Several of the fundamental theorems about algebraic K, and Kr are concerned with finding unimodular elements, that is, elements of a projective module which generate a free summand. In this paper we use the notion of a basic element (in the terminology of Swan [22]) to extend these theorems to the context of finitely generated modules. Our techniques allow a simplification and strengthening of ...

Abstract Pirashvili’s Dold–Kan type theorem for finite pointed sets follows from the identification in terms of surjections morphisms between tensor powers a functor playing role augmentation ideal; these functors are projective. We give an unpointed analogue this result: namely, we compute corresponding context. also calculate Ext groups such objects, particular showing that not projective; is...

emph{ Smooth Projective Hash Functions } ( SPHFs ) as a specific pattern of zero knowledge proof system are fundamental tools to build many efficient cryptographic schemes and protocols. As an application of SPHFs, emph { Password - Based Authenticated Key Exchange } ( PAKE ) protocol is well-studied area in the last few years. In 2009, Katz and Vaikuntanathan described the first lattice-based ...

Definition 1. Let k be a field. An algebraic variety over k is a k-scheme X such that there exists a covering by a finite number of affine open subschemes Xi which are affine varieties over k, i.e. each Xi is the affine scheme associated to a finitely generated algebra over k. A projective variety over k is a projective scheme over k, i.e. a k-scheme isomorphic to Proj k[T0, . . . , Tn]/I for a...

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrica...

Serre [5] has recently proved a general theorem about projective modules over commutative rings. This theorem has the following consequence : If 7T is a finite abelian group, any finitely generated projective module over the integral group ring Zir is the direct sum of a free module and an ideal of Zir. The question naturally arises as to whether this result holds for nonabelian groups x. Serre...

The Drinfel’d Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the canonical embedding of the Lagrangian Grassmannian. We show that the defining ideal of any Schubert subvariety of the Drinfel’d Lagrangian Grassmannian is generated by polynomials which gi...

Let V be a closed subscheme of a projective space P. We give an algorithm to compute the Chern-Schwartz-MacPherson class, and the Euler characteristic of V and an algorithm to compute the Segre class of V . The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a h...

New constructions of 3-dimensional optical orthogonal codes are presented. In each case the codes have ideal autocorrelation λa = 0, and cross correlation of λc = 1. All codes produced are demonstrated to be optimal. The constructions utilize a particular automorphism (a Singer cycle) of PG(k,q), the finite projective geometry of dimension k over the field of order q, or its affine analogue in ...

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H∞(X) of bounded holomorph...