Manfred Knebusch

Grothendieck- und Wittringe von nichtausgearteten symmetrischen Bilinearformen

Diese Arbeit besteht aus drei Teilen. In § 1–4 entwickeln wir die Definition des Wittringes W (X) der nichtausgearteten symmetrischen Bilinearraume uber einem beliebigen Schema X (= Praschema in der Terminologie der [EGA]1). In § 5–10 studieren wir Wittringe uber lokalen Ringen, in § 11–14 uber Dedekindringen und damit verwandten Schemata.

On Algebraic Curves over Real Closed Fields. II.

The contents of this second part of our study on smooth algebraic curves over a real closed fieldK have roughly been indicated in part I [K3] at the end of the introduction. Throughout we use the terminology, notations, and results developed in part I.

Structure of Witt rings, quotients of abelian group rings and orderings of fields

In particular, Theorems 5 and 6 yield the results of [5, §3] for Witt rings of formally real fields and Theorem 7 those of [5, §5] for Witt rings of nonreal fields. By studying subrings of the rings described in Theorems 5-7 and using the results of [2] for symmetric bilinear forms over a Dedekind ring

Manis valuations and Prüfer extensions

Overrings and PM-Spectra.- Approximation Theorems.- Kronecker extensions and star operations.- Basics on Manis valuations and Prufer extensions.- Multiplicative ideal theory.- PM-valuations and valuations of weaker type.- Overrings and PM-Spectra.- Approximation Theorems.- Kronecker extensions and star operations.- Appendix.- References.- Index.

On valuation spectra

If K is an ordered field then every convex subring of K is a valuation ring of K. This easy but fundamental observation has made valuation theory a very natural and important tool in real algebraic geometry. In particular many topological phenomena of semialgebraic sets and of constructible subsets of real spectra are best explained by use of valuations. We have seen in recent years how important it is to switch from the consideration of particular orderings of fields to a study of the set of all orderings of all residue class fields of a commutative ring A, i.e. the real spectrum SperA of A. Now why not do the same with valuations? This leads to the definition of valuation spectra. In principle the points of the valuation spectrum SpevA should be pairs (p, v) consisting of a prime ideal p of A, i.e. a point of SpecA, and a Krull valuation v of the residue class field Quot(A/p). Here one has to made a decision whether one should distinguish between different valuations of Quot(A/p) which have the same valuation ring or not. One further has to choose a topology on SpevA, where again at least two reasonable choices can be made. Finally one should look for sheaves of “functions” on SpevA and some prominent subsets of SpevA. In recent years various authors have defined valuation spectra and/or related spaces. (Brumfiel, de la Puente, Berkovich, Robson, Huber, Schwartz). To my opinion the question which valuation spectrum is the “right” one depends on the applications one has in mind. Certain valuation spectra are important both for real algebraic and for p-adic geometry. In want to stress here a direction followed by Roland Huber which leads to a new foundation of rigid analytic geometry. Huber defines for A in a certain class of topological rings, which he calls “f -adic rings”, a ringed space SpaA, the analytic spectrum of A. The points of SpaA are those points (p, v) of the valuation spectrum SpevA such that a homomorphism form A to a valued field K inducing v is continuous. Analytic spectra are the building blocks of “adic

Semialgebraic topology over a real closed field I: Paths and components in the set of rational points of an algebraic variety

We fix once and for all a real closed base field R. By a variety X over R we mean a separated algebraic scheme X over R. For all problems attacked here we could equally well assume that X is also reduced, since we are only interested in the set X(R) of rational points of X. Notice that for every closed point x of X the residue class field K(x) = OJmx either coincides with R, i.e. x is rational, or is isomorphic to R(>/--T). We call the points x of X with K(X) = R the real points of X and the other closed points the complex points of X. R is a topological field, a basis of open sets being given by the open intervals

Über die Grade quadratischer Formen

Sei q> eine quadratische Form uber einem Korper K einer Charakteristik + 2, die wir stillschweigend stets als nicht ausgeartet voraussetzen. Ist q> nicht hyperbolisch, so betrachten wir zu allen Korpererweiterungen L von K in einem Universalkorper, fur die . Einer hyperbolischen Form ordnen wir den Grad oo zu. Der Grad von q> hangt ersichtlich nur von der Klasse {<p} von cp im Wittring W(K) ab. Wir haben also eine numerische Funktion

Categories of Layered Semirings

We generalize the constructions of layered domains† to layered semirings, in order to enrich the structure, and in particular to provide finite examples for applications in arithmetic. The layered category theory is extended accordingly, to cover noncancellative monoids, which are examined in their own right.

Annullatoren von Pfisterformen über semilokalen Ringen

w 1. Einleitung Mit der vorliegenden Untersuchung setzen wit unser in den Arbeiten [6], [-1, w 1] und [2] begonnenes Studium der Pfisterformen fiber semilokalen Ringen fort. Wit benutzen hier dieselben Bezeichnungen wie in den drei genannten Arbeiten bis auf den unwesentlichen Unterschied, dab alle symmetrisch bilinearen und quadratischen R~iume in dieser Arbeit nicht nut projektive, sondern freie Moduln sein sollen. Ohne Zweifel hat auch fiber semilokalen Ringen eine m6glichst genaue Kenntnis der Eigenschaften von Pfisterformen zentrale Bedeutung fiir den weiteren Aufbau einer Theorie der quadratischen und (symmetrisch) bilinearen Formen, wie dies fiber K6rpern eindrucksvoll dutch Arbeiten von Pfister, Arason, Elman und Lam belegt wird. A ist in dieser Arbeit stets ein kommutativer Ring mit Eins, der semilokai ist, d.h. nur endlich viele maximale Ideale hat. W(A) bezeichnet den Wittring der (freien) bilinearen Riiume fiber A und Wq(A) die Wittgruppe der (freien) quadratischen R~iume fiber A. Wie in [1] bezeichnen wit fiir Elemente a, b aus A mit 1-4ab Einheit den

Signatures on semilocal rings

Most of the results of this paper have been announced in [31, Section 3] and, in slightly simplified form, in [32]. The reader is advised to consult these announcements for an outline of the contents of the present work. One of our main purposes here is to extend part of the Artin-Schreier theory of real closed fields to commutative semilocal rings with involution. The central concept that enables us to accomplish this goal is that of a signature: Let C denote a semilocal ring with an involution / whose fixed ring we denote by A. A signature is a homomorphism a from A*, the group of units of A, to {±1} with certain properties (Definition 2.1 and Proposition 2.4). If A is a field the signatures correspond byectively with the set of total orderings of A for which all the norms N(c) = c J(c) for c in C * are positive. In particular then, if / is the identity, this latter set consists of all orderings of A. In Section 2 we study the notion of signature. By definition a signature a on (C, /) corresponds with a unique homomorphism 5 from the Witt ring WF(C, J) of free Hermitian spaces over (C, /) to Z and conversely. For the case / the identity and C = A a field this correspondence between orderings