Sisterhoods of flat Laguerre planes

Every flat Laguerre plane of shear type over a pair of skew parabolae is related to a flat Laguerre plane of translation type over a pair of skew parabolae and vice versa. The relationship is defined using the connection between flat Laguerre planes and three-dimensional generalized quadrangles.Dedicated to Prof. H. R. Salzmann on his 65th birthday     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

Strongly Flat and Condition (P) Covers of Acts Over Monoids

In this article we characterize monoids over which every right S-act has a strongly flat (condition (P)) cover. Similar to the perfect monoids, such monoids are characterized by condition (A) and having strongly flat (condition (P)) cover for each cyclic right S-act. We also give a new characterization for perfect monoids as monoids over which every strongly flat right S-act has a projective cover.     

On torsionless and dense acts

Torsionless acts over a monoid S are investigated, in particular torsionless factor acts of 2 -free and 1 -free acts. Monoids over which free or projective acts are torsionless and vice versa are characterized. Some necessary conditions for torsionless acts to be principally weakly flat, weakly flat or strongly flat are given. First results on dense acts are mentioned and several examples, mostly on the basis of cofree acts, are presented to illustrate these concepts. August 15, 2000     

Response of heat transfer from a moving flat plate in a parabolic flow

This paper deals with the solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations and the study of the response of heat transfer when there is a parabolic flow over a moving flat plate. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse thermal boundary layer equations. A general analysis is made from which we deduce (i) Solutions of two-dimensional thermal boundary layer on a moving flat plate, (ii) Solutions of thermal boundary layer on a yawed flat plate, (iii) Solutions of thermal boundary layer when there is a parabolic flow over a moving flat plate by giving different values to β and Cx. Solutions are developed for large and small times and curves are drawn representing the variations of heat transfer from the plate with time for all the cases. The limiting time is also calculated.     

On Gorenstein FP-Injective and Gorenstein Flat Complexes

In this article, the concept of Gorenstein FP-injective modules and some related known results are generalized to Gorenstein FP-injective complexes. Moreover, some new characterizations of Gorenstein flat complexes are given. It is also proved that every complex has a Gorenstein flat preenvelope over coherent rings with finite self-FP-injective dimension.     

Exact borel subalgebras of quasi-hereditary algebras. II

In this article, we consider the class of flat G-modules in the category of discrete modules over a profinite group G. We will appeal to a recent result of Enochs to prove that we have flat covers in this situation.     

On strongly flat and condition (P) <Emphasis Type="Italic">S</Emphasis>-posets

In this paper, we first present some homological classifications of pomonoids by using condition (P) and strongly flat properties. Unlike the case for acts, condition (P) and strongly flat coincide for cyclic right S-posets when all weakly right reversible convex subpomonoids of a pomonoid S are left collapsible. Thereby we characterize pomonoids over which strong flatness and condition (P) imply some other flatness properties. Furthermore, we characterize a pomonoid over which every right S-poset has a strongly flat (condition (P)) cover.     

The localisation of the discharge channel in a glow discharge

The localisation of the discharge channel under the conditions of a glow discharge in air, at different distances from the cathode and at various points in the cathode dark space for different voltages, has been studied by measuring the radial distribution of the light intensity in the discharge channel. It has been shown that the radial ion distribution which has a flat maximum at lower voltages changes over into an approximately Gaussian one as the discharge voltage is increased. In the former case the width of the flat maximum decreases as the ion-channel proceeds towards the cathode. At higher voltages (~ 6 kV.) the flat maximum type of distribution in the ion-channel progressively changes over into the Gaussian as the channel approaches the cathode.     

Gorenstein Projective Precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.     

Flat Comodules and Perfect Coalgebras

Stenström introduced the notion of flat object in a locally finitely presented Grothendieck category 𝒜. In this article we investigate this notion in the particular case of the category 𝒜 = C-Comod of left C-comodules, where C is a coalgebra over a field K. Several characterizations of flat left C-comodules are given and coalgebras having enough flat left C-comodules are studied. It is shown how far these coalgebras are from being left semiperfect. As a consequence, we give new characterizations of a left semiperfect coalgebra in terms of flat comodules. Left perfect coalgebras are introduced and characterized in analogy with Bass's Theorem P. Coalgebras whose injective left C-comodules are flat are discussed and related to quasi-coFrobenius coalgebras.     

Coherent rings of finite weak global dimension

The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, Martínez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.

     

Perfect monoids revisited

A new characterization of perfect monoids, i.e., monoids over which every act has a projective cover, is given. As was shown by Fountain [1] a monoid is perfect if and only if all strongly flat acts over it are projective. Using our new condition, an alternative version is given of a recent result, of Liu [7] describing monoids over which all strongly flat right acts are projective generators, or are free. This research has been supported by the Estonian Science Foundation, Grant No. 930. I would like to thank the School of Mathematical and Computational Sciences of the University of St. Andrews for excellent working conditions.     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate

A numerical study of magnetic effect on the formation of longitudinal vortices in natural convection flow over a rotating heated flat plate is presented. The onset position characterized by the local Grashof number, depends on the rotational Reynolds number, the Prandtl number, the Hartmann number, and the wave number. The Coriolis force and the Lonertz force have significant effects on the formation of longitudinal vortices and the associated instability. Positive rotation stabilizes the flow on the rotating flat surface. On the contrary, a negative rotation destabilizes the flow. The flow is found more stable as the value of Hartmann number increases. The numerical data show reasonable agreement with the experimental results with the case of thermal instability in natural convection over a flat plate heated from below.     

Representing homology classes by locally flat 2-spheres

We prove an existence result for topologically locally flat embeddings of 2-spheres in simply connected 4-manifolds. This topological result is deduced from a splitting theorem for pointed Hermitian modules over a cyclic group ring. A stability result for such modules is also proved. This applies to the isotopy classification of locally flat embeddings.Partially supported by the NSF.     

On Holomorphic Principal Bundles Over a Compact Riemann Surface Admitting a Flat Connection, II

Holomorphic principal bundles over a compact Riemann surfaceX that admits a flat connection are considered. A holomorphicG-bundle over X, where G is a connected semisimple linear algebraicgroup over C, admits a flat connection if and only if the adjointvector bundle admits one. More generally, for a complex reductivegroup G, the necessary and sufficient condition on a G-bundleto admit a flat connection is described. This simplifies thecriterion obtained by the authors and given in Math. Ann. 322(2002) 333–346. 2000 Mathematics Subject Classification53C05, 32L05.     

Riemann-Roch-Grothendieck and torsion for foliations

In this article we prove a Riemann-Roch-Grothendieck theorem for the characteristic classes of a flat vector bundle over a foliation whose graph is Hausdorff. We assume that the strong foliation Novikov-Shubin invariants of the flat bundle are greater than three times the codimension of the foliation. Using transgression, we define a torsion form which in the odd acyclic case determines a Haefliger cohomology class which only depends on the foliation and the flat bundle. We construct examples where this torsion class is highly non-trivial.