### A Bertini type theorem in analytic geometry.

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We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence ${\mathbb{L}}_{1}$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/\left(p\right)$. Further, we show that $(\mathbb{Z},{\mathbb{L}}_{1}^{*})$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

I extend the Hasse–Arf theorem from residually separable extensions of complete discrete valuation rings to monogenic extensions.

Let a be an ideal of a commutative ring A. There is a kind of duality between the left derived functors Uia of the a-adic completion functor, called local homology functors, and the local cohomology functors Hai.Some dual results are obtained for these Uia, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the Ua and Ha-global dimensions of A are finite.

Using lattice-ordered algebras it is shown that a totally ordered field which has a unique total order and is dense in its real closure has the property that each of its positive semidefinite rational functions is a sum of squares.