389 research outputs found
Bounding the dimensions of rational cohomology groups
Let be an algebraically closed field of characteristic , and let
be a simple simply-connected algebraic group over that is defined and
split over the prime field . In this paper we investigate
situations where the dimension of a rational cohomology group for can be
bounded by a constant times the dimension of the coefficient module. We then
demonstrate how our results can be applied to obtain effective bounds on the
first cohomology of the symmetric group. We also show how, for finite Chevalley
groups, our methods permit significant improvements over previous estimates for
the dimensions of second cohomology groups.Comment: 13 page
Patterns of primes in arithmetic progressions
After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic progressions of primes with the property that p′ = p + d is the prime following p for each element of the progression. This was a common generalization of the results of Zhang and Green-Tao. In the present work it is shown that for every m we have a bounded m-tuple of primes such that this configuration (i.e. the integer translates of this m-tuple) appear as arbitrarily long arithmetic progressions in the sequence of all primes. In fact we show that this is true for a positive proportion of all m-tuples. This is a common generalization of the celebrated works of Green-Tao and Maynard/Tao.
Dedicated to the 60th birthday of Robert F. Tich
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