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It is shown how abstract localization theory may be applied in order to associate to a not necessarily noetherian pi algebra a ringed space (Spec(R),O R), which behaves functorially with respect to extensions and which possesses suitable features allowing one to study this type of ring from a geometric point of view. These results generalize previous ones, obtained by F. Van Oystaeyen and the author in the noetherian case.

We show that that if every real has a sharp and there are Δ 2 1 -definable prewellorderings of ℝ of ordinal ranks unbounded inω 2, then there is an inner model for a strong cardinal. Similarly, assuming the same sharps, the Core ModelK is Σ 3 1 -absolute unless there is an inner model for a strong cardinal.

Using an ergodic transformation defined on an infinite measure space, we discuss complements in ℤ of the setA consisting of finite sums of odd powers of 2.

For an odd prime p, let K/k be a Galois p-extension and S be a set of primes of k containing the primes lying over p. For the p r th roots $${\mu _{{p^r}}}\left( K \right)$$ of unity in K, we describe the so-called Sha group Sha S (G(K/k), $${\mu _{{p^r}}}\left( K \right)$$ ) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields $${\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0$$ where $${k_{{T^{i + 1}}}}$$ is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of $${k_{{T^i}}}$$ over $${k_{{T^i}}}$$ . This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees $${\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}$$ over the towers. Using this formula we investigate whether the towers are stationary or not.

LetY be a Gorenstein trigonal curve withg:=pa(Y)≥0. Here we study the theory of special linear systems onY, extending the classical case of a smoothY given by Maroni in 1946. As in the classical case, to study it we use the minimal degree surface scroll containing the canonical model ofY. The answer is different if the degree 3 pencil onY is associated to a line bundle or not. We also give the easier case of special linear series on hyperelliptic curves. The unique hyperelliptic curve of genusg which is not Gorenstein has no special spanned line bundle.

Let (G n) be a sequence which is dense (in the sense of the Banach-Mazur distance coefficient) in the class of all finite dimensional Banach spaces. Set $$C_p = (\Sigma G_n )_{l_p } (1< p< \infty ) = (\Sigma G_n )_{c_0 } $$ . It is shown that a Banach spaceX is isomorphic to a subspace ofC p (1