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Type: report × search.filters.applied.f.series: Oberwolfach Preprints (OWP), Volume 2012-12, ISSN 1864-7596 ×

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    K-triviality, Oberwolfach randomness, and differentiability
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Bienvenu, Laurent; Greenberg, Noam; Kucera, Antonín; Nies, André; Turetsky, Dan
    We show that a Martin-Löf random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12(3):390-410, 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set. The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a "smart" K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.
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    Module categories for group algebras over commutative rings
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Stevenson, Greg
    We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.