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Notes on the Riemann Hypothesis
These notes were written from a series of lectures given in March 2010 at the
Universidad Complutense of Madrid and then in Barcelona for the centennial
anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an
introduction to the Riemann Hypothesis and a panoramic view of the world of
zeta and L-functions. We first review Riemann's foundational article and
discuss the mathematical background of the time and his possible motivations
for making his famous conjecture. We discuss some of the most relevant
developments after Riemann that have contributed to a better understanding of
the conjecture.Comment: 2 sections added, 55 pages, 6 figure
Relative FP-injective and FP-flat complexes and their model structures
In this paper, we introduce the notions of -injective and -flat complexes in terms of complexes of type . We show that
some characterizations analogous to that of injective, FP-injective and flat
complexes exist for -injective and -flat complexes. We
also introduce and study -injective and -flat
dimensions of modules and complexes, and give a relation between them in terms
of Pontrjagin duality. The existence of pre-envelopes and covers in this
setting is discussed, and we prove that any complex has an -flat
cover and an -flat pre-envelope, and in the case that
any complex has an -injective cover and an -injective
pre-envelope. Finally, we construct model structures on the category of
complexes from the classes of modules with bounded -injective and
-flat dimensions, and analyze several conditions under which it is
possible to connect these model structures via Quillen functors and Quillen
equivalences.Comment: 41 page
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