119 research outputs found
Galois cohomology of a number field is Koszul
We prove that the Milnor ring of any (one-dimensional) local or global field
K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions
that are only needed in the case l=2, we also prove various module Koszulity
properties of this algebra. This provides evidence in support of Koszulity
conjectures that were proposed in our previous papers. The proofs are based on
the Class Field Theory and computations with quadratic commutative Groebner
bases (commutative PBW-bases).Comment: LaTeX 2e, 25 pages; v.2: minor grammatic changes; v.3: classical
references added, remark inserted in subsection 1.6, details of arguments
added in subsections 1.4, 1.7 and sections 5 and 6; v.4: still more misprints
corrected, acknowledgement updated, a sentence inserted in section 4, a
reference added -- this is intended as the final versio
Contramodules
Contramodules are module-like algebraic structures endowed with infinite
summation (or, occasionally, integration) operations satisfying natural axioms.
Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras
over commutative rings, contramodules experience a small renaissance now after
being all but forgotten for three decades between 1970-2000. Here we present a
review of various definitions and results related to contramodules (drawing
mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621,
arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230,
arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings,
topological associative rings, topological Lie algebras and topological groups,
semicontramodules over semialgebras, and a "contra version" of the
Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the
comodule-contramodule correspondence phenomenon are discussed.Comment: LaTeX 2e with pb-diagram and xy-pic; 93 pages, 3 commutative
diagrams; v.4: updated to account for the development of the theory over the
four years since Spring 2015: introduction updated, references added, Remark
2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 adde
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