Pullback Diagram Research Articles

On GPW-Flat Acts

In this article, we present GPW-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right S-act A_{S} is GPW-flat if for every s in S, there exists a natural number n = n_ {(s, A_{S})} in mathbb{N} such that the functor A_{S} otimes {}_{S}- preserves the embedding of the principal left ideal {}_{S}(Ss^n) into {}_{S}S. We show that a right S-act A_{S} is GPW-flat if and only if for every s in S there exists a natural number n = n_{(s, A_{S})} in mathbb{N} such that the corresponding varphi is surjective for the pullback diagram P(Ss^n, Ss^n, iota, iota, S), where iota : {}_{S}(Ss^n) rightarrow {}_{S}S is a monomorphism of left S-acts. Also we give some general properties and a characterization of monoids for which this condition of their acts implies some other properties and vice versa.

Pullback diagrams and Kronecker function rings

Knebusch and Kaiser introduced the notion of Kronecker function ring of a ring extension with respect to a star operation to generalize the classical notion of Kronecker function ring. Let $\star $ be a star operation on the extension $R\subseteq S$. Let $Kr ( \star)$ be the set of all quotients ${f}/{g}\in S(X)$ such that $f, g\in S[X]$, $c_{S}(g) = S$ and $(c_{R}(f)H)^{\star } \subseteq (c_{R}(g)H)^{\star }$ for some finitely generated $S$-regular $R$-submodule $H$ of $S$, where for each ring $A$ such that $R\subseteq A \subseteq S$, $c_{A}(g)$ denotes the content of $g\in S[X]$. The ring $Kr (\star )$ is a Kronecker subring of $S(X)$ by a theorem of Knebusch and Kaiser. We study properties of the ring $Kr (\star )$ in pullback diagrams.

Pullbacks of maximally (strong) Prüfer rings

We investigate the behavior of (strong) Prüfer rings in a general pullback diagram. We also consider three new Prüfer conditions introduced by Klingler, Lucas, and Sharma.

Pullback diagrams, syzygy finite classes and Igusa–Todorov algebras

For an abelian category A, we define the category PEx(A) of pullback diagrams of short exact sequences in A, as a subcategory of the functor category Fun(Δ,A) for a fixed diagram category Δ. For any object M in PEx(A), we prove the existence of a short exact sequence 0→K→P→M→0 of functors, where the objects are in PEx(A) and P(i)∈Proj(A) for any i∈Δ. As an application, we prove that if (C,D,E) is a triple of syzygy finite classes of objects in mod Λ satisfying some special conditions, then Λ is an Igusa–Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa–Todorov.

An example of an atomic pullback without the ACCP

We find necessary and sufficient conditions on a pullback diagram in order that every nonzero nonunit in its pullback ring admits a finite factorization into irreducible elements. As a result, we can describe a method of easily producing atomic domains that do not satisfy the ascending chain condition on principal ideals.

Multiplicative canonical ideals of ring extensions

We study properties of multiplicative canonical (m-canonical) ideals of ring extensions. Let [Formula: see text] be a ring extension. A nonzero [Formula: see text]-regular ideal [Formula: see text] of [Formula: see text] is called an m-canonical ideal of the extension [Formula: see text] if [Formula: see text] for all [Formula: see text]-regular ideal [Formula: see text] of [Formula: see text]. We study m-canonical ideals for pullback diagrams, and we use the notion of m-canonical ideal to characterize Prüfer extensions.

Pullback Crossed Modules in the Category of Racks

In this paper, we define the pullback crossed modules in the category of racks which mainly based on a pullback diagram of rack morphisms with extra crossed module data on some of its arrows. Furthermore we prove that the conjugation functor, which is defined between the category of crossed modules of groups and of racks, preserves the pullback crossed modules.

On congruence extension properties for ordered algebras

We introduce four extension properties (CEP, QEP, SCEP and SQEP) for ordered algebras, similar to the congruence extension property (CEP) and the strong congruence extension property of usual (unordered) algebras. All four properties turn out to have a description in terms of commutative squares or pullback diagrams. We then use these categorical descriptions to prove an ordered analogue of the well-known relation TP = AP + CEP, namely that a variety of ordered algebras has the ordered transferability property if and only if it has the ordered amalgamation property and QEP.

Pullback diagram of Hilbert modules over H*-algebras

In this paper, we generalize the construction of a pullback diagram in the framework of Hilbert modules over H*-algebras. More precisely we prove that if a commutative diagram of Hilbert H*-modules and morphisms X2←X1→Ф1→Y1→Y2 is pullback and ψ2 is a surjection, then (i) ψ 1 is a surjection and (ii) kerф1 ∩ ker ψ1 = {0}. Conversely, if (i) and (ii) hold, ψ1(т(A1)) is тA2 -closed and ψ2 is injective, then the above diagram is pullback.

Subpullbacks and Po-flatness Properties of S-posets

In (Golchin A. and Rezaei P., Subpullbacks and flatness properties of S-posets. Comm. Algebra. 37: 1995-2007 (2009)) study was initiated of flatness properties of right -posets over a pomonoid that can be described by surjectivity of corresponding to certain (sub)pullback diagrams and new properties such as and were discovered. In this article first of all we describe po-flatness properties of -posets over pomonoids by po-surjectivity of corresponding to certain subpullback diagrams. Then we introduce three new Conditions , and and investigate the relation between them and Conditions , , , , and . Also we describe these properties by po-surjectivity of corresponding to certain subpullback diag In (Golchin A. and Rezaei P., Subpullbacks and flatness properties of S-posets. Comm. Algebra. 37: 1995-2007 (2009)) study was initiated of flatness properties of right -posets over a pomonoid that can be described by surjectivity of corresponding to certain (sub)pullback diagrams and new properties such as and were discovered. In this article first of all we describe po-flatness properties of -posets over pomonoids by po-surjectivity of corresponding to certain subpullback diagrams. Then we introduce three new Conditions , and and investigate the relation between them and Conditions , , , , and . Also we describe these properties by po-surjectivity of corresponding to certain subpullback diagrams and describe Conditions , and by weak po-surjectivity of corresponding to certain subpullback diagrams. rams and describe Conditions , and by weak po-surjectivity of corresponding to certain subpullback diagrams.

Pullback diagram of $H^*$-algebras

In this paper we obtain some properties for the pullback diagram of H -algebras. More precisely, we prove that the commutative diagram of H -algebras and morphisms A 1 φ 1 ! B 1 ? y 1 ? y 2

Ranks of Low K-Groups of Fibre Products

In this article, we discuss the ranks of low K-groups in pullback diagrams and give the upper and lower bounds on ranks of K-groups of a fibre product (or pullback) under some suitable conditions.

Torsion-free modules of finite balanced-projective dimension over valuation domains

Let R be a valuation domain. Several characterizations are obtained of torsion-free R-modules having finite balanced-projective dimension ( bpd) by using the new concept of an n-balanced submodule and a special pull-back diagram. Noteworthy among the various results are: For torsion-free R-modules M, bpd( M)≤ pd( M), where pd( M) denotes the projective dimension of M. If M is a torsion-free R-module which is not balanced-projective, then bpd( M)= pd( M/ B) for any basic submodule of M.

Toric degenerations and vector bundles

There are many affine subalgebras of polynomial rings with highly non-trivial projective modules, whose initial algebras (toric degenerations) are still finitely generated and have all projective modules free. Let k[X1, . . . , Xn] be a polynomial algebra (k a field, n ∈ N) and A an affine k-subalgebra. Let ≺ denote a term order on the multiplicative semigroup of monomials in the Xi and let in≺(A) denote the monomial subalgebra of k[X1, . . . , Xn], generated by the leading monomials of elements f ∈ A with respect to ≺. In case the initial algebra in≺(A) is finitely generated, one can obtain many properties of A by checking them for in≺(A) (called sometimes a toric degeneration of A) (see [CHV], [RS]). However, this is not the case for the property ‘all projective modules are free’ – thanks to Bernd Sturmfels for asking me this question. Theorem 1. Let A = k[X, Y, Z, Z − XY Z] and ≺ be the lexicographic term order corresponding to Z ≺ Y ≺ X. Then SK0(A) = Ker(K0(A) det → Pic(A)) is not trivial (equivalently, there are projective A-modules which are not even stably of type free⊕rank 1), while in≺(A) is finitely generated and all projective in≺(A)-modules are free. (Here ‘projective’ includes ‘finitely generated’.) Proof. Since (Z2−XY )k[X, Y, Z] ⊂ A, the following diagram with the upper horizontal identity embedding is a pull-back diagram (all the letters refer to variables) A −−−−→ k[X, Y, Z] y y k[U, V ] −−−−→ k[S, ST, T ] , where X 7→ S, Y 7→ T , Z 7→ ST , U 7→ S, V 7→ T . It is similarly easy to show that in≺(A) = k[X, Y, Z, XY Z] – a seminormal monomial algebra (as a kvector space it is generated by the normal monomial subalgebras k[X, Y ], k[X, Z], k[Y, Z] and k[{XaY bZc|a > 0, b > 0, c > 0}]). So by [Gu1] projective modules over in≺(A) are free, while the Mayer-Vietoris sequence (see [Bass], p.490), applied to the diagram above, implies SK0(A) = SK1(k[S, ST, T ]). Hence, by [Gu2] SK0(A) 6= 0. Received by the editors February 20, 1998. 1991 Mathematics Subject Classification. Primary 13D15, 19A49. This research was supported in part by the Alexander von Humboldt Foundation and CRDF grant #GM1-115. c ©1999 American Mathematical Society

On genus and embeddings of torsion-free nilpotent groups of class two

We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N0 of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N0, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank.

Stable homotopy monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback.

Exponentiable morphisms, partial products and pullback complements

In a category K with finite limits, the exponentiability of a morphisms s is (rather easily) characterised in terms of K admitting partial products (essentially those of Pasynkov) over s; and that of a monomorphism is characterised in terms of the new concept of a pullback complement (a universal construction of a pullback diagram whose top and right sides are given). Then, characterisations, previously given by the first author for the category Sp of topological spaces, of the notions of totally reflective subcategory and of hereditary factorisation system are shown to be instances of simple results on adjointness and factorisations.

Correction and supplement to “On a pull-back diagram for orthodox semigroups”

Correction and supplement to “On a pull-back diagram for orthodox semigroups”

On a pull-back diagram for orthodox semigroups

In the present paper we deal with two problems concerning orthodox semigroups. M. Yamada raised the questions in [6] whether there exists an orthodox semigroup T with band of idempotents E and greatest inverse semigroup homomorphic image S for every band E and inverse semigroup S which have the property that Open image in new window is isomorphic to the semilattice of idempotents of S, and if T exists then whether it is always unique up to isomorphism. T. E. Hall [1] has published counter-examples in connection with both questions and, moreover, he has given a necessary and sufficient condition for existence. Now we prove a more effective necessary and sufficient condition for existence and deal with uniqueness, too. On the other hand, D. B. McAlister's theorem in [4] saying that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup is generalized for orthodox semigroups. The proofs of these results are based on a theorem concerning a special type of pullback diagrams. In verifying this theorem we make use of the results in [5] which we draw up in Section 1. The main theorems are stated in Section 2. For the undefined notions and notations the reader is referred to [2].

A Mayer-Vietoris sequence in non-linear K-theory

1.1. The purpose of this discussion is to provide another link in the K-theory introduced in [8] and subsequently developed in [9], [6] and [51. Given a commutative ring R, this theory is a study of the K-groups KA,(R) arising from the category 9(R) of invertible’ R-algebras, i.e. R-algebras A for which there exists an R-algebra B such that A ORB is isomorphic to some polynomial algebra R[X,, . . . , X,,]. Here (Theorem 9.16) we will show the existence, under certain hypotheses, of the “Mayer-Vietoris sequence”; that is, given a pullback diagram