Flat and Stably Flat Modules

We study Kropholler's generalisation of Lazard's criterion for a module to be flat. The context is complete cohomology and modules of type (FP)_∞. Let G be a group in the class ₁ of groups which act on a finite dimensional contractible cell complex with finite stabilisers and let R be a strongly G-graded algebra. We provide a characterisation of the stably flat R-modules under the assumption that R₁ is coherent of finite global dimension.     

Flat Curves

We characterize the sextic trefoil among plane curves of low degree: first as a complex curve with compact, flat geometry; and then as a curve with meromorphic arc length parameterization.     

Flat Circle Planes as Integrals of Flat Affine Planes

In a previous article (Arch. Math. {64} (1995), 75–85) we showed that flat Laguerre planes can be constructed by'integrating' flat affine planes. It turns out that'most' of the known flat Laguerre planes arise in this manner. In this paper we show that similar constructions are also possible in the case of the other two kinds of flat circle planes, that is, the flat Möbius planes and the flat Minkowski planes. In particular, we show that many of the known flat Möbius planes can be constructed by integrating a closed strip taken from a flat affine plane. We also show how flat Minkowski planes arise as integrals of two flat affine planes. All known flat Minkowski planes can be constructed in this manner.     

On Strongly Copure Flat Modules and Copure Flat Dimensions

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext ⁱ (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings.     

Flat cyclic modules

Necessary and sufficient conditions are established for cyclic -modules to be flat, and rings over which any flat cyclic -module is free are investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 245–250, August, 1970.The author wishes to thank L. A. Skornyakov for his advice and suggestions for improving this article.     

Flat regular polytopes

At the centre of the theory of abstract regular polytopes lies the amalgamation problem: given two regularn-polytopesP ₁ andP ₂, when does there exist a regular (n+1)-polytopeP whose facets are isomorphic toP ₁ and whose vertex-figures are isomorphic toP ₂? The most general circumstances known hitherto which lead to a positive answer involve flat polytopes, which are such that each vertex lies in each facet. The object of this paper is to describe an analogous but wider class of constructions, which generalize the previous results.     

Flat ideals II

This paper is concerned with the relationships that hold between finiteness and divisibility properties of flat ideals of integral domains. Brought out often is the role of the arithmetic of a ring in the finiteness of its flat ideals.Supported in part by NSF grant GP-33133X.     

(i1i)*(b12k0)及(i1i)*(b31k0)在π*V(1)中的收敛性

对连通有限型谱X,y,存在Adams谱序列{Es,tr,dr},满足(1)dr:Es,t,r→Es+r,t+r-1,r是谱序列的微分, (2)Es,t,2≌Exts,t,A(H*X,H*y),(3)收敛到[∑t-sY,X].当X分别是球谱S,Moore谱M,Toda-smith谱V(1)时,(πt-X)p分别是S,M,V(1)的稳定同伦群.利用Adams 谱序列,证明了(i1i)*(62,1k0)及(i1i)*(b3,1k0)是永久循环但不是边缘,因此收敛到π*V(1)中的非零元,其中P为奇索数,q=2p-2.     

The k-record processes are i.i.d.

Let X ₁, X ₂, ... be i.i.d. positive random variables, and let _n be the initial rank of X _n (that is, the rank of X _n among X ₁, ..., X _n). Those observations whose initial rank is k are collected into a point process N ^k on ⁺, called the k-record process. The fact that {itN^k; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N ¹, ..., N ^k, the lifetimes in rank k of all observations of initial rank at most k are independent geometric random variables.These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a time-space Poisson process. Initially, we think of this Poisson process as having values in ⁺, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes.     

RSUV isomorphisms for TAC i,TNC i and TLS

Summary We investigate the second order bounded arithmetical systems which is isomorphic to TAC ⁱ , TNC ⁱ or TLS.     

An Equivalent Flat Condition

We show that on an 8-dimensional manifold with Euler characteristic zero every semiflat metric must be flat.     

Isospectral Flat 3-Manifolds

A well-known problem about Riemannian manifolds asks whether the isospectral manifolds are isometric. We settle this problem for compact flat 3-manifolds.