73 research outputs found
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
Pseudo-dualizing complexes and pseudo-derived categories
The definition of a pseudo-dualizing complex is obtained from that of a
dualizing complex by dropping the injective dimension condition, while
retaining the finite generatedness and homothety isomorphism conditions. In the
specific setting of a pair of associative rings, we show that the datum of a
pseudo-dualizing complex induces a triangulated equivalence between a
pseudo-coderived category and a pseudo-contraderived category. The latter terms
mean triangulated categories standing "in between" the conventional derived
category and the coderived or the contraderived category. The constructions of
these triangulated categories use appropriate versions of the Auslander and
Bass classes of modules. The constructions of derived functors providing the
triangulated equivalence are based on a generalization of a technique developed
in our previous paper arXiv:1503.05523.Comment: LaTeX 2e with pb-diagram, xy-pic, and tikz-cd, 60 pages, 14+3
commutative diagrams; v.4: sections 10-12 added, new subsections 0.8 and 0.10
inserted in the Introduction; v.5: subsection 0.2 shortened, basic
definitions added in section 1, explanations added in the proof of Theorem
4.2, several references added; v.7: misprints corrected, references updated
-- this is intended as the final versio
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