A generalized Second Main Theorem for closed subschemes

On the Second Main Theorem of Cartan

The possibility of reversion of the inequality in the Second Main Theorem of Cartan in the theory of holomorphic curves in projective space is discussed. A new version of this theorem is proved that becomes an asymptotic equality for a class of holomorphic curves defined by solutions of linear differential equations. 2010 MSC: 30D35, 32A22.

Sheaves of Modules and Closed Subschemes

Definition 1.1. If X is a topological space with a sheaf of ring OX , an OX-module is a sheaf F of abelian groups on X, together with the structure of an OX(U)-module on each F (U), which is compatible with restriction, in the sense that if V ⊆ U , for any s ∈ F (U) and f ∈ OX(U) we have ρUV (fs) = ρUV (f) · ρUV (s) in F (V ). A morphism between OX -modules F and G is a morphism φ of the underl...

GENERALIZED PRINCIPAL IDEAL THEOREM FOR MODULES

The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

Compactness Theorem for Some Generalized Second-Order Language

For the first-order language the compactness theorem was proved by K. Gödel and A. I. Mal’cev in 1936. In 1955, it was proved by J. Łoś (1955) by means of the method of ultraproducts. Unfortunately, for the usual second-order language the compactness theorem does not hold. Moreover, the method of ultraproducts is also inapplicable to second-order models. A possible way out of this situation is ...

The Second Main Theorem of Hypersurfaces in the Projective Space