Categories with Projective Functors

We introduce a notion of a category with full projective functors.It encodes certain common properties of categories appearingin representation theory of Lie groups, Lie algebras and quantumgroups. We describe the left or right exact functors which naturallycommute with projective functors and provide a unified approachto the verification of relations between such functors. 2000Mathematics Subject Classification 17B10.     

Projective manifolds with small pluridegrees

Let be a very ample line bundle on a connected complex projective manifold of dimension . Except for a short list of degenerate pairs , and there exists a morphism expressing as the blowup of a projective manifold at a finite set , with nef and big for the ample line bundle . The projective geometry of is largely controlled by the pluridegrees for , of . For example, , where is the genus of a curve section of , and is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of . In this article, a detailed analysis is made of the pluridegrees of . The restrictions found are used to give a new lower bound for the dimension of the space of sections of . The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to .

     

Projective spaces with Cayley-Klein metrics

Projective metrics were first introduced by A. Cayley and F. Klein who constructed analytical models over the field of complex numbers. The aim of this paper is to give for the first time a purely synthetic definition of all projective spaces with Cayley-Klein metrics and to develop the synthetic foundation of projective-metric geometry to a level of generality including metrics over arbitrary fields of characteristic 2.     

Projective geometry with Clifford algebra

Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to projective geometry and facilitates connections with the rest of mathematics.This work was partially supported by NSF grant MSM-8645151.     

Projective surfaces with bi-elliptic hyperplane sections

We study projective surfaces X which have a bi-elliptic curve (i.e. 2∶1 covering of an elliptic curve) among their hyperplane sections . We give a complete characterization of those surfaces when their degree d is d≥17 (only conic bundles and scrolls if d≥19, three possible exception otherwise) and when d≤8. A conjecture is given for the remaining cases. The main tool we use is the study of the adjunction mapping on X.     

Some Ideals with Large Projective Dimension

For an ideal in a polynomial ring over a field, a monomial support of is the set of monomials that appear as terms in a set of minimal generators of . Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projective dimension of grows at least exponentially with the size of a monomial support. The ideal we construct is generated by monomials and binomials.

     

On Projective Modules with Constant Ranks

On Projective Modules with Constant RanksIn this paper,we investigate module structures of rings over which every finitely generated projective module with constant rank is stably free. As applications,we give characterizations of some related rings.     

Complex Projective Structures with Schottky Holonomy

Let S be a closed orientable surface of genus at least two. Let ?? be a Schottky group whose rank is equal to the genus of S, and ?? be the domain of discontinuity of ??. Pick an arbitrary epimorphism : ${{\rho : \pi_{1}(S) \rightarrow \Gamma}}$ . Then ??/?? is a surface homeomorphic to S carrying a (complex) projective structure with holonomy ??. We show that every projective structure with holonomy ?? is obtained by (2??)grafting ??/?? along a multiloop on S.     

Projective structures with discrete holonomy representations

Let denote the set of projective structures on a compact Riemann surface whose holonomy representations are discrete. We will show that each component of the interior of is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of is a quasifuchsian group.

     

Projective surfaces with many skew lines

We give an example of a smooth surface of degree that contains pairwise disjoint lines. In particular, our example shows that the degree in Miyaoka's bound is sharp.

     

Projective homogeneity

We consider aC simply connected manifoldM endowed with a projective structureP and under an additional hypothesis on the projective curvature tensor, we find necessary and sufficient conditions in order thatM turns out to be a reductive homogeneous spaceG/H whereG is a Lie group acting onM as a group of automorphisms ofP.     

Projective semimodules

In this paper we represent a projective semimodule as a retract of a direct sum of its countably generated projective retracts with zero intersection. A characterization by means of congruences is also given. Received May 17, 1999; accepted in final form March 13, 2002. RID="h1" ID="h1"This research was supported by ESF grant 4912 and the Fulbright Fellow award.?The author would like to thank the referee for helpful remarks.     

Projective polynomials

Certain nice trinomials have the projective linear groups as their Galois groups. This was proved using considerable group theory. Here is an easier proof based on the observation that the said trinomials are what may be called projective polynomials. It extends the results to a local situation.

     

Projective bichains

An algebra with two binary operations · and +  that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) =  x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation +  and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Projective algebras

Presented by R. Freese.