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Register a new userThe torsionfree part of the Ziegler spectrum of orders over Dedekind domains
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We study the R-torsionfree part of the Ziegler spectrum of an order Lambda over a Dedekind domain R. We underline and comment on the role of lattices over Lambda. We describe the torsionfree part of the spectrum when Lambda is of finite lattice representation type.
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Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $mathcal{O}$ is the integral closure of $R$ in $L$, then $mathcal{O}$ contains infinitely many prime ideals. In particular, if $mathcal{O}$ is further a unique factorization domain, then $mathcal{O}$ contains infinitely many non-associate prime elements.
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We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of $eta$-periodic motivic stable homotopy theory established in [arxiv:2005.06778] from fields to arbitrary base schemes, and use this to determine (among other things) the $eta$-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1/2.
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