivalent to PrBFrm. We also show that PrBFrm is equivalent to the category PrFrm of proximity frames, and give a simple description of the concept of regularization using the language of proximity biframes. Finally, we describe the dual equivalence of PrBFrm and the category Nach of Nachbin spaces, which provides a direct way to construct compactifications of ordered spaces.

A BIRKHOFF TYPE THEOREM FOR STRONG VARIETIES

Algebraic systems with partial operations have different ways to interpret equality between two terms of the language. A strong identity is a formula which says that two terms are equal in the algebra if the existence of one of them implies the existence of the other one and in the case of existence their values are equal. A class of partial algebras defined by a set of strong identities is called a strong variety. In the characterization of strong varieties in the case of partial algebras by means of a Birkhoff-type theorem there appeared a new concept, regularity of partial homomorphisms and partial subalgebras. Here we define and study these operators from two different perspectives. Firstly, in their relation with other well known concecpts of partial homomorphisms and partial subalgebras, as well as with the po-monoid of Pigozzi for the H, S and P operators. Secondly, in regard to the preservation of the different types of formulae that represent equality in the case of partial algebras for these operators. Finally, we give a characterization of the strong varieties as classes closed under regular homomorphisms, regular subalgebras, direct products and that satisfy a closure condition.

A Note on Non-Closure Property of Sublogarithmic Space-Bounded 1-Inkdot Alternating Pushdown Automata with Only Existential (Universal) States

1-inkdot alternating pushdown automaton is a slightly modified alternating pushdown automaton with the additional power of marking at most 1 tape-cell on the input (with an inkdot) once. This paper investigates the closure property of sublogarithmic space-bounded 1-inkdot alternating pushdown automata with only existential (universal) states, and shows, for example, that for any function L(n) such that L(n) ≥ log log n and L(n) = o(log n), the class of sets accepted by weakly (strongly) L(n) space-bounded 1-inkdot two-way alternating pushdown automata with only existential (universal) states is not closed under concatenation with regular sets, length-preserving homomorphism, and Kleene closure.

Some Remarks on Relative Tor and Regular Homomorphisms

Some Remarks on Relative Tor and Regular Homomorphisms

Some categorical aspects of BCH‐algebras

We show that the category BCH of BCH‐algebras and BCH‐homomorphisms is complete. We also show that it has coequalizers, kernel pairs, and an image factorization system. It is also proved that onto homomorphisms and coequalizers, and monomorphisms and one‐to‐one homomorphisms coincide, respectively, in BCH. It is shown that MBCI is a coreflexive subcategory of BCH. Regular homomorphisms have been defined and their properties are studied. An open problem has been posed.

Nahm’s equations, configuration spaces and flag manifolds

We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T . This map is constructed via solutions to Nahm’s equations and it is compatible with the S O(3) action, where S O(3) acts on G/T via a regular homomorphism from SU(2) to G. We then generalize this picture to include an arbitrary homomorphism from SU(2) to G. This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.

Regular Homomorphisms and Regular Maps

Regular homomorphisms of oriented maps essentially arise from a factorization by a subgroup of automorphisms. This kind of map homomorphism is studied in detail, and generalized to the case when the induced homomorphism of the underlying graphs is not valency preserving. Reconstruction is treated by means of voltage assignments on angles, a natural extension of the common assignments on darts. Lifting and projecting groups of automorphisms along regular homomorphisms is studied in some detail. Finally, the split-extension structure of lifted groups is analysed.

The universal regular quotient of the Chow group of points on projective varieties

Let X be a projective variety of dimension n de®ned over an algebraically closed ®eld k. For X irreducible and non-singular, Matsusaka [Ma] constructed an abelian variety Alb X † and a morphism a : X ! Alb X † (called the Albanese variety and mapping respectively), depending on the choice of a base-point on X , which is universal among the morphisms to abelian varieties (see Lang [La], Serre [Se] for other constructions). Over the ®eld of complex numbers the existence of Alb X † and a was known before, and has a purely Hodge-theoretic description (see Igusa [I] for the Hodge theoretic construction). Incidentally, the terminology ``Albanese variety'' was introduced by A. Weil, for reasons explained in his commentary on the article [1950a] of Volume I of his collected works (see [W]), one of which is that the paper [Alb] of Albanese de®nes it (for a surface) as a quotient of the group of 0-cycles of degree 0 modulo an equivalence relation. Let CHn X †deg 0 denote the Chow group of 0-cycles of degree 0 on X modulo rational equivalence. When X is irreducible and non-singular, a remarkable feature of the Albanese morphism a is that it factors through a regular homomorphism u : CHn X †deg 0 ! Alb X †, that is a homomorphism, which when composed with the cycle map c : X ! CHn X †deg 0, gives an algebraic morphism. This follows immediately from the fact that an abelian variety does not contain any rational curve. Thus one can reformulate Matsusaka's theorem as the Invent. math. 135, 595±664 (1999) DOI 10.1007/s002229900890

A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms

We give a new proof of D. Popescu’s theorem which says that if σ : A → B \sigma :A\rightarrow B is a regular homomorphism of noetherian rings, then B B is a filtered inductive limit of smooth finite type A A -algebras. We strengthen Popescu’s theorem in two ways. First, we show that a finite type A A -algebra C C , mapping to B B , has a desingularization C → D C\rightarrow D which is smooth wherever possible (roughly speaking, above the smooth locus of C C ). Secondly, we give sufficient conditions for B B to be a filtered inductive limit of its smooth finite type A A -subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.

Regular homomorphisms of generalized projective planes

A generalized projective plane π is an incidence structure together with a relation “distant” on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism ϕ of π into an incidence structure\(\widetilde\pi \) is called regular if the following condition and its dual are valid: A distant B and cϕ IAϕ,Bϕ implies cϕ=(A,B)ϕ. We shall prove the following two theorems. Let π be a generalized projective plane satisfying a richness condition called (U). Let M I m. If ϕ and ψ are regular homomorphisms of π such that Xϕ = Mϕ ⇔ Xψ = Mψ for each point X of the line m then Aϕ = Bϕ ⇔ Aψ = Bψ for any two points A,B. If π is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of π are induced by the ideals of the ring; in particular, the image of π under a regular homomorphism is again a projective plane over a ring, and ϕ preserves “distant”.

A Class of Homomorphisms of Pre-Hjelmslev Groups

E. Salow [8] introduced the concept of pre-Hjelmslev groups, a generalization of F. Bachmann's Hjelmslev groups [1] which leads to a more natural theory of homomorphisms and permits a simpler construction of algebraic models. Basically, both types of groups are the groups of motions of a metric plane, the so-called group plane. In such a plane there is a unique perpendicular through any point to any line and the product of three collinear points (three copunctal lines) is a point (a line). Our first section contains the precise definitions and some basic facts.The homomorphic image of a pre-Hjelmslev group can be more complicated than the pre-image. For instance, there may always be a unique line through two distinct points of the pre-image but not of the image. We study regular homomorphisms of pre-Hjelmslev groups, i.e., homomorphisms with the following property: If two lines intersect at exactly one point, their images will also have precisely one point in common.

Constant-to-one and onto global maps of homomorphisms between strongly connected graphs

AbstractThe global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.

Some Remarks on Regular and Strongly Regular Rings

This article presents some new algebraic and module theoretic characterizations of strongly regular rings. The latter uses Lambek’s notion of symmetry. Strongly regular rings are shown to admit an involution and form an equational category. An example due to Paré shows that the category of regular rings and ring homomorphisms between them is not equational. Remarks on quasiinverses and the generalized inverse of a matrix are included. The author acknowledges support from the NRC (A7752) and improvements from W. Blair received after announcement of the results.