Download Link Theory in Manifolds by Uwe Kaiser PDF

By Uwe Kaiser

Any topological idea of knots and hyperlinks will be according to basic principles of intersection and linking. during this e-book, a common conception of hyperlink bordism in manifolds and common buildings of linking numbers in orientated 3-manifolds are built. during this approach, classical suggestions of hyperlink idea within the 3-spheres are generalized to a definite category of orientated 3-manifolds (submanifolds of rational homology 3-spheres). The innovations wanted are defined within the publication yet easy wisdom in topology and algebra is thought. The ebook could be of interst to these operating in topology, particularly knot idea and low-dimensional topology.

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The case of characteristic > 0 is standard, since the prime ring is a field (Noether's Normalization Theorem). The case of interest to us here is that of characteristic O. Denote by R Q the algebra generated over Q by R. By the normalization theorem, there exist elements t 1 , ••• ,tr E R algebraically independent over Q such that R Q is integral over Q[t l ' ... ,tr]. • ,tr]. Since this latter ring is integrally closed, its integral closure in the quotient field of R is a finite module over it, say by lAG, Theorem 2 of Chapter V, §l.

Now let L be a projective module of rank lover o. As o-module, such L is isomorphie to some fractional ideal 0, but we prefer to keep L separate from any embedding in K. Alternatively, if the reader does not wish to use the words projective or rank, we let L be a finitely generated torsion free module over o. I caU such L a line module over 0 (to avoid caUing L a line bundle, since the bundle itself would require further terminology to define). Let v be archimedean. Let K v be the completion, wh ich can be identified with C or R.

4. Divisors on Schemes To a large extent, the constant field in the last section was irrelevant. We used it mostly in connection with the projective embedding, and to get the product formula. For the general theory of divisors, it played no role, and so we go through once again the general discussion of divisors in the context ofschemes. We first start in the analogue of the affine case. Let R be a Noetherian ring which we ass urne normal, meaning that it has no divisors of zero and is integrally closed in its quotient field, which we denote by K.

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