Witt morphisms (Q2903264)

The present book is a thorough presentation of the theory of function fields of curves over a real closed field, their Witt rings and Witt equivalence between such fields, where fields are called Witt equivalent if their Witt rings are isomorphic, and injectivity and surjectivity questions when passing from a Witt ring of a domain to that of certain ring extensions. There are many related aspects that are also presented in detail such as Witt equivalence of real holomorphy rings. Some of the sections consist of work published earlier by the author, some contain new material.NEWLINENEWLINENEWLINESection 1 gives an introduction to the basic notions and relevant known results such as Witt rings, orderings and valuations, and the Knebusch-Milnor exact sequence of Witt rings NEWLINE\[NEWLINE0\to WP\to WK\to \bigoplus WK(\mathfrak{p})NEWLINE\]NEWLINE where \(P\) is a Dedekind domain, \(K\) its quotient field, \(K(\mathfrak{p})\) the residue field at a maximal ideal \(\mathfrak{p}\), and where the sum ranges over all maximal ideals of \(P\). A short introduction to real curves is also included. In the situation where \(K\) is the function field of a real curve \(\gamma\) over a real closed field and \(R\) is the ring of regular functions on \(\gamma\), it is shown how the Knebusch-Milnor exact sequence extends to Knebusch's exact sequence NEWLINE\[NEWLINE0\to WR\to WK\to \bigoplus WK(\mathfrak{p})\to\mathbb{Z}^N\to 0NEWLINE\]NEWLINE where \(N\) is the number of semi-algebraically connected components of \(\gamma\). In Section 4, this exact sequence is revisited and it is shown that it splits, more precisely, that there is an exact sequence going from right to left in with the maps are suitably compatible with the ones in the original Knebusch sequence, i.e. in homological language, that the Knebusch sequence slices into and is patched by two split short exact sequences. It is furthermore shown that if \(P\) is the ring of polynomial functions on \(\gamma\) and if \(\gamma\) is affine semi-algebraically compact and semi-algebraically connected, then \(WP\) is a direct summand of \(WR\). These results from section 4 have appeared in earlier papers by the author [J. Algebra 301, No. 2, 616--626 (2006; Zbl 1158.11019); Int. J. Pure Appl. Math. 45, No. 1, 5--11 (2008; Zbl 1142.11329)].NEWLINENEWLINENEWLINESection 2 contains new results. It deals with versions of the Scharlau transfer and Scharlau's norm theorem in the case of bilinear forms over a local domain \(P\) with \(2\) invertible and where the extension is given by \(P[\sqrt{d}\,]\) for \(d\in P\) a non-square. The main problem in defining a transfer is the fact that \(P[\sqrt{d}\,]\) need not be free as a \(P\)-module. The transfer thus only yields a map \(s_*:WP[\sqrt{d}\,]\to WP/\mathfrak{c}\) where \(\mathfrak{c}=\{ a\in P\,|\,aP[\sqrt{d}\,]\subseteq P\}\) is the conductor. This can be used to show that the map \(WP\to WP[\sqrt{d}\,]\) need not be an epimorphism. One also obtains a version of Scharlau's norm theorem stating that if \(\xi\) is a bilinear form over \(P\), then \(a+b\sqrt{d}\in P[\sqrt{d}\,]\) is a similarity factor of \(\xi\otimes P[\sqrt{d}\,]\) if and only if the norm \(N(a+b\sqrt{d})\) (defined as an element in \(P/\mathfrak{c}\)) is a similarity factor of the bilinear form \(\overline{\xi}\) over \(P/\mathfrak{c}\).NEWLINENEWLINESection 3 also contains new results. The question here is as follows. Let \(P\) be a domain and \(R\) be its integral closure. What can one say about the injectivity of \(WP\to WR\)? The main result of this section reads as follows. Let \(P\) be a Noetherian domain of dimension \(1\), \(R\) its integral closure, and suppose further that the units of \(R\) are in \(P\) and that the kernel of the natural Picard group map \(\text{Pic}(P)\to\text{Pic}(R)\) contains elements of order \(2\), then \(WP\to WR\) is not a monomorphism. It is shown how this result applies to explicit situations where \(P\) is the coordinate ring of a certain curve.NEWLINENEWLINENEWLINEIn the final Section 5, the author gives a proof of a result by the reviewer and Grenier-Boley [Forum Math., to appear] stating that two real fields \(K\) and \(L\) with \(u\)-invariant \(\leq 2\) (e.g. function fields of real curves over a real closed field) have isomorphic Witt rings if and only if \(K\) and \(L\) are \(\mathcal{X}\)-equivalent, i.e. there is a homeomorphism \(T:\mathcal{X}_K\to \mathcal{X}_L\) of the respective spaces of orderings and an isomorphism of square class groups \(t:\dot{K}/\dot{K}^2\to \dot{L}/\dot{L}^2\) such that for all \(\alpha\in\mathcal{X}_K\) one has \(a>0\) at \(\alpha\) iff \(t(a)>0\) at \(T(\alpha)\). This result is then extended to the real holomorphy rings \(\mathcal{H}_K\) and \(\mathcal{H}_L\) of \(K\) and \(L\), respectively, and it is shown that in the case where \(\mathcal{H}_K\) and \(\mathcal{H}_L\) are Dedekind domains, then they are Witt equivalent if \(K\) and \(L\) are tamely \(\mathcal{X}\)-equivalent, where ``tame'' means that the \(\mathcal{X}\)-equivalence is compatible with real valuations in a suitable sense. The section concludes with an appendix in which the author summarizes his earlier results on Witt equivalence between function fields of real curves over real closed fields and Witt equivalence between their rings of regular functions.