Lecture Notes in Mathematics

5 primary works

Locally Semialgebraic Spaces

by Hans Delfs and Manfred Knebusch

Published 1 December 1985

Weakly Semialgebraic Spaces

by Manfred Knebusch

Published 22 February 1989

The book is the second part of an intended three-volume treatise on semialgebraic topology over an arbitrary real closed field R. In the first volume (LNM 1173) the category LSA(R) or regular paracompact locally semialgebraic spaces over R was studied. The category WSA(R) of weakly semialgebraic spaces over R - the focus of this new volume - contains LSA(R) as a full subcategory. The book provides ample evidence that WSA(R) is "the" right cadre to understand homotopy and homology of semialgebraic sets, while LSA(R) seems to be more natural and beautiful from a geometric angle. The semialgebraic sets appear in LSA(R) and WSA(R) as the full subcategory SA(R) of affine semialgebraic spaces. The theory is new although it borrows from algebraic topology. A highlight is the proof that every generalized topological (co)homology theory has a counterpart in WSA(R) with in some sense "the same", or even better, properties as the topological theory. Thus we may speak of ordinary (=singular) homology groups, orthogonal, unitary or symplectic K-groups, and various sorts of cobordism groups of a semialgebraic set over R. If R is not archimedean then it seems difficult to develop a satisfactory theory of these groups within the category of semialgebraic sets over R: with weakly semialgebraic spaces this becomes easy. It remains for us to interpret the elements of these groups in geometric terms: this is done here for ordinary (co)homology.

Homology of Locally Semialgebraic Spaces

by Hans Delfs

Published 23 October 1991

Locally semialgebraic spaces serve as an appropriate
framework for studying the topological properties of
varieties and semialgebraic sets over a real closed field.
This book contributes to the fundamental theory of
semialgebraic topology and falls into two main parts.
The first dealswith sheaves and their cohomology on spaces
which locally look like a constructible subset of a real
spectrum. Topics like families of support, homotopy, acyclic
sheaves, base-change theorems and cohomological dimension
are considered.
In the second part a homology theory for locally complete
locally semialgebraic spaces over a real closed field is
developed, the semialgebraic analogue of classical
Bore-Moore-homology. Topics include fundamental classes of
manifolds and varieties, Poincare duality, extensions of the
base field and a comparison with the classical theory.
Applying semialgebraic Borel-Moore-homology, a semialgebraic
("topological") approach to intersection theory on varieties
over an algebraically closed field of characteristic zero is
given. The book is addressed to researchers and advanced
students in real algebraic geometry and related areas.

Manis Valuations and Prufer Extensions I

by Manfred Knebusch and Digen Zhang

Published 20 August 2002

The present book is devoted to a study of relative Prufer rings and Manis valuations, with an eye to application in real and p-adic geometry. If one wants to expand on the usual algebraic geometry over a non-algebraically closed base field, e.g. a real closed field or p-adically closed field, one typically meets lots of valuation domains. Usually they are not discrete and hence not noetherian. Thus, for a further develomemt of real algebraic and real analytic geometry in particular, and certainly also rigid analytic and p-adic geometry, new chapters of commutative algebra are needed, often of a non-noetherian nature. The present volume presents one such chapter.

Manis Valuations and Prufer Extensions II

by Manfred Knebusch and Tobias Kaiser

Published 2 April 2014

This volume is a sequel to "Manis Valuation and Prufer Extensions I," LNM1791. The Prufer extensions of a commutative ring A are roughly those commutative ring extensions R / A, where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=Prufer-Manis) valuations. While in Volume I Prufer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Grater's work in 1980, a far-reaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called "Kronecker extensions," where PM valuations are put to use in arbitrary commutative ring extensions in a way that ultimately goes back to the work of Leopold Kronecker.