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The structure theory of abelian -groups does not depend on the properties of the ring of integers, in general. The substantial portion of this theory is based on the fact that a finitely generated -group is a direct sum of cyclics. Given a hereditary torsion theory on the category -Mod of unitary left -modules we can investigate torsionfree modules having the corresponding property for all torsionfree factor-modules (and a natural requirement concerning extensions of some homomorphisms). This...
We introduce a quantization of the graded algebra of functions on the canonical cone of
an algebraic curve , based on the theory of formal pseudodifferential operators. When
is a complex curve with Poincaré uniformization, we propose another, equivalent
construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a
presentation of the quantum algebra when is a rational curve, and discuss the problem
of constructing algebraically “differential liftings”.
Using quantum sections of filtered rings and the associated Rees rings one can lift the scheme structure on Proj of the associated graded ring to the Proj of the Rees ring. The algebras of interest here are positively filtered rings having a non-commutative regular quadratic algebra for the associated graded ring; these are the so-called gauge algebras obtaining their name from special examples appearing in E. Witten's gauge theories. The paper surveys basic definitions and properties but concentrates...
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