Flat Algebra Research Articles

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

English

We investigate the triangulated hull of the orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull will correspond to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao's result when the ring is a finite dimensional algebra with finite global dimension over a field. As the first application, if $A,B$ are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of $n$-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.

Symmetries of a Flat Cosymbol Algebra of Differential Operators

In this paper, a structure theorem for the symmetries of a graded flat cosymbol algebra of differential operators is proved. Together with a lemma on equivariant polynomials also proved in the paper, this theorem gives an upper bound on the dimension of the graded Lie algebra associated with the symmetries of geodesic flow on a smooth variety. Bibliography: 14 titles

The structure of a Laurent polynomial fibration in n variables

Bass, Connell and Wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain R is isomorphic to the symmetric algebra of a finitely generated projective R-module of rank n. In this paper we prove a corresponding structure theorem for a ring A which is a locally Laurent polynomial algebra in n variables over an integral domain R, viz., we show that A is isomorphic to an R-algebra of the form ( Sym R ( Q ) ) [ I − 1 ] , where Q is a direct sum of n finitely generated projective R-modules of rank one and I is a suitable invertible ideal of the symmetric algebra Sym R ( Q ) . Further, we show that any faithfully flat algebra over a Noetherian normal domain R, whose generic and codimension-one fibres are Laurent polynomial algebras in n variables, is a locally Laurent polynomial algebra in n variables over R.

The Division Relation: Congruence Conditions and Axiomatisability

We examine a universal algebraic abstraction of the semigroup theoretic concept of “divides:” a divides b in an algebra A if for some n ∈ ω, there is a term t(x, y 1,…, y n ) involving all of the listed variables, and elements c 1,…, c n such that t A (a, c 1,…, c n ) = b. The first order definability of this relation is shown to be a very broad generalisation of some familiar congruence properties, such as definability of principal congruences. The algorithmic problem of deciding when a finitely generated variety has this relation definable is shown to be equivalent to an open problem concerning flat algebras. We also use the relation as a framework for establishing some results concerning the finite axiomatisability of finitely generated varieties.

REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

Flat algebras and the translation of universal Horn logic to equational logic

AbstractWe describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, partial algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown to be equivalent to the finite identity basis problem for the finite members of a finiteiy based variety with definable principal congruences.

Some results on codimension-one A1-fibrations

We prove a few results on the sufficiency of generic and codimension-one fibre conditions for determination of the structure of algebras of transcendence degree one. We first show that over a Noetherian normal domain R, a faithfully flat subalgebra of a finitely generated algebra whose generic and codimension-one fibres are A 1 is necessarily the symmetric algebra of an invertible ideal of R. We next prove a structure theorem for a faithfully flat algebra over a locally factorial Krull domain R whose generic and codimension-one fibres are A 1 . For R local, we deduce a minimal sufficient condition for the algebra to be finitely generated and hence A 1 .

Quasiequational Theories of Flat Algebras

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q-based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q-based.

Flat and weakly flat projection algebras

The notion of a projection algebra was first introduced in [4] by Ehrig et al. as an algebraic version of ultrametric spaces. Computer scientists use this notion as a convenient means of algebraic specification of process algebras. Some algebraic notions regarding these algebras have been studied in [1], [2], [5]. The flat projection algebras have been investigated by the authors in [1]. Here we completely characterize flat and weakly flat (m-separated and separated) projection algebras.

COMPUTATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM FOR VARIETIES

We exhibit finite algebras each generating a variety with NP-complete finite algebra membership problem. The smallest of these algebras is the flat graph algebra belonging to the tetrahedral graph, a graph of 6 vertices obtained by cutting and spreading out the surface of a tetrahedron on the plane. The sequence of graphs we use to build up our flat graph algebras is similar to the sequence exhibited by Wheeler in [36] , 1979, to describe the first order theory of k-colorable graphs. Graph algebras were introduced by Shallon in [34] , 1979, and investigated, among others, by Baker, McNulty and Werner in [2] , 1987. Flat algebras were constructed and used by McKenzie in [27] , 1996, to settle some open questions related to decidability, like Tarski's Finite Basis Problem. Flat graph algebras were also discussed by Willard in [37] , 1996, and Delić in [8] , 1998.

Algebraically flat or projective algebras

We define and study algebraically flat algebras in order to have a better understanding of algebraically projective algebras of finite type (the projective algebras of literature). A close examination of the differential properties of these algebras leads to our main structure theorem. As a corollary, we get that an algebraically projective algebra of finite type over a field is either a polynomial ring or the affine algebra of a complete intersection.

Full duality among graph algebras and flat graph algebras

We prove that among finite graph algebras and among finite flat graph algebras, dualizability, full dualizability, strong dualizability and entropicity are all equivalent. Any finite (flat) graph algebra which is not dualizable must be inherently non-\( \kappa \)-dualizable for every infinite cardinal \( \kappa \). A new, general method for proving strong duality is presented.

THE GROUP OF HOPF-GALOIS EXTENSIONS WITH CENTRAL INVARIANTS

In this paper, for an algebra A and a flat Hopf algebra H, in a symmetric closed category C with equalizers and coequalizers, we define the product of H-extensions of A with central invariants. This product is a generalization of the product of H-Galois objects. Finally, we define the Hopf-Galois H-extensions of A with central invariants and we prove that, if A is a commutative faithfully flat algebra and H is cocommutative and faithfully flat, the set of isomorphism classes of Hopf-Galois H-extensions of A with central invariants, with the product of H-extensions, is an abelian group denoted by Gal C (A, H).

On [formula omitted]-fibrations

In this paper we investigate minimal sufficient fibre conditions for a finitely generated flat algebra over a noetherian integral domain to be locally A ∗ or at least an A ∗ -fibration. We also describe the structure of finitely generated locally A ∗ -algebras.

A variation of sweedler's complex and the group of galois objects of an infinite hopf algebra

Let H be a cocommutative faithfully flat Hopf algebra over a commutative ring R. We give a four term exact sequence describing the group of H-Galois objects. This generalizes an exact sequence due to Early and Kreimer and Yokogawa. We have to introduce the Picard group of a coalgebra, and a new type of Sweedler cohomology. Using methods of Villamayor and Zelinsky and Hattori, we can extend the four term exact sequence to a sequence of infinite length.

Converting nested algebra expressions into flat algebra expressions

Nested relations generalize ordinary flat relations by allowing tuple values to be either atomic or set valued. The nested algebra is a generalization of the flat relational algebra to manipulate nested relations. In this paper we study the expressive power of the nested algebra relative to its operation on flat relational databases. We show that the flat relational algebra is rich enough to extract the same “flat information” from a flat database as the nested algebra does. Theoretically, this result implies that recursive queries such as the transitive closure of a binary relation cannot be expressed in the nested algebra. Practically, this result is relevant to (flat) relational query optimization.

A comparison between algebraic query languages for flat and nested databases

Recently, much attention has been paid to query languages for nested relations. In the present paper, we consider the nested algebra and the powerset algebra, and compare them both mutually as well as to the traditional flat algebra. We show that either nest or difference can be removed as a primitive operator in the powerset algebra. While the redundancy of the nest operator might have been expected, the same cannot be said of the difference. Basically, this result shows that the presence of one nonmonotonic operator suffices in the powerset algebra. As an interesting consequence of this result, the nested algebra without the difference remains complete in the sense of Bancilhon and Paredaens. Finally, we show there are both similarities and fundamental differences between the expressiveness of query languages for nested relations and that of their counterparts for flat relations.

When is a Flat Algebra of Finite Type?

When is a Flat Algebra of Finite Type?

When is a flat algebra of finite type?

Let A A denote a commutative Noetherian domain. For an intermediate ring A ⊆ B ⊆ A x A \subseteq B \subseteq {A_x} flat over A A , it is shown that B B is an A A -algebra of finite type. This is followed by an intrinsic description of the flatness of B B over A A and the asymptotic behavior of certain prime divisors. As an application, flat ideal-transforms are characterized.